On universality of critical behaviour in the focusing
nonlinear Schrödinger equation, elliptic umbilic
catastrophe and the tritronquée solution to the
Painlevé-I equation.
B.Dubrovin∗, T.Grava∗, C.Klein∗∗
∗ SISSA, Trieste; ∗∗ Max-Planck Institute, Leipzig
November 7, 2018
Abstract
We argue that the critical behaviour near the point of “gradient catastrophe” of
the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation
i�Ψt + �
Ψxx + |Ψ|2Ψ = 0, � � 1, with analytic initial data of the form Ψ(x, 0; �) =
A(x) e
S(x) is approximately described by a particular solution to the Painlevé-I equa-
tion.
1 Introduction
The focusing nonlinear Schrödinger (NLS) equation for the complex valued function Ψ =
Ψ(x, t)
iΨt +
Ψxx + |Ψ|2Ψ = 0 (1.1)
has numerous physical applications in the description of nonlinear waves (see, e.g., the books
[47, 35, 36]). It can be considered as an infinite dimensional analogue of a completely
integrable Hamiltonian system [48], where the Hamiltonian and the Poisson bracket is given
Ψt + {Ψ(x),H} = 0
{Ψ(x),Ψ∗(y)} = i δ(x− y) (1.2)
|Ψx|2 − |Ψ |4
(here Ψ∗ stands for the complex conjugate function). Properties of various classes of so-
lutions to this equation have been extensively studied both analytically and numerically
[5, 6, 7, 8, 16, 21, 25, 26, 29, 32, 34, 43, 44]. One of the striking features that distinguishes
this equation from, say, the defocusing case
iΨt +
Ψxx − |Ψ|2Ψ = 0
is the phenomenon of modulation instability [1, 11, 16]. Namely, slow modulations of the
plane wave solutions
Ψ = Aei(kx−ωt), ω =
k2 − A2
develop fast oscillations in finite time.
The appropriate mathematical framework for studying these phenomena is the theory of
the initial value problem
Ψ(x, 0; �) = A(x) e
S(x) (1.3)
for the �-dependent NLS
i �Ψt +
Ψxx + |Ψ|2Ψ = 0. (1.4)
Here � > 0 is a small parameter, A(x) and S(x) are real-valued smooth functions. Introduc-
ing the slow variables
u = |Ψ|2, v =
(1.5)
the equation can be recast into the following system:
ut + (u v)x = 0
(1.6)
vt + v vx − ux +
The initial data for the system (1.6) coming from (1.3) do not depend on �:
u(x, 0) = A2(x), v(x, 0) = S ′(x). (1.7)
The simplest explanation of the modulation instability then comes from considering the so-
called dispersionless limit �→ 0. In this limit one obtains the following first order quasilin-
ear system
ut +v ux + u vx = 0
vt − ux + v vx = 0
 . (1.8)
This is a system of elliptic type because of the condition u > 0. Indeed, the eigenvalues of
the coefficient matrix (
are complex conjugate, λ = v ± i
u. So, the Cauchy problem for the system (1.8) is ill-
posed in the Hadamard sense (cf. [33, 7]). Even for analytic initial data the life span of a
typical solution is finite, t < t0. The x- and t-derivatives explode at some point x = x0 when
the time approaches t0. This phenomenon is similar to the gradient catastrophe of solutions
to nonlinear hyperbolic PDEs [2].
For the full system (1.6) the Cauchy problem is well-posed for a suitable class of �-
independent initial data (see details in [17, 46]). However, the well-posedness is not uniform
in �. In practical terms that means that the solution to (1.6) behaves in a very irregular way
in some regions of the (x, t)-plane when � → 0. Such an irregular behaviour begins near
the points (x = x0, t = t0) of the “gradient catastrophe” of the solution to the dispersionless
limit (1.8). The solutions to (1.8) and (1.6) are essentially indistinguishable for t < t0; the
situation changes dramatically near x0 when approaching the critical point. Namely, when
approaching t = t0 the peak near a local maximum1 of u becomes more and more narrow
due to self-focusing; the solution develops a zone of rapid oscillations for t > t0. They
have been studied both analytically and numerically in [8, 16, 21, 23, 25, 26, 34, 43, 44].
However, no results are available so far about the behaviour of the solutions to the focusing
NLS at the critical point (x0, t0).
The main subject of this work is the study of the behaviour of solutions to the Cauchy
problem (1.6), (1.7) near the point of gradient catastrophe of the dispersionless system (1.8).
In order to deal with the Cauchy problem for (1.8) we will assume analyticity2 of the initial
data u(x, 0), v(x, 0). Then the Cauchy problem for (1.8) can be solved for t < t0 via a
suitable version of the hodograph transform (see Section 2 below). An important feature of
the gradient catastrophe for this system is that it happens at an isolated point of the (x, t)-
plane, unlikely the case of KdV or defocusing NLS where the singularity of the hodograph
solution takes place on a curve. We identify this singularity for a generic solution to (1.8)
as the elliptic umbilic singularity (see Section 4 below) in the terminology of R.Thom [42].
This codimension 2 singularity is one of the real forms labeled by the root system of the D4
type in the terminology of V.Arnold et al. [3].
Our main goal is to find a replacement for the elliptic umbilic singularity when the disper-
sive terms are added, i.e., we want to describe the leading term of the asymptotic behaviour
for �→ 0 of the solution to (1.6) near the critical point (x0, t0) of a generic solution to (1.8).
Thus, our study can be considered as a continuation of the programme initiated in [13]
to study critical behaviour of Hamiltonian perturbations of nonlinear hyperbolic PDEs; the
fundamental difference is that the non perturbed system (1.8) is not hyperbolic! However,
many ideas and methods of [13] (see also [12]) play an important role in our considerations.
The most important of these is the idea of universality of the critical behaviour. The
1Regarding initial data with local minima we did not observe cusps related to minima in numerical simula-
tions. We believe they do not exist because of the focusing effect in the NLS that pushes maxima to cusps but
seems to smoothen minima.
2We believe that the main conclusions of this paper must hold true also for non analytic initial data; the
numerical experiments of [8] do not show much difference in the properties of solution between analytic and
non analytic cases. However, the precise formulation of our Main Conjecture has to be refined in the non
analytic case.
general formulation of the universality suggested in [13] for the case of Hamiltonian pertur-
bations of the scalar nonlinear transport equation says that the leading term of the multiscale
asymptotics of the generic solution near the critical point does not depend on the choice of
the solution, modulo Galilean transformations and rescalings. This leading term was identi-
fied in [13] via a particular solution to the fourth order analogue of the Painlevé-I equation
(the so-called P 21 equation). The existence of the needed solution to P
1 has been rigorously
established in [9]. Moreover, it was argued in [13] that this behaviour is essentially indepen-
dent on the choice of the Hamiltonian perturbation. Some of these universality conjectures
have been partially confirmed by numerical analysis carried out in [21].
The main message of this paper is the formulation of the Universality Conjecture for
the critical behaviour of the solutions to the focusing NLS. Our considerations suggest the
description of the leading term in the asymptotic expansion of the solution to (1.6) near the
critical point via a particular solution to the classical Painlevé-I equation (P-I)
Ωζζ = 6 Ω
2 − ζ.
The so-called tritronquée solution to P-I was discovered by P.Boutroux [4] as the unique
solution having no poles in the sector | arg ζ| < 4π/5 for sufficiently large |ζ|. Remarkably,
the very same solution3 arises in the critical behaviour of solutions to focusing NLS!
The paper is organized as follows. In Section 2 we develop a version of the hodograph
transform for integrating the dispersionless system (1.8) before the catastrophe t < t0. We
also establish the shape of the singularity of the solution near the critical point; the latter is
identified in Section 4 with the elliptic umbilic catastrophe of Thom. In Section 3 we develop
a method of constructing formal perturbative solutions to the full system (1.6) before the
critical time. In Section 5 we collect the necessary information about the tritronquée solution
of P-I and formulate the Main Conjecture of this paper. Such a formulation relies on a much
stronger property of the tritronquée solution: namely, we need this solution to be pole-free
in the whole sector | arg ζ| < 4π/5. Numerical evidence for the absence of poles in this
sector is given in Section 6. In Section 7 we analyze numerically the agreement between the
critical behaviour of solutions to focusing NLS and its conjectural description in terms of the
tritronquée solution restricted on certain lines in the complex ζ-plane. In the final Section 8
we give some additional remark and outline the programme of future research.
Acknowledgments. The authors thank K.McLaughlin for a very instructive discussion. One
of the authors (B.D.) thanks R.Conte for bringing his attention to the tritronquées solutions
of P-I. The results of this paper have been presented by one of the authors (T.G.) at the
Conference “The Theory of Highly Oscillatory Problems”, Newton Institute, Cambridge,
March 26 - 30, 2007. T.G. thanks A.Fokas and S.Venakides for the stimulating discussion
after the talk. The present work is partially supported by the European Science Foundation
Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM),
3It is interesting that the same tritronquée solution (for real ζ only) appeares also in the study of certain
critical phenomena in plasma [41]. In the theory of random matrices and orthogonal polynomials a different
solution to P-I arises; see, e.g., [14].
Marie Curie RTN “European Network in Geometry, Mathematical Physics and Applications”
(ENIGMA). The work of B.D. and T.G. is also partially supported by Italian Ministry of
Universities and Researches (MUR) research grant PRIN 2006 “Geometric methods in the
theory of nonlinear waves and their applications”.
2 Dispersionless NLS, its solutions and critical behaviour
The equations (1.6) are a Hamiltonian system
ut + {u(x), H} = 0
vt + {v(x), H} = 0
with respect to the Poisson bracket originated in (1.2)
{u(x), v(y)} = δ′(x− y), (2.1)
other brackets vanish, with the Hamiltonian
u v2 − u2
dx. (2.2)
Let us first describe the general analytic solution to the dispersionless system (1.8).
Lemma 2.1 Let uI(x), vI(x) be two real valued analytic functions of the real variable x
satisfying
(uI,x)
+ (vI,x)
2 6= 0.
Then the solution u = u(x, t), v = v(x, t) to the Cauchy problem
u(x, 0) = uI(x), v(x, 0) = vI(x) (2.3)
for the system (1.8) for sufficiently small t can be determined from the following system
x = v t+ fu
0 = u t+ fv
 (2.4)
where f = f(u, v) is an analytic solution to the following linear PDE:
fvv + u fuu = 0. (2.5)
Conversely, given any solution to (2.5) satisfying u2f 2uu + f
uv 6= 0 at some point (u = ũ, v =
ṽ) such that fv(ũ, ṽ) = 0, the system (2.4) determines a solution to (1.8) defined locally near
the point x = x̃ := fu(ũ, ṽ) for sufficiently small t.
Remark 2.2 The solutions to the linear PDE (2.5) correspond to the first integrals of dis-
persionless NLS:
f(u, v) dx,
F = 0. (2.6)
Taking them as the Hamiltonians
us + {u(x), F} ≡ ut + (fv)x = 0
vs + {v(x), F} ≡ vt + (fu)x = 0
 (2.7)
yields infinitesimal symmetries of the dispersionless NLS:
(ut)s = (us)t , (vt)s = (vs)t . (2.8)
One of the first integrals will be extensively used in this paper. It corresponds to the
Hamiltonian density
g = −
v2 + u(log u− 1). (2.9)
The associated Hamiltonian flow reads
us + vx = 0
 (2.10)
Eliminating the dependent variable v one arrives at the elliptic version of the long wave limit
of Toda lattice:
uss + (log u)xx = 0.
Due to commutativity (2.8) the systems (1.8) and (2.10) admit a simultaneous solution u =
u(x, t, s), v = v(x, t, s). Any such solution can be locally determined from a system similar
to (2.4)
x = v t+ fu
s = u t+ fv
 (2.11)
where f = f(u, v), as above, solves the linear PDE (2.5).
The system (2.11) determines a solution u = u(x, t, s), v = v(x, t, s) provided applica-
bility of the implicit function theorem. The conditions of the latter fail to hold at the critical
point (x0, s0, t0, u0, v0) such that
x0 = v0t0 + fu(u0, v0)
s0 = u0t0 + fv(u0, v0)
fuu(u0, v0) = fvv(u0, v0) = 0, fuv(u0, v0) = −t0
(2.12)
In sequel we adopt the following system of notations: the values of the function f and of its
derivatives at the critical point will be denoted by f 0 etc. E.g., the last line of the conditions
(2.12) will read
f 0uu = f
vv = 0, f
uv = −t0.
Definition 1. We say that the critical point is generic if at this point:
f 0uuv 6= 0.
Let us the introduce real parameters r, ψ determined by the third derivatives of the function
f evaluated at the critical point,
(cosψ − i sinψ) = f 0uuv + i
uuu. (2.13)
Due to the genericity assumption
+ πk.
In order to describe the local behaviour of a solution to the dispersionless NLS/Toda
equations we define a function R(X;S, ψ) of real variables X , S depending on the real
parameter ψ satisfying
(S + cosψ)2 + (X + sinψ)2 6= 0 (2.14)
by the following formula
R(X,S, ψ) = sign [cosψ] (2.15)
1 +X sinψ + S cosψ +
1 + 2(X sinψ + S cosψ) +X2 + S2
P0(X,S, ψ) =
R(X,S, ψ) cosψ −
(X cosψ − S sinψ) sinψ
R(X,S, ψ)
− cosψ
(2.16)
Q0(X,S, ψ) =
(X cosψ − S sinψ) cosψ
R(X,S, ψ)
+R(X,S, ψ) sinψ
− sinψ.
Observe that P0(X,S, ψ) and Q0(X,S, ψ) are smooth functions of the real variable X pro-
vided validity of the inequality (2.14).
Lemma 2.3 Given an analytic solution u(x, s, t), v(x, s, t) to the dispersionless NLS/Toda
equations with a generic critical point (x0, s0, t0, u0, v0), and arbitrary real numbers X , S
satisfying (2.14), T < 0, then there exist the following limits
λ−1/2
u(x0 + λ
1/2v0T +
r X T 2, s0 + λ
1/2u0T +
rS T 2, λ1/2T )− u0
= r T P0 (X,S, ψ)
(2.17)
λ−1/2
v(x0 + λ
1/2v0T +
r X T 2, s0 + λ
1/2u0T +
rS T 2, λ1/2T )− v0
T Q0 (X,S, ψ)
where the parameters r, ψ are defined by (2.13).
Proof From the linear PDE (2.5) it follows that
fuvv = −ufuuu − fuu, fvvv = −ufuuv.
Using these formulae we expand the implicit function equations (2.11) near the critical point
in the form
x̄− v0t̄ = v̄ t̄+
f 0uuu(ū
2 − u0v̄2) + 2f 0uuvū v̄
(|ū|2 + |v̄|2)3/2
(2.18)
s̄− u0t̄ = ū t̄+
f 0uuv(ū
2 − u0v̄2)− 2u0f 0uuuū v̄
(|ū|2 + |v̄|2)3/2
where we introduce the shifted variables
x̄ = x− x0, s̄ = s− s0, t̄ = t− t0
ū = u− u0, v̄ = v − v0.
The rescaling
x̄− v0t̄ 7→ λ(x̄− v0t̄)
s̄− u0t̄ 7→ λ(s̄− u0t̄)
t̄ 7→ λ1/2t̄
ū 7→ λ1/2ū
v̄ 7→ λ1/2v̄
(2.19)
in the limit λ→ 0 yields the quadratic equation
z = t̄ w +
aw2, t̄ 6= 0 (2.20)
where the complex independent and dependent variables z and w read
z = s̄+ i
u0x̄− (u0 + i
u0v0)t̄, w = ū+ i
u0v̄ (2.21)
and the complex constant a is defined by
a = f 0uuv + i
uuu, (2.22)
therefore
= r eiψ.
The substitution
X = 2
x̄− v0t̄
r t̄2
, S = 2
s̄− u0t̄
r t̄2
reduces the quadratic equation to(
w + t̄ r eiψ
= r2t̄2e2iψ
1 + e−iψ(S + iX)
For t̄ < 0 we choose the following root
w = r t̄ei ψ
1 + e−i ψ(S + iX)− 1
(2.23)
where the branch of the square root is obtained by the analytic continuation of the one taking
positive values on the positive real axis. Equivalently,
w = t̄ r eiψ
sign (cosψ)
∆ + 1 + S cosψ +X sinψ + i
X cosψ − S sinψ
∆ + 1 + S cosψ +X sinψ
where
1 + 2(S cosψ +X sinψ) +X2 + S2.
This gives the formulae (2.15).
The result of the lemma describes the local structure of generic solutions to the dis-
persionless NLS/Toda equations near the critical point. It can also be represented in the
following form
u(x, s, t) ' u0 + r T P0(X,S, ψ)
(2.24)
v(x, s, t) ' v0 +
r T Q0(X,S, ψ)
where
X = 2
x̄− v0t̄
r t̄2
, S = 2
s̄− u0t̄
r t̄2
, T = t̄. (2.25)
We want to emphasize that the approximation (2.24) works only near the critical point.
Indeed, for large x→ ±∞ the function u(x, s, t) and v(x, s, t) have the following behaviour
u = −
r |x|u1/40
1∓ sinψ + u0 − r t̄ cosψ +O
(2.26)
v = ∓
r |x|
u01/4
sign (cosψ)
1± sinψ + v0 −
t̄ sinψ +O
. (2.27)
So, for sufficiently large |x| the function u(x, s, t) defined by (2.24) becomes negative.
The function u has a maximum at the point X = S tanψ, so locally
u ≤ u0 + r T cosψ −
r | cosψ|
+ r T 2 < u0. (2.28)
At the critical point (x0, s0, t0, u0, v0) the function u develops a cusp. Let us consider only
the particular case S = 0 in order to avoid complicated expressions. In this case the local
behaviour of the function u near the critical point is given by
t̄→−0
r |x̂|
1− sinψ, x̂ > 0
r |x̂|
1 + sinψ, x̂ < 0
(2.29)
(here x̂ =
u0(x̄ − v0t̄)). Thus the parameters r, ψ describe the shape of the cusp at the
critical point.
3 First integrals and solutions of the NLS/Toda equations
Let us first show that any first integral (2.6) of the dispersionless equations can be uniquely
extended to a first integral of the full equations.
Lemma 3.1 Given a solution f = f(u, v) to the linear PDE (2.5), there exists a unique, up
to a total derivative, formal power series in �
hf = f +
f (u, v;ux, vx, . . . , u
(2k), v(2k))
such that the integral
hf dx
commutes with the Hamiltonian of NLS equation:
{H,Hf} = 0
at every order in �. Explicitly,
hf = f −
fuuu +
u2x + 2fuuvuxvx − ufuuuv
(3.1)
fuuuu +
u2xx + 2fuuuvuxxvxx − ufuuuuv
fuuuuuxxv
fuuuvvxxu
3456u3
30fuuu − 9ufuuuu + 12u2f5u + 4u3f6u
432u2
−3fuuuv + 6ufuuuuv + 2u2f5u v
u3xvx +
9fuuuu + 9uf5u + 2u
(9fuuuuv + 10uf5u v)uxv
(18f5u + 5uf6u) v
+O(�6)
Here we use short notations
f5u :=
, f6u :=
, f5u v :=
∂u5∂v
Example 1. Taking f = 1
(u v2 − u2) one obtains the Hamiltonian of the NLS equation
(u v2 − u2) +
In this case the infinite series truncates. It is easy to see that the series in � truncates if
and only if f(u, v) is a polynomial in u. Polynomial in u solutions to the linear PDE (2.5)
correspond to the standard first integrals of the NLS hierarchy.
Example 2. Taking g = −1
v2 + u(log u− 1) (cf. (2.9)) one obtains the Hamiltonian of
Toda equation
hg = −
v2 + u(log u− 1)−
u2x + 2u v
240u3
144u5
360u3
+O(�6) (3.2)
written in terms of the function φ = log u in the form
�2φxx + e
φ(s+�) − 2eφ(s) + eφ(s−�) = 0.
Lemma 3.2 Any solution to the NLS/Toda equations in the class of formal power series in �
can be obtained from the equations
x = v t+
δu(x)
(3.3)
s = u t+
δv(x)
where f = f(u, v; �) is an arbitrary admissible solution to the linear PDE (2.5) in the class
of formal power series in �,
hf dx.
Now, we can apply to the system (3.3) the rescaling (2.19) accompanied by the transfor-
mation
� 7→ λ5/4�. (3.4)
At the limit λ→ 0 we arrive at the following system of equations
s̄− u0t̄ = ū t̄+ f 0uuv
(ū2 − u0v̄2) +
− u0f 0uuu
ū v̄ +
(3.5)
x̄− v0t̄ = v̄ t̄+ f 0uuu
(ū2 − u0v̄2) +
+ f 0uuv
ū v̄ +
Using the complex variables z, w defined in (2.21) we can rewrite the system in the following
form:
z reiψ = w t̄ reiψ +
wxx. (3.6)
The last observation is that the Toda equations generated by the Hamiltonian Hg =
hg dx
(see Example 2 above) after the scaling limit (2.19), (3.4) yield the Cauchy - Riemann equa-
tions for the function w = w(z),
∂w/∂z̄ = 0.
Therefore the system (3.5) can be recast into the form equivalent to the Painlevé-I (P-I)
equation (see (5.1) below)
z reiψ = w t̄ reiψ +
u0wzz. (3.7)
Choosing
λ = �4/5
we eliminate � from the equation.
In the Section 5 below we will write explicitly the reduction of (3.7) to the Painlevé-I
equation and give a conjectural characterization of the particular solution of the latter.
4 Critical behaviour and elliptic umbilic catastrophe
Separating again the real and complex parts of (3.6) one obtains a system of ODEs
UXX +
(U2 − V 2) + r t̄ (U cosψ − V sinψ)− r (S cosψ −X sinψ) = 0
(4.1)
VXX + UV + r t̄ (U sinψ + V cosψ)− r (S sinψ +X cosψ) = 0
that can be identified with the Euler - Lagrange equations
δS = 0, S =
L(U, V, UX , VX) dX
with the Lagrangian
V 2X − U
U3 − 3U V 2
(U2 − V 2) cosψ − 2U V sinψ
(4.2)
+r (X sinψ − S cosψ)U + r (S sinψ +X cosψ)V.
In the “dispersionless limit” � → 0 the Euler - Lagrange equations reduce to the search of
stationary points of a function (let us also set t̄ = 0)
U3 − 3U V 2
+ a+U + a−V (4.3)
where we redenote
a+ = r (X sinψ − S cosψ) , a− = r (S sinψ +X cosψ) .
At a+ = a− = 0 the function F has an isolated singularity at the origin U = V = 0 of the
type D4,− also called elliptic umbilic singularity, according to R.Thom [42]. This singular-
ity appears in various physical problems; we mention here the caustics in the collisionless
dark matter [40] to give just an example. The parameters a+ and a− define two particular
directions on the base of the miniversal unfolding of the elliptic umbilic; the full unfolding
depending on 4 parameters reads
U3 − 3U V 2
b(U2 + V 2) + a+U + a−V + c. (4.4)
It would be interesting to study the properties of the modified Euler - Lagrange equations for
the Lagrangian
L̂ = L+
b(U2 + V 2).
This deformation does not seem to arrive from considering solutions to the NLS hierarchy.
5 The tritronquée solution to the Painlevé-I equation and
the Main Conjecture
In this section we will select a particular solution to the Painlevé-I (P-I) equation
Ωζζ = 6Ω
2 − ζ (5.1)
Recall [22] that an arbitrary solution to this equation is a meromorphic function on the com-
plex ζ-plane. According to P. Boutroux [4] the poles of the solutions accumulate along the
arg ζ =
, n = 0, ±1, ±2. (5.2)
Boutroux proved that, for each ray there is a one-parameter family of particular solutions
called intégrales tronquées whose lines of poles truncate for large ζ . He proved that the
intégrale tronquée has no poles for large |ζ| within two consecutive sectors of the angle
2π/5 around the ray, and, moreover it has the asymptotic behaviour of the form
Ω = −
)1/2 [
(1−ε)
(5.3)
for a suitable choice of the branch of the square root (see below) and a sufficiently small
ε > 0.
Furthemore, if a solution truncates along any two of the rays (5.2) then it truncates along
three of them. These particular solutions to P-I are called tritronquées. They have no poles
for large |ζ| in four consecutive sectors; their asymptotics for large ζ is given by (5.3). It
suffices to know the tritronquée solution Ω0(ζ) for the sector
| arg ζ| <
. (5.4)
In this case the branch of the square root in (5.3) is obtained by the analytic continuation
of the principal branch taking positive values on the positive half axis ζ > 0. Other four
tritronquées solutions are obtained by applying the symmetry
Ωn(ζ) = e
, n = ±1, ±2. (5.5)
The properties of the tritronquées solutions in the finite part of the complex plane were
studied in the important paper of N.Joshi and A.Kitaev [24].
A. Kapaev [27] obtained a complete characterization of the tritronquées solutions in
terms of the Riemann - Hilbert problem associated with P-I. We will briefly sketch here
the main steps of his construction.
The equation (5.1) can be represented as the compatibility condition of the following
system of linear differential equations for a two-component vector valued function Φ =
Φ(λ, ζ)
 Ωζ 2λ2 + 2Ωλ− ζ + 2Ω2
2(λ− Ω) −Ωζ
 Φ (5.6)
Φζ = −
 0 λ+ 2Ω
 Φ. (5.7)
The canonical matrix solutions Φk(λ, ζ) to the system (5.6) - (5.7) are uniquely determined
by their asymptotic behaviour
Φk(λ, ζ) ∼
λ1/4 λ1/4
λ−1/4 −λ−1/4
(5.8)
λ−3/2
eθ(λ,ζ)σ3 , |λ| → ∞, λ ∈ Σk
in the sectors
λ ∈ C |
< arg λ <
, k ∈ Z. (5.9)
θ(λ, ζ) =
λ5/2 − ζ λ1/2, σ3 =
, H =
Ω2ζ − 2 Ω
3 + ζ Ω, (5.10)
the branch cut on the complex λ-plane for the fractional powers of λ is chosen along the
negative real half-line.
The Stokes matrices Sk are defined by
Φk+1(λ, ζ) = Φk(λ, ζ)Sk, λ ∈ Σk ∩ Σk+1. (5.11)
They have the triangular form
S2k−1 =
1 s2k−1
, S2k =
s2k 1
(5.12)
and satisfy the constraints
Sk+5 = σ1 Sk σ1, k ∈ Z; S1S2S3S4S5 = i σ1 (5.13)
where
Due to (5.13) two of the Stokes multipliers sk determine all others; they depend neither on λ
nor on ζ provided Ω(ζ) satisfies (5.1).
In order to obtain a parametrization of solutions to the P-I equation (5.1) by Stokes mul-
tipliers of the linear differential equation (5.6) one has to reformulate the above definitions
as a certain Riemann - Hilbert problem. The solution of the Riemann - Hilbert problem de-
pends on ζ through the asymptotics (5.8). If the Riemann - Hilbert problem has a unique
solution for the given ζ0 ∈ C then the canonical matrices Φk(λ, ζ) depend analytically on
ζ for sufficiently small |ζ − ζ0|; the coefficient Ω = Ω(ζ) will then satisfy (5.1). The poles
of the meromorphic function Ω(ζ) correspond to the forbidden values of the parameter ζ for
which the Riemann - Hilbert problem admits no solution.
We will now consider a particular solution to the P-I equation specified by the following
Riemann - Hilbert problem. Denote four oriented rays γ0, γ±1, ρ in the complex λ-plane
defined by
γk = {λ ∈ C | arg λ =
}, k = 0, ±1
(5.14)
ρ = {λ ∈ C | arg λ = π}
directed towards infinity. The rays divide the complex plane in four sectors. We are looking
for a piecewise analytic function Φ(λ, ζ) on
λ ∈ C \ (γ−1 ∪ γ0 ∪ γ1 ∪ ρ)
depending on the parameter ζ continuous up to the boundary with the asymptotic behaviour
at |λ| → ∞ of the form (5.8) satisfying the following jump conditions on the rays
Φ+(λ, ζ) = Φ−(λ, ζ)Sk, λ ∈ γk
(5.15)
Φ+(λ, ζ) = Φ−(λ, ζ)Sρ, λ ∈ ρ.
Here the plus/minus subscripts refer to the boundary values of Φ respectively on the left/right
sides of the corresponding oriented ray, the jump matrices are given by
, S±1 =
, Sρ =
. (5.16)
The following result is due to A.Kapaev4 .
Theorem 5.1 The solution to the above Riemann - Hilbert problem exists and it is unique
| arg λ| <
, |λ| > R (5.17)
for a sufficiently large positive number R. The associated function
Ω0(ζ) :=
dH(ζ)
(5.18)
H(ζ) :=
σ3Φ(λ, ζ) e−θ(λ,ζ)σ3 − 1
is analytic in the domain (5.17), it satisfies P-I and enjoys the asymptotic behaviour
Ω0(ζ) ∼ −
, |ζ| → ∞, | arg λ| <
. (5.19)
Moreover, any solution of P-I having no poles in the sector (5.17) for some large R > 0
coincides with Ω0(ζ).
4Our solution Ω0(ζ) coincides with y3(x) ≡ y−2(x), x = −ζ, of [27] (see eq. (2.73) of [27]; Kapaev uses
a different normalization y′′ = 6y2 + x of the P-I equation).
Joshi and Kitaev proved that the tritronquée solution has no poles on the positive real
axis. They found a numerical estimate for the position of the first pole ζ0 of the tritronquée
solution Ω0(ζ) on the negative real axis:
ζ0 ' −2.3841687
(cf. also [10]). Besides this estimate very little is known about the location of poles of Ω0(ζ).
Our numerical experiments (see below) suggest the following
Main Conjecture. Part 1. The tritronquée solution Ω0(ζ) has no poles in the sector
| arg λ| <
. (5.20)
We are now ready to describe the conjectural universal structure behind the critical be-
haviour of generic solutions to the focusing NLS. For simplicity of the formulation let us
assume cosψ > 0.
Main Conjecture. Part 2. Any generic solution to the NLS/Toda equations near the
critical point behaves as follows
u(x, s, t; �) + i
u0v(x, s, t; �) ' u0 + i
u0v0 − t̄ reiψ + 2 �2/5(3r
5 Ω0(ζ) +O
(5.21)
s̄− u0t̄+ i
u0(x̄− v0t̄) + 12re
iψ t̄2
where Ω0(ζ) is the tritronquée solution to the Painlevé-I equation (5.1).
The above considerations can actually be applied replacing the NLS time flow by any
other flow of the NLS/Toda hierarchy. The local description of the critical behaviour remains
unchanged.
Remark 5.2 Note that the angle of the line ζ(x̄) in (5.21) for t̄ fixed is equal to ψ/5 + π/2,
ψ ∈ [−π, π]. Thus the maximal value of argζ is equal to 7π/10 < 4π/5. The lines in (5.21)
consequently do not get close to the critical lines of the tritronquée solution.
6 Numerical analysis of the tritronquée solution of P-I
In this section we will numerically construct the tritronquée solution Ω0, i.e. the tritronquée
solution with asymptotic behavior (5.3). We will drop the index 0 in the following. The
solution will be first constructed on a straight line in the complex plane. In a second step
we will then explore global properties of these solutions within the limitations imposed by a
numerical approach5.
5Cf. [15] where a similar technique was applied to solve numerically the Painlevé-II equation in the complex
domain.
Let the straight line in the complex plane be given by ζ = ay + b with a, b ∈ C constant
(we choose a to have a non-negative imaginary part) and y ∈ R. The asymptotic conditions
Ω ∼ −
, (6.1)
for y → ±∞. The root is defined to have its cut along the negative real axis and to as-
sume positive values on the positive real axis. This choice of the root implies the following
symmetry for the solution:
Ω(ζ∗) = Ω∗(ζ). (6.2)
Thus Ω is real on the real axis, see [24].
Numerically it is not convenient to impose boundary conditions at infinity. We thus
assume that the wanted solution can be expanded in a Laurent series in
ζ around infinity.
Such an asymptotic expansion is possible for the considered tritronquée solution in the sector
| arg ζ| < 4π/5. The formal series can be written there (see [24]) in the form
Ωf = −
ζ5k/2
, (6.3)
where a0 = 1, and where the remaining coefficients follow from the recurrence relation for
k ≥ 0
ak+1 =
25k2 − 1
amak+1−m. (6.4)
This formal series is divergent, the coefficients ak behave asymptotically as ((k − 1)!)2, see
[24] for a detailed discussion.
It is known that divergent series can be used to get numerically acceptable approxima-
tions to the values of the sum by properly truncating the series. Generally the best approx-
imations for the sum result from truncating the series where the terms take the smallest
absolute values (see e.g. [18]). Since we work in Matlab with a precision of 16 digits and
with values of |ζ| ≥ 10, we typically consider up to 10 terms in the series. In this case the
terms corresponding to a10 are of the order of machine precision (10−14 and below).
Thus we have constructed approximations to the numerical values of the tritronquée
solution for large values of |ζ|. These can be used as in [24] to set up an initial value problem
for the P-I equation and to solve this with a standard ODE solver. In fact the approach works
well on the real axis starting from positive values until one reaches the first singularity on
the negative real axis. It is straightforward to check the results of [24] with e.g. ode45, the
Runge-Kutta solver in Matlab corresponding to the Maple solver used in [24]. If one solves
P-I on a line that avoids the sector | arg ζ| > 4π/5, one could integrate until one reaches once
more large values of |ζ| for which the asymptotic conditions are known. This would provide
a control on the numerical accuracy of this so-called shooting approach. Shooting methods
are problematic if the second solution to the initial value problem has poles as is the case for
P-I. In this case the numerical errors in the initial data (here due to the asymptotic conditions)
and in the time integration will lead to a large contribution of the unwanted solution close
to its poles which will make the numerical solution useless. It is obvious that P-I has such
poles from the numerical results in [24] and the property (5.5). In [24] the task was to locate
poles in the tritronquée solution, and in this case the shooting approach seems to be the only
available. Here we are studying, however, the solution on a line in the complex plane where
we know the asymptotic conditions for the affine parameter tending to ±∞.
Thus we use as in [15] the asymptotic conditions on lines avoiding the sector | arg ζ| >
4π/5 to set up a boundary value problem for y = ±y0, y0 ≥ 10. The solution in the
interval [−y0, y0] is numerically obtained with a finite difference code based on a collocation
method. The code bvp4c distributed with Matlab, see [39] for details, uses cubic polynomials
in between the collocation points. The P-I equation is rewritten in the form of a first order
system. With some initial guess (we use Ω = −
ζ/6 as the initial guess), the differential
equation is solved iteratively by linearization. The collocation points (we use up to 10000)
are dynamically adjusted during the iteration. The iteration is stopped when the equation is
satisfied at the collocation points with a prescribed relative accuracy, typically 10−10. The
solution for a = i and b = 0 is shown in Fig. 1. The values of Ω in between the collocation
!20 !15 !10 !5 0 5 10 15 20
Figure 1: Real (blue) and imaginary part (red) of the tritronquée solution to the Painlevé I
equation for ζ = iy.
points are obtained by interpolation via the cubic polynomials in terms of which the solution
has been constructed. This interpolation leads to a loss in accuracy of roughly one order of
magnitude with respect to the precision at the collocation points. To test this we determine
the numerical solution via bvp4c for P-I on Chebychev collocation points and check the
accuracy with which the equation is satisfied via Chebychev differentiation, see e.g. [45].
It is found that the numerical solution with a relative tolerance of 10−10 on the collocation
points satisfies the ODE to roughly the same order except at the boundary points where it
is of the order 10−8, see Fig. 2 where we show the residual ∆ by plugging the numerical
solution into the differential equation for the above example. It is straightforward to achieve
a prescribed accuracy by requiring a certain value for the relative tolerance. Notice that we
are not interested in a high precision solution of P-I here, but in a comparison of solutions
to the NLS equation close to the point of gradient catastrophe of the semiclassical system
with an asymptotic solution in terms of P-I transcendents. For this purpose an accuracy of
the solution of the order of 10−4 will be sufficient in all studied cases.
!20 !15 !10 !5 0 5 10 15 20
Figure 2: Error in the solution of the Painlevé I equation.
The quality of the used boundary conditions via the asymptotic behavior can be checked
by computing the solution for different values of y0. One finds that the difference between
the asymptotic square root and the tritronquée solution is only visible near the origin, see
Fig. 3. For large x it can be seen that the difference between the square root asymptotics
and the tritronquée solution reaches quickly values below the aimed at threshold of 10−4.
It is interesting to note that this difference is actually smaller than the difference between
the tritronquée solution and the truncated formal asymptotic series except at the boundary,
where the latter condition is implemented (see Fig. 3).
The dominant behavior of the square root changes if one approaches the critical lines
a = exp(4πi/5), b = 0. As can be seen from Fig. 4, the solution shows oscillations on top
of the square root. The closer one comes to the critical lines, the slower is the fall off of the
amplitude of the oscillations. We conjecture that these oscillations will have on the critical
10 12 14 16 18 20
!20 !10 0 10 20
Figure 3: The plot on the left side shows the absolute value of the difference between the
tritronquée solution and the asymptotic condition −
ζ/6 for a = i and b = 0. The plot on
the right side shows in blue the same difference for x > 10 and in red the difference between
the tritronquée solution and the truncated asymptotic series.
lines only a slow algebraic fall off towards infinity.
The above approach thus allows the computation of the tritronquée solution for a line
avoiding the sector | arg ζ| > 4π/5 with high accuracy. The picture one obtains by com-
puting Ω along several such lines is that there are indeed no singularities in the sector
| arg ζ| < 4π/5, and that the square root behavior is followed for large |ζ|. To obtain a
more complete picture, we compute the tritronquée solution for | arg ζ| < 4π/5 − 0.05 and
|ζ| < R ( we choose R = 20). The boundary data for |ζ| = R follow as before from the
truncated asymptotic series, the data for arg ζ = ±4π/5 − 0.05 are obtained by computing
the tritronquée solution on the respective lines as above.
To solve the resulting boundary value problem for the P-I equation is, however, computa-
tionally expensive since we have to solve an equation in 2 real dimensions iteratively. Since
the solution we want to construct is holomorphic there, we can instead solve the harmonicity
condition (the two dimensional Laplace equation) for the given boundary conditions. To this
end we introduce polar coordinates r, φ and use a spectral grid as described in [45]: the main
point is a doubling of the interval r ∈ [0, R] to [−R,R] to allow for a better distribution of
the Chebychev collocation points. Since we work with values of φ < φ0, we cannot use the
usual Fourier series approach for the azimuthal coordinate. Instead we use again a Cheby-
chev collocation method. The found solution in the considered domain is shown in Fig. 5
!20 !15 !10 !5 0 5 10 15 20
Figure 4: Real (blue) and imaginary part (red) of the tritronquée solution close to the critical
line (for a = exp(i(4π/5− .05))) with oscillations of slowly decreasing amplitude.
and Fig. 6. The quality of the solution can be tested by plugging the found solution to the
Laplace equation into the P-I equation. Due to the low resolution and problems at the bound-
ary, the accuracy is considerably lower in the two dimensional case than on the lines. This
is, however, not a problem since we need only the one dimensional solutions for quantitative
comparisons with NLS solutions. The two dimensional solutions give nonetheless strong
numerical evidence for the conjecture that the tritronquée solution has globally no poles in
the sector | arg ζ| < 4π/5.
7 Critical behavior in NLS and the tritronquée solution of
P-I: numerical results
In this section we will compare the numerical solution of the focusing NLS equation for
two examples of initial data for values of � between 0.1 and 0.025 with the asymptotic so-
lutions discussed in the previous sections, the semiclassical solution up to the breakup and
the tritronquée solution to the Painlevé I equation. The numerical approach to solve the NLS
equation is discussed in detail in [29]. For values of � below 0.04 we have to use Krasny
filtering [30] (Fourier coefficients with an absolute value below 10−13 are put equal to zero
to avoid the excitation of unstable modes). With double precision arithmetic we could thus
reach � = 0.025, but could not go below.
Figure 5: Real part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5− 0.05.
7.1 Initial data
We consider initial data where u(x, 0) has a single positive hump, and where v(x, 0) is
monotonously decreasing. For initial data of the form u(x, 0) = A2(x) and v(x, 0) = 0
where the function A(x) is analytic with a single positive hump with maximum value A0,
the semiclassical solution of NLS follows from (2.11) with f(u, v) given by
f(u, v) = 2=
dη ρ(η)
v)2 + u
 (7.1)
where
ρ(η) =
∫ x+(η)
x−(η)
A2(x) + η2
and where x±(η) are defined by A(x±(η)) = iη. The formula (7.1) follows from results by
S.Kamvissis, K.McLaughlin and P.Miller in [26].
From f(u, v) it is straightforward to recover the initial data from the equations
x = fu, fv = 0. (7.2)
Numerically we study the critical behavior of two classes of initial data, one symmetric with
respect to x which were used in [34], and initial data without symmetry with respect to x
Figure 6: Imaginary part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5 −
0.05.
which are built from the initial data studied in [43]. For the former class the corresponding
exact solution of focusing NLS is known in terms of a determinant. Nonetheless we integrate
the NLS equation for these initial data numerically since this approach is not limited to
special cases, but can be used for general smooth Schwartzian initial data as in the latter
case.
7.1.1 Symmetric initial data
We consider the particular class of initial data data
u(x, t = 0) = A20sech
2 x, v(x, t = 0) = −µ tanhx, µ ≥ 0. (7.3)
Introducing the quantity
− A20,
we find that the semiclassical solution for these initial data follows from (2.11) with
f(u, v) =
(v − 2M)∆+ −
(v + 2M)∆− −
u log u
u log
v +M + ∆+)(−
v −M + ∆−)
] (7.4)
where
v ±M)2 + u
For µ = 0 we recover the Satsuma-Yajima [37] initial data that were studied numerically in
[34]. The function f(u, v) takes the form
f(u, v) = <
+ i A0
+ i A0
+ u log
+ i A0)
+ iA0)2 + u√
(7.5)
which can also be recovered from (7.1) by setting ρ = i. The critical point is given by
u0 = 2A
0, v0 = 0, x0 = 0, t0 =
. (7.6)
Furthermore we have
f 0uuu = 0, f
uuv =
, r = 4A30, ψ = 0, (7.7)
where r, ψ are defined in (2.13). For A0 = 1 the initial data (7.3) coincides with the one
studied in [43] by A.Tovbis, S.Venakides and X.Zhou. In the particular case µ = 2, A0 = 1
the function f(u, v) in (7.1) simplifies to
f(u, v) = v −
v2 + u+ u log
−12v +
v2 + u
 . (7.8)
In this case the critical point is given by
v0 = 0, u0 = 2 + µ, t0 =
2 + µ
, x0 = 0.
Furthermore,
f 0uuu = 0, f
uuv =
(µ+ 2)3
, r =
(µ+ 2)3
, ψ = 0.
7.1.2 Non-symmetric initial data
Recall that we are interested here in Cauchy data in the Schwartz class of rapidly decreasing
functions. The above initial data are symmetric with respect to x, u is an even and v an odd
function in x. To obtain a situation which is manifestly not symmetric, we use the fact that if
f is a solution to (2.5), the same holds for derivatives and anti-derivatives of f with respect
to v and for any linear combination of those. If fv is an even function in v, this will obviously
not be the case for a linear combination of f and fv.
As a specific example, we consider the linear combination
f = f1 + αf2, α = const,
where f1 coincides with (7.8) and
f2 = 2u
v2 + u−
v2 + u
+ u v log
−12v +
v2 + u
 . (7.9)
The function f2 is obtained from the integration f2,v = f1 − v. The critical point is given in
this case by
u0 = 4(1− 16α2), v0 = −16α, x0 =
1 + 4α
1− 4α
, t0 =
1 + 4α
1− 4α
thus we have |α| < 1/4. Furthermore,
f 0uuv = −
4α2 − 1/8
1− 16α2
, f 0uuu =
1− 16α2
such that
r = 8u0, ψ = − arctan
1− 16α2
1/8− 4α2
We determine the initial data corresponding to f for a given value of |α| < 1/4 by
solving (7.2) for u, v in dependence of x. This is done numerically by using the algorithm of
[31] which is implemented as the function fminsearch in Matlab. The algorithm provides an
iterative approach which converges in our case rapidly if the starting values are close enough
to the solution, which is achieved by choosing u and v corresponding to f0 as an initial guess.
For α close to 1/4 we observe numerically a steepening of the initial pulse which will lead
to a shock front in the limit α→ 1/4. For the computations presented here, we consider the
case α = 0.1 which leads to the initial data shown in Fig. 7.
The initial data are computed in the way described above to the order of the Krasny filter
on the interval x ∈ [−15, 11] on Chebychev collocation points. Standard interpolation via
Chebychev polynomials is then used to interpolate the resulting data to a Fourier grid. To
avoid a Gibbs phenomenon at the interval ends due to the non-periodicity of the data, we use
a Fourier grid on the interval [−10π, 10π] to ensure that the function u takes values of the
order of the Krasny filter. For x < −15 and x > 11, the function u is exponentially small
which implies the zero-finding algorithm will no longer provide the needed precision. Thus
we determine the exponential tails of the solution to leading order analytically. We find for
x→ −∞
u = v2+ exp
2(x− αv+)
αv+ + 1
v = v+ − v+ exp
2(x− αv+)
αv+ + 1
2 log(v+) +
2(x− αv+)
αv+ + 1
, (7.10)
!4 !3 !2 !1 0 1 2 3 4
!4 !3 !2 !1 0 1 2 3 4
Figure 7: Initial data for the NLS equations without symmetry with respect to x.
and for x→ +∞
u = v2− exp
2(x+ α)
αv− + 1
v = v− − v− exp
2(x+ α)
αv− + 1
2 log(−v−)−
2(x+ α)
αv− + 1
, (7.11)
where v± = (
1± α − 1)/α. The initial data for the NLS equation in the form Ψ =√
u exp(iS/�) are then found by integrating v on the Chebychev grid by standard integration
of Chebychev polynomials. The exponential tails for S follow from (7.10) and (7.11). The
matching of the tails to the Chebychev interpolant is not smooth and leads to a small Gibbs
phenomenon. The Fourier coefficients decrease, however, to the order of the Krasny filter
which is sufficient for our purposes. Thus we obtain the non-symmetric initial data with
roughly the same precision as the analytic symmetric data.
7.2 Semiclassical solution
For times t� tc, the semiclassical solution gives a very accurate asymptotic description for
the NLS solution. The situation is similar to the Hopf and the KdV equation [19]. We find
for the symmetric initial data for t = tc/2 that the L∞ norm of the difference between the
solutions decreases as �2. More precisely a linear regression analysis in the case of symmetric
initial data (for the values � = 0.03, 0.04, . . . , 0.1) for the logarithm of this norm leads to an
error proportional to �a with a = 1.94, a correlation coefficient r = 0.9995 and standard error
σa = 0.03. In the non-symmetric case, we find a = 1.98, r = 0.999996 and σa = 0.003.
Close to the critical time the semiclassical solution only provides a satisfactory descrip-
tion of the NLS solution for large values of |x − xc|. In the breakup region it fails to be
accurate since it develops a cusp at xc whereas the NLS solution stays smooth. This behavior
can be well seen in Fig. 8 for the symmetric initial data. The largest difference between the
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 8: The blue line is the function u of the solution to the focusing NLS equation for
the initial data u(x, 0) = 2 sechx and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
semiclassical and the NLS solution is always at the critical point. We find that the L∞ norm
of the difference scales roughly as �2/5 as suggested by the Main Conjecture. More precisely
we find a scaling proportional to �a with a = 0.38 and r = 0.999997 and σa = 4.2 ∗ 10−4.
For the non-symmetric initial data, we find a = 0.36, r = 0.9999 and σa = 0.002. The
corresponding plot for u can be seen in Fig. 9.
The function v for the same situation as in Fig. 8 is shown in Fig. 10. It can be seen that
the semiclassical solution is again a satisfactory description for |x − xc| large, but fails to
be accurate close to the breakup point. The phase for the non-symmetric initial data can be
seen in Fig. 11. In the following we will always study the scaling for the function u without
further notice.
0.2 0.3 0.4 0.5 0.6 0.7
Figure 9: The blue line is the function u of the solution to the focusing NLS equation for
the non-symmetric initial data and � = 0.04 at the critical time, and the red line is the corre-
sponding semiclassical solution given by formulas (2.4). The green line gives the multiscales
solution via the tritronquée solution of the Painlevé I equation.
7.3 Multiscales solution
It can be seen in Fig. 8 and Fig. 10 that the multiscales solution (5.21) in terms of the
tritronquée solution to the Painlevé I equation gives a much better asymptotic description
to the NLS solution at breakup close to the breakup point than the semiclassical solution
for the symmetric initial data. For larger values of |x − xc|, the semiclassical solution pro-
vides, however, the better approximation. The rescaling of the coordinates in (5.21) sug-
gests to consider the difference between the NLS and the multiscales solution in an interval
[−γ�4/5, γ�4/5] (we choose here γ = 1, but within numerical accuracy the result does not
depend on varying γ around this value). These intervals can be seen in Fig. 12. We find that
the L∞ norm of the difference between these solutions in this interval scales roughly like
�4/5. More precisely we have a scaling �a with a = 0.76 (r = 0.998 and σa = 0.019).
For the non-symmetric initial data, the situation at the critical point can be seen in Fig. 9
and Fig. 11. Again the multiscales solution (5.21) gives a much better description close to
the critical point than the semiclassical solution. However, the approximation is here much
better on the side with weak slope for u than on the side with strong slope. We consider
again the L∞-norm of the difference between the multiscales and the NLS solution in the
interval [−γ�4/5, γ�4/5]. The scaling behavior of the solution can be seen in Fig. 13. For
γ = 1 we find a = 0.71, r = 0.998 and σa = 0.02. These values do not change much for
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
Figure 10: The blue line is the function v of the solution to the focusing NLS equation
for the initial data u0(x) = 2 sechx and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
larger γ. For smaller γ there are not enough points to provide a valid statistics. The value
of a smaller than the predicted 4/5 is seemingly due to the strong asymmetry in the quality
of the approximation of NLS by the multiscales solution as can be seen from Fig. 9. In the
considered interval, the deviation is already so big that the scaling no longer holds as in the
symmetric case. To study the scaling with a reliable statistics would, however, require the
use of a considerably higher resolution which would be computationally too expensive.
Going beyond the critical time, one finds that the real part of the NLS solution continues
to grow before the central hump breaks up into several humps. Notice that the multiscales
solution always leads to a function u that is smaller than the corresponding function of the
NLS solution at breakup and before. This changes for times after the breakup as can be
inferred from Fig. 14 which shows the time dependence of the NLS and the corresponding
multiscales solution for the non-symmetric initial data. The approximation is always best at
the critical time.
To study the quality of the approximation (5.21), we use rescaled times. The scaling of
the coordinates in (5.21) suggests to consider the NLS solution close to breakup at the times
t±(�) with
t±(�) = tc + u0/r −
(u0/r)2 ± �4/5β, (7.12)
0.2 0.3 0.4 0.5 0.6 0.7
Figure 11: The blue line is the function v of the solution to the focusing NLS equation
for the non-symmetric initial data and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
where β is a constant (we consider β = 0.1). We will only study the symmetric initial data in
this context. Before breakup we obtain the situation shown in Fig. 15. It can be seen that the
multiscales solution always provides a better description close to xc than the semiclassical
solution, and that the quality improves in this respect with decreasing �. We find that the L∞
norm of the difference scales in this case as �a with a = 0.55 (r = 0.994 and σa = 0.03).
The situation for times after breakup can be inferred from Fig. 16. Close to the central
region the multiscales solution shows a clear difference to the NLS solution. But it is inter-
esting to note that the ripples next to the central hump are well approximated by the Painlevé
I solution. The L∞ norm of the difference between the two solutions scales roughly like �.
More precisely we find a scaling �a with a = 1.02 (r = 0.9999 and σa = 7.7 ∗ 10−3).
8 Concluding remarks
In this paper we have started the study of the critical behavior of generic solutions of the fo-
cusing nonlinear Schrödinger equation. We have formulated the conjectural analytic descrip-
tion of this behavior in terms of the tritronquée solution to the Painlevé-I equation restricted
to certain lines in the complex plane. We provided analytical as well as numerical evidence
−0.05 0 0.05
ε=0.03
−0.1 0 0.1
ε=0.1
Figure 12: The blue line is the solution to the focusing NLS equation for the initial data
u0(x) = 2 sechx at the critical time, and the green line gives the multiscales solution via the
tritronquée solution of the Painlevé I equation. The plots are shown for two values of � at the
critical time.
supporting our conjecture. In subsequent publications we plan to further study the Main
Conjecture of the present paper by applying techniques based, first of all, on the Riemann -
Hilbert problem method [26, 43, 44] and the theory of Whitham equations (see [19] for the
numerical implementation of the Whitham procedure in the analysis of oscillatory behavior
of solutions to the KdV equations). The latter will also be applied to the asymptotic descrip-
tion of solutions inside the oscillatory zone. Furthermore we plan to study the possibility
of extending the Main Conjecture to the critical behavior of solutions to the Hamiltonian
perturbations of more general first order quasilinear systems of elliptic type. Last but not
least, it would be of interest to study the distribution of poles of the tritronquée solution in
the sector | arg ζ| > 4π
and to compare these poles with the peaks of solutions to NLS inside
the oscillatory zone. The elliptic asymptotics obtained by Kitaev [28] might be useful for
studying these poles for large |ζ|.
In this paper we did not study the behaviour of solutions to NLS near the boundary u = 0.
Such a study is postponed for a subsequent publication.
0.35 0.4 0.45 0.5
!=0.03
0.2 0.3 0.4 0.5 0.6 0.7
!=0.1
Figure 13: The blue line is the solution to the focusing NLS equation for the non-symmetric
initial data at the critical time, and the green line gives the multiscales solution via the
tritronquée solution of the Painlevé I equation. The plots are shown for two values of �
at the critical time.
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	Introduction
	Dispersionless NLS, its solutions and critical behaviour
	First integrals and solutions of the NLS/Toda equations
	Critical behaviour and elliptic umbilic catastrophe
	The tritronquée solution to the Painlevé-I equation and the Main Conjecture
	Numerical analysis of the tritronquée solution of P-I
	Critical behavior in NLS and the tritronquée solution of P-I: numerical results
	Initial data
	Symmetric initial data
	Non-symmetric initial data
	Semiclassical solution
	Multiscales solution
	Concluding remarks

  We argue that the critical behaviour near the point of ``gradient
catastrophe" of the solution to the Cauchy problem for the focusing nonlinear
Schr\"odinger equation $ i\epsilon \psi_t +\frac{\epsilon^2}2\psi_{xx}+
|\psi|^2 \psi =0$ with analytic initial data of the form $\psi(x,0;\epsilon)
=A(x) e^{\frac{i}{\epsilon} S(x)}$ is approximately described by a particular
solution to the Painlev\'e-I equation.

Introduction
The focusing nonlinear Schrödinger (NLS) equation for the complex valued function Ψ =
Ψ(x, t)
iΨt +
Ψxx + |Ψ|2Ψ = 0 (1.1)
has numerous physical applications in the description of nonlinear waves (see, e.g., the books
[47, 35, 36]). It can be considered as an infinite dimensional analogue of a completely
integrable Hamiltonian system [48], where the Hamiltonian and the Poisson bracket is given
Ψt + {Ψ(x),H} = 0
{Ψ(x),Ψ∗(y)} = i δ(x− y) (1.2)
|Ψx|2 − |Ψ |4
(here Ψ∗ stands for the complex conjugate function). Properties of various classes of so-
lutions to this equation have been extensively studied both analytically and numerically
[5, 6, 7, 8, 16, 21, 25, 26, 29, 32, 34, 43, 44]. One of the striking features that distinguishes
this equation from, say, the defocusing case
iΨt +
Ψxx − |Ψ|2Ψ = 0
is the phenomenon of modulation instability [1, 11, 16]. Namely, slow modulations of the
plane wave solutions
Ψ = Aei(kx−ωt), ω =
k2 − A2
develop fast oscillations in finite time.
The appropriate mathematical framework for studying these phenomena is the theory of
the initial value problem
Ψ(x, 0; �) = A(x) e
S(x) (1.3)
for the �-dependent NLS
i �Ψt +
Ψxx + |Ψ|2Ψ = 0. (1.4)
Here � > 0 is a small parameter, A(x) and S(x) are real-valued smooth functions. Introduc-
ing the slow variables
u = |Ψ|2, v =
(1.5)
the equation can be recast into the following system:
ut + (u v)x = 0
(1.6)
vt + v vx − ux +
The initial data for the system (1.6) coming from (1.3) do not depend on �:
u(x, 0) = A2(x), v(x, 0) = S ′(x). (1.7)
The simplest explanation of the modulation instability then comes from considering the so-
called dispersionless limit �→ 0. In this limit one obtains the following first order quasilin-
ear system
ut +v ux + u vx = 0
vt − ux + v vx = 0
 . (1.8)
This is a system of elliptic type because of the condition u > 0. Indeed, the eigenvalues of
the coefficient matrix (
are complex conjugate, λ = v ± i
u. So, the Cauchy problem for the system (1.8) is ill-
posed in the Hadamard sense (cf. [33, 7]). Even for analytic initial data the life span of a
typical solution is finite, t < t0. The x- and t-derivatives explode at some point x = x0 when
the time approaches t0. This phenomenon is similar to the gradient catastrophe of solutions
to nonlinear hyperbolic PDEs [2].
For the full system (1.6) the Cauchy problem is well-posed for a suitable class of �-
independent initial data (see details in [17, 46]). However, the well-posedness is not uniform
in �. In practical terms that means that the solution to (1.6) behaves in a very irregular way
in some regions of the (x, t)-plane when � → 0. Such an irregular behaviour begins near
the points (x = x0, t = t0) of the “gradient catastrophe” of the solution to the dispersionless
limit (1.8). The solutions to (1.8) and (1.6) are essentially indistinguishable for t < t0; the
situation changes dramatically near x0 when approaching the critical point. Namely, when
approaching t = t0 the peak near a local maximum1 of u becomes more and more narrow
due to self-focusing; the solution develops a zone of rapid oscillations for t > t0. They
have been studied both analytically and numerically in [8, 16, 21, 23, 25, 26, 34, 43, 44].
However, no results are available so far about the behaviour of the solutions to the focusing
NLS at the critical point (x0, t0).
The main subject of this work is the study of the behaviour of solutions to the Cauchy
problem (1.6), (1.7) near the point of gradient catastrophe of the dispersionless system (1.8).
In order to deal with the Cauchy problem for (1.8) we will assume analyticity2 of the initial
data u(x, 0), v(x, 0). Then the Cauchy problem for (1.8) can be solved for t < t0 via a
suitable version of the hodograph transform (see Section 2 below). An important feature of
the gradient catastrophe for this system is that it happens at an isolated point of the (x, t)-
plane, unlikely the case of KdV or defocusing NLS where the singularity of the hodograph
solution takes place on a curve. We identify this singularity for a generic solution to (1.8)
as the elliptic umbilic singularity (see Section 4 below) in the terminology of R.Thom [42].
This codimension 2 singularity is one of the real forms labeled by the root system of the D4
type in the terminology of V.Arnold et al. [3].
Our main goal is to find a replacement for the elliptic umbilic singularity when the disper-
sive terms are added, i.e., we want to describe the leading term of the asymptotic behaviour
for �→ 0 of the solution to (1.6) near the critical point (x0, t0) of a generic solution to (1.8).
Thus, our study can be considered as a continuation of the programme initiated in [13]
to study critical behaviour of Hamiltonian perturbations of nonlinear hyperbolic PDEs; the
fundamental difference is that the non perturbed system (1.8) is not hyperbolic! However,
many ideas and methods of [13] (see also [12]) play an important role in our considerations.
The most important of these is the idea of universality of the critical behaviour. The
1Regarding initial data with local minima we did not observe cusps related to minima in numerical simula-
tions. We believe they do not exist because of the focusing effect in the NLS that pushes maxima to cusps but
seems to smoothen minima.
2We believe that the main conclusions of this paper must hold true also for non analytic initial data; the
numerical experiments of [8] do not show much difference in the properties of solution between analytic and
non analytic cases. However, the precise formulation of our Main Conjecture has to be refined in the non
analytic case.
general formulation of the universality suggested in [13] for the case of Hamiltonian pertur-
bations of the scalar nonlinear transport equation says that the leading term of the multiscale
asymptotics of the generic solution near the critical point does not depend on the choice of
the solution, modulo Galilean transformations and rescalings. This leading term was identi-
fied in [13] via a particular solution to the fourth order analogue of the Painlevé-I equation
(the so-called P 21 equation). The existence of the needed solution to P
1 has been rigorously
established in [9]. Moreover, it was argued in [13] that this behaviour is essentially indepen-
dent on the choice of the Hamiltonian perturbation. Some of these universality conjectures
have been partially confirmed by numerical analysis carried out in [21].
The main message of this paper is the formulation of the Universality Conjecture for
the critical behaviour of the solutions to the focusing NLS. Our considerations suggest the
description of the leading term in the asymptotic expansion of the solution to (1.6) near the
critical point via a particular solution to the classical Painlevé-I equation (P-I)
Ωζζ = 6 Ω
2 − ζ.
The so-called tritronquée solution to P-I was discovered by P.Boutroux [4] as the unique
solution having no poles in the sector | arg ζ| < 4π/5 for sufficiently large |ζ|. Remarkably,
the very same solution3 arises in the critical behaviour of solutions to focusing NLS!
The paper is organized as follows. In Section 2 we develop a version of the hodograph
transform for integrating the dispersionless system (1.8) before the catastrophe t < t0. We
also establish the shape of the singularity of the solution near the critical point; the latter is
identified in Section 4 with the elliptic umbilic catastrophe of Thom. In Section 3 we develop
a method of constructing formal perturbative solutions to the full system (1.6) before the
critical time. In Section 5 we collect the necessary information about the tritronquée solution
of P-I and formulate the Main Conjecture of this paper. Such a formulation relies on a much
stronger property of the tritronquée solution: namely, we need this solution to be pole-free
in the whole sector | arg ζ| < 4π/5. Numerical evidence for the absence of poles in this
sector is given in Section 6. In Section 7 we analyze numerically the agreement between the
critical behaviour of solutions to focusing NLS and its conjectural description in terms of the
tritronquée solution restricted on certain lines in the complex ζ-plane. In the final Section 8
we give some additional remark and outline the programme of future research.
Acknowledgments. The authors thank K.McLaughlin for a very instructive discussion. One
of the authors (B.D.) thanks R.Conte for bringing his attention to the tritronquées solutions
of P-I. The results of this paper have been presented by one of the authors (T.G.) at the
Conference “The Theory of Highly Oscillatory Problems”, Newton Institute, Cambridge,
March 26 - 30, 2007. T.G. thanks A.Fokas and S.Venakides for the stimulating discussion
after the talk. The present work is partially supported by the European Science Foundation
Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM),
3It is interesting that the same tritronquée solution (for real ζ only) appeares also in the study of certain
critical phenomena in plasma [41]. In the theory of random matrices and orthogonal polynomials a different
solution to P-I arises; see, e.g., [14].
Marie Curie RTN “European Network in Geometry, Mathematical Physics and Applications”
(ENIGMA). The work of B.D. and T.G. is also partially supported by Italian Ministry of
Universities and Researches (MUR) research grant PRIN 2006 “Geometric methods in the
theory of nonlinear waves and their applications”.
2 Dispersionless NLS, its solutions and critical behaviour
The equations (1.6) are a Hamiltonian system
ut + {u(x), H} = 0
vt + {v(x), H} = 0
with respect to the Poisson bracket originated in (1.2)
{u(x), v(y)} = δ′(x− y), (2.1)
other brackets vanish, with the Hamiltonian
u v2 − u2
dx. (2.2)
Let us first describe the general analytic solution to the dispersionless system (1.8).
Lemma 2.1 Let uI(x), vI(x) be two real valued analytic functions of the real variable x
satisfying
(uI,x)
+ (vI,x)
2 6= 0.
Then the solution u = u(x, t), v = v(x, t) to the Cauchy problem
u(x, 0) = uI(x), v(x, 0) = vI(x) (2.3)
for the system (1.8) for sufficiently small t can be determined from the following system
x = v t+ fu
0 = u t+ fv
 (2.4)
where f = f(u, v) is an analytic solution to the following linear PDE:
fvv + u fuu = 0. (2.5)
Conversely, given any solution to (2.5) satisfying u2f 2uu + f
uv 6= 0 at some point (u = ũ, v =
ṽ) such that fv(ũ, ṽ) = 0, the system (2.4) determines a solution to (1.8) defined locally near
the point x = x̃ := fu(ũ, ṽ) for sufficiently small t.
Remark 2.2 The solutions to the linear PDE (2.5) correspond to the first integrals of dis-
persionless NLS:
f(u, v) dx,
F = 0. (2.6)
Taking them as the Hamiltonians
us + {u(x), F} ≡ ut + (fv)x = 0
vs + {v(x), F} ≡ vt + (fu)x = 0
 (2.7)
yields infinitesimal symmetries of the dispersionless NLS:
(ut)s = (us)t , (vt)s = (vs)t . (2.8)
One of the first integrals will be extensively used in this paper. It corresponds to the
Hamiltonian density
g = −
v2 + u(log u− 1). (2.9)
The associated Hamiltonian flow reads
us + vx = 0
 (2.10)
Eliminating the dependent variable v one arrives at the elliptic version of the long wave limit
of Toda lattice:
uss + (log u)xx = 0.
Due to commutativity (2.8) the systems (1.8) and (2.10) admit a simultaneous solution u =
u(x, t, s), v = v(x, t, s). Any such solution can be locally determined from a system similar
to (2.4)
x = v t+ fu
s = u t+ fv
 (2.11)
where f = f(u, v), as above, solves the linear PDE (2.5).
The system (2.11) determines a solution u = u(x, t, s), v = v(x, t, s) provided applica-
bility of the implicit function theorem. The conditions of the latter fail to hold at the critical
point (x0, s0, t0, u0, v0) such that
x0 = v0t0 + fu(u0, v0)
s0 = u0t0 + fv(u0, v0)
fuu(u0, v0) = fvv(u0, v0) = 0, fuv(u0, v0) = −t0
(2.12)
In sequel we adopt the following system of notations: the values of the function f and of its
derivatives at the critical point will be denoted by f 0 etc. E.g., the last line of the conditions
(2.12) will read
f 0uu = f
vv = 0, f
uv = −t0.
Definition 1. We say that the critical point is generic if at this point:
f 0uuv 6= 0.
Let us the introduce real parameters r, ψ determined by the third derivatives of the function
f evaluated at the critical point,
(cosψ − i sinψ) = f 0uuv + i
uuu. (2.13)
Due to the genericity assumption
+ πk.
In order to describe the local behaviour of a solution to the dispersionless NLS/Toda
equations we define a function R(X;S, ψ) of real variables X , S depending on the real
parameter ψ satisfying
(S + cosψ)2 + (X + sinψ)2 6= 0 (2.14)
by the following formula
R(X,S, ψ) = sign [cosψ] (2.15)
1 +X sinψ + S cosψ +
1 + 2(X sinψ + S cosψ) +X2 + S2
P0(X,S, ψ) =
R(X,S, ψ) cosψ −
(X cosψ − S sinψ) sinψ
R(X,S, ψ)
− cosψ
(2.16)
Q0(X,S, ψ) =
(X cosψ − S sinψ) cosψ
R(X,S, ψ)
+R(X,S, ψ) sinψ
− sinψ.
Observe that P0(X,S, ψ) and Q0(X,S, ψ) are smooth functions of the real variable X pro-
vided validity of the inequality (2.14).
Lemma 2.3 Given an analytic solution u(x, s, t), v(x, s, t) to the dispersionless NLS/Toda
equations with a generic critical point (x0, s0, t0, u0, v0), and arbitrary real numbers X , S
satisfying (2.14), T < 0, then there exist the following limits
λ−1/2
u(x0 + λ
1/2v0T +
r X T 2, s0 + λ
1/2u0T +
rS T 2, λ1/2T )− u0
= r T P0 (X,S, ψ)
(2.17)
λ−1/2
v(x0 + λ
1/2v0T +
r X T 2, s0 + λ
1/2u0T +
rS T 2, λ1/2T )− v0
T Q0 (X,S, ψ)
where the parameters r, ψ are defined by (2.13).
Proof From the linear PDE (2.5) it follows that
fuvv = −ufuuu − fuu, fvvv = −ufuuv.
Using these formulae we expand the implicit function equations (2.11) near the critical point
in the form
x̄− v0t̄ = v̄ t̄+
f 0uuu(ū
2 − u0v̄2) + 2f 0uuvū v̄
(|ū|2 + |v̄|2)3/2
(2.18)
s̄− u0t̄ = ū t̄+
f 0uuv(ū
2 − u0v̄2)− 2u0f 0uuuū v̄
(|ū|2 + |v̄|2)3/2
where we introduce the shifted variables
x̄ = x− x0, s̄ = s− s0, t̄ = t− t0
ū = u− u0, v̄ = v − v0.
The rescaling
x̄− v0t̄ 7→ λ(x̄− v0t̄)
s̄− u0t̄ 7→ λ(s̄− u0t̄)
t̄ 7→ λ1/2t̄
ū 7→ λ1/2ū
v̄ 7→ λ1/2v̄
(2.19)
in the limit λ→ 0 yields the quadratic equation
z = t̄ w +
aw2, t̄ 6= 0 (2.20)
where the complex independent and dependent variables z and w read
z = s̄+ i
u0x̄− (u0 + i
u0v0)t̄, w = ū+ i
u0v̄ (2.21)
and the complex constant a is defined by
a = f 0uuv + i
uuu, (2.22)
therefore
= r eiψ.
The substitution
X = 2
x̄− v0t̄
r t̄2
, S = 2
s̄− u0t̄
r t̄2
reduces the quadratic equation to(
w + t̄ r eiψ
= r2t̄2e2iψ
1 + e−iψ(S + iX)
For t̄ < 0 we choose the following root
w = r t̄ei ψ
1 + e−i ψ(S + iX)− 1
(2.23)
where the branch of the square root is obtained by the analytic continuation of the one taking
positive values on the positive real axis. Equivalently,
w = t̄ r eiψ
sign (cosψ)
∆ + 1 + S cosψ +X sinψ + i
X cosψ − S sinψ
∆ + 1 + S cosψ +X sinψ
where
1 + 2(S cosψ +X sinψ) +X2 + S2.
This gives the formulae (2.15).
The result of the lemma describes the local structure of generic solutions to the dis-
persionless NLS/Toda equations near the critical point. It can also be represented in the
following form
u(x, s, t) ' u0 + r T P0(X,S, ψ)
(2.24)
v(x, s, t) ' v0 +
r T Q0(X,S, ψ)
where
X = 2
x̄− v0t̄
r t̄2
, S = 2
s̄− u0t̄
r t̄2
, T = t̄. (2.25)
We want to emphasize that the approximation (2.24) works only near the critical point.
Indeed, for large x→ ±∞ the function u(x, s, t) and v(x, s, t) have the following behaviour
u = −
r |x|u1/40
1∓ sinψ + u0 − r t̄ cosψ +O
(2.26)
v = ∓
r |x|
u01/4
sign (cosψ)
1± sinψ + v0 −
t̄ sinψ +O
. (2.27)
So, for sufficiently large |x| the function u(x, s, t) defined by (2.24) becomes negative.
The function u has a maximum at the point X = S tanψ, so locally
u ≤ u0 + r T cosψ −
r | cosψ|
+ r T 2 < u0. (2.28)
At the critical point (x0, s0, t0, u0, v0) the function u develops a cusp. Let us consider only
the particular case S = 0 in order to avoid complicated expressions. In this case the local
behaviour of the function u near the critical point is given by
t̄→−0
r |x̂|
1− sinψ, x̂ > 0
r |x̂|
1 + sinψ, x̂ < 0
(2.29)
(here x̂ =
u0(x̄ − v0t̄)). Thus the parameters r, ψ describe the shape of the cusp at the
critical point.
3 First integrals and solutions of the NLS/Toda equations
Let us first show that any first integral (2.6) of the dispersionless equations can be uniquely
extended to a first integral of the full equations.
Lemma 3.1 Given a solution f = f(u, v) to the linear PDE (2.5), there exists a unique, up
to a total derivative, formal power series in �
hf = f +
f (u, v;ux, vx, . . . , u
(2k), v(2k))
such that the integral
hf dx
commutes with the Hamiltonian of NLS equation:
{H,Hf} = 0
at every order in �. Explicitly,
hf = f −
fuuu +
u2x + 2fuuvuxvx − ufuuuv
(3.1)
fuuuu +
u2xx + 2fuuuvuxxvxx − ufuuuuv
fuuuuuxxv
fuuuvvxxu
3456u3
30fuuu − 9ufuuuu + 12u2f5u + 4u3f6u
432u2
−3fuuuv + 6ufuuuuv + 2u2f5u v
u3xvx +
9fuuuu + 9uf5u + 2u
(9fuuuuv + 10uf5u v)uxv
(18f5u + 5uf6u) v
+O(�6)
Here we use short notations
f5u :=
, f6u :=
, f5u v :=
∂u5∂v
Example 1. Taking f = 1
(u v2 − u2) one obtains the Hamiltonian of the NLS equation
(u v2 − u2) +
In this case the infinite series truncates. It is easy to see that the series in � truncates if
and only if f(u, v) is a polynomial in u. Polynomial in u solutions to the linear PDE (2.5)
correspond to the standard first integrals of the NLS hierarchy.
Example 2. Taking g = −1
v2 + u(log u− 1) (cf. (2.9)) one obtains the Hamiltonian of
Toda equation
hg = −
v2 + u(log u− 1)−
u2x + 2u v
240u3
144u5
360u3
+O(�6) (3.2)
written in terms of the function φ = log u in the form
�2φxx + e
φ(s+�) − 2eφ(s) + eφ(s−�) = 0.
Lemma 3.2 Any solution to the NLS/Toda equations in the class of formal power series in �
can be obtained from the equations
x = v t+
δu(x)
(3.3)
s = u t+
δv(x)
where f = f(u, v; �) is an arbitrary admissible solution to the linear PDE (2.5) in the class
of formal power series in �,
hf dx.
Now, we can apply to the system (3.3) the rescaling (2.19) accompanied by the transfor-
mation
� 7→ λ5/4�. (3.4)
At the limit λ→ 0 we arrive at the following system of equations
s̄− u0t̄ = ū t̄+ f 0uuv
(ū2 − u0v̄2) +
− u0f 0uuu
ū v̄ +
(3.5)
x̄− v0t̄ = v̄ t̄+ f 0uuu
(ū2 − u0v̄2) +
+ f 0uuv
ū v̄ +
Using the complex variables z, w defined in (2.21) we can rewrite the system in the following
form:
z reiψ = w t̄ reiψ +
wxx. (3.6)
The last observation is that the Toda equations generated by the Hamiltonian Hg =
hg dx
(see Example 2 above) after the scaling limit (2.19), (3.4) yield the Cauchy - Riemann equa-
tions for the function w = w(z),
∂w/∂z̄ = 0.
Therefore the system (3.5) can be recast into the form equivalent to the Painlevé-I (P-I)
equation (see (5.1) below)
z reiψ = w t̄ reiψ +
u0wzz. (3.7)
Choosing
λ = �4/5
we eliminate � from the equation.
In the Section 5 below we will write explicitly the reduction of (3.7) to the Painlevé-I
equation and give a conjectural characterization of the particular solution of the latter.
4 Critical behaviour and elliptic umbilic catastrophe
Separating again the real and complex parts of (3.6) one obtains a system of ODEs
UXX +
(U2 − V 2) + r t̄ (U cosψ − V sinψ)− r (S cosψ −X sinψ) = 0
(4.1)
VXX + UV + r t̄ (U sinψ + V cosψ)− r (S sinψ +X cosψ) = 0
that can be identified with the Euler - Lagrange equations
δS = 0, S =
L(U, V, UX , VX) dX
with the Lagrangian
V 2X − U
U3 − 3U V 2
(U2 − V 2) cosψ − 2U V sinψ
(4.2)
+r (X sinψ − S cosψ)U + r (S sinψ +X cosψ)V.
In the “dispersionless limit” � → 0 the Euler - Lagrange equations reduce to the search of
stationary points of a function (let us also set t̄ = 0)
U3 − 3U V 2
+ a+U + a−V (4.3)
where we redenote
a+ = r (X sinψ − S cosψ) , a− = r (S sinψ +X cosψ) .
At a+ = a− = 0 the function F has an isolated singularity at the origin U = V = 0 of the
type D4,− also called elliptic umbilic singularity, according to R.Thom [42]. This singular-
ity appears in various physical problems; we mention here the caustics in the collisionless
dark matter [40] to give just an example. The parameters a+ and a− define two particular
directions on the base of the miniversal unfolding of the elliptic umbilic; the full unfolding
depending on 4 parameters reads
U3 − 3U V 2
b(U2 + V 2) + a+U + a−V + c. (4.4)
It would be interesting to study the properties of the modified Euler - Lagrange equations for
the Lagrangian
L̂ = L+
b(U2 + V 2).
This deformation does not seem to arrive from considering solutions to the NLS hierarchy.
5 The tritronquée solution to the Painlevé-I equation and
the Main Conjecture
In this section we will select a particular solution to the Painlevé-I (P-I) equation
Ωζζ = 6Ω
2 − ζ (5.1)
Recall [22] that an arbitrary solution to this equation is a meromorphic function on the com-
plex ζ-plane. According to P. Boutroux [4] the poles of the solutions accumulate along the
arg ζ =
, n = 0, ±1, ±2. (5.2)
Boutroux proved that, for each ray there is a one-parameter family of particular solutions
called intégrales tronquées whose lines of poles truncate for large ζ . He proved that the
intégrale tronquée has no poles for large |ζ| within two consecutive sectors of the angle
2π/5 around the ray, and, moreover it has the asymptotic behaviour of the form
Ω = −
)1/2 [
(1−ε)
(5.3)
for a suitable choice of the branch of the square root (see below) and a sufficiently small
ε > 0.
Furthemore, if a solution truncates along any two of the rays (5.2) then it truncates along
three of them. These particular solutions to P-I are called tritronquées. They have no poles
for large |ζ| in four consecutive sectors; their asymptotics for large ζ is given by (5.3). It
suffices to know the tritronquée solution Ω0(ζ) for the sector
| arg ζ| <
. (5.4)
In this case the branch of the square root in (5.3) is obtained by the analytic continuation
of the principal branch taking positive values on the positive half axis ζ > 0. Other four
tritronquées solutions are obtained by applying the symmetry
Ωn(ζ) = e
, n = ±1, ±2. (5.5)
The properties of the tritronquées solutions in the finite part of the complex plane were
studied in the important paper of N.Joshi and A.Kitaev [24].
A. Kapaev [27] obtained a complete characterization of the tritronquées solutions in
terms of the Riemann - Hilbert problem associated with P-I. We will briefly sketch here
the main steps of his construction.
The equation (5.1) can be represented as the compatibility condition of the following
system of linear differential equations for a two-component vector valued function Φ =
Φ(λ, ζ)
 Ωζ 2λ2 + 2Ωλ− ζ + 2Ω2
2(λ− Ω) −Ωζ
 Φ (5.6)
Φζ = −
 0 λ+ 2Ω
 Φ. (5.7)
The canonical matrix solutions Φk(λ, ζ) to the system (5.6) - (5.7) are uniquely determined
by their asymptotic behaviour
Φk(λ, ζ) ∼
λ1/4 λ1/4
λ−1/4 −λ−1/4
(5.8)
λ−3/2
eθ(λ,ζ)σ3 , |λ| → ∞, λ ∈ Σk
in the sectors
λ ∈ C |
< arg λ <
, k ∈ Z. (5.9)
θ(λ, ζ) =
λ5/2 − ζ λ1/2, σ3 =
, H =
Ω2ζ − 2 Ω
3 + ζ Ω, (5.10)
the branch cut on the complex λ-plane for the fractional powers of λ is chosen along the
negative real half-line.
The Stokes matrices Sk are defined by
Φk+1(λ, ζ) = Φk(λ, ζ)Sk, λ ∈ Σk ∩ Σk+1. (5.11)
They have the triangular form
S2k−1 =
1 s2k−1
, S2k =
s2k 1
(5.12)
and satisfy the constraints
Sk+5 = σ1 Sk σ1, k ∈ Z; S1S2S3S4S5 = i σ1 (5.13)
where
Due to (5.13) two of the Stokes multipliers sk determine all others; they depend neither on λ
nor on ζ provided Ω(ζ) satisfies (5.1).
In order to obtain a parametrization of solutions to the P-I equation (5.1) by Stokes mul-
tipliers of the linear differential equation (5.6) one has to reformulate the above definitions
as a certain Riemann - Hilbert problem. The solution of the Riemann - Hilbert problem de-
pends on ζ through the asymptotics (5.8). If the Riemann - Hilbert problem has a unique
solution for the given ζ0 ∈ C then the canonical matrices Φk(λ, ζ) depend analytically on
ζ for sufficiently small |ζ − ζ0|; the coefficient Ω = Ω(ζ) will then satisfy (5.1). The poles
of the meromorphic function Ω(ζ) correspond to the forbidden values of the parameter ζ for
which the Riemann - Hilbert problem admits no solution.
We will now consider a particular solution to the P-I equation specified by the following
Riemann - Hilbert problem. Denote four oriented rays γ0, γ±1, ρ in the complex λ-plane
defined by
γk = {λ ∈ C | arg λ =
}, k = 0, ±1
(5.14)
ρ = {λ ∈ C | arg λ = π}
directed towards infinity. The rays divide the complex plane in four sectors. We are looking
for a piecewise analytic function Φ(λ, ζ) on
λ ∈ C \ (γ−1 ∪ γ0 ∪ γ1 ∪ ρ)
depending on the parameter ζ continuous up to the boundary with the asymptotic behaviour
at |λ| → ∞ of the form (5.8) satisfying the following jump conditions on the rays
Φ+(λ, ζ) = Φ−(λ, ζ)Sk, λ ∈ γk
(5.15)
Φ+(λ, ζ) = Φ−(λ, ζ)Sρ, λ ∈ ρ.
Here the plus/minus subscripts refer to the boundary values of Φ respectively on the left/right
sides of the corresponding oriented ray, the jump matrices are given by
, S±1 =
, Sρ =
. (5.16)
The following result is due to A.Kapaev4 .
Theorem 5.1 The solution to the above Riemann - Hilbert problem exists and it is unique
| arg λ| <
, |λ| > R (5.17)
for a sufficiently large positive number R. The associated function
Ω0(ζ) :=
dH(ζ)
(5.18)
H(ζ) :=
σ3Φ(λ, ζ) e−θ(λ,ζ)σ3 − 1
is analytic in the domain (5.17), it satisfies P-I and enjoys the asymptotic behaviour
Ω0(ζ) ∼ −
, |ζ| → ∞, | arg λ| <
. (5.19)
Moreover, any solution of P-I having no poles in the sector (5.17) for some large R > 0
coincides with Ω0(ζ).
4Our solution Ω0(ζ) coincides with y3(x) ≡ y−2(x), x = −ζ, of [27] (see eq. (2.73) of [27]; Kapaev uses
a different normalization y′′ = 6y2 + x of the P-I equation).
Joshi and Kitaev proved that the tritronquée solution has no poles on the positive real
axis. They found a numerical estimate for the position of the first pole ζ0 of the tritronquée
solution Ω0(ζ) on the negative real axis:
ζ0 ' −2.3841687
(cf. also [10]). Besides this estimate very little is known about the location of poles of Ω0(ζ).
Our numerical experiments (see below) suggest the following
Main Conjecture. Part 1. The tritronquée solution Ω0(ζ) has no poles in the sector
| arg λ| <
. (5.20)
We are now ready to describe the conjectural universal structure behind the critical be-
haviour of generic solutions to the focusing NLS. For simplicity of the formulation let us
assume cosψ > 0.
Main Conjecture. Part 2. Any generic solution to the NLS/Toda equations near the
critical point behaves as follows
u(x, s, t; �) + i
u0v(x, s, t; �) ' u0 + i
u0v0 − t̄ reiψ + 2 �2/5(3r
5 Ω0(ζ) +O
(5.21)
s̄− u0t̄+ i
u0(x̄− v0t̄) + 12re
iψ t̄2
where Ω0(ζ) is the tritronquée solution to the Painlevé-I equation (5.1).
The above considerations can actually be applied replacing the NLS time flow by any
other flow of the NLS/Toda hierarchy. The local description of the critical behaviour remains
unchanged.
Remark 5.2 Note that the angle of the line ζ(x̄) in (5.21) for t̄ fixed is equal to ψ/5 + π/2,
ψ ∈ [−π, π]. Thus the maximal value of argζ is equal to 7π/10 < 4π/5. The lines in (5.21)
consequently do not get close to the critical lines of the tritronquée solution.
6 Numerical analysis of the tritronquée solution of P-I
In this section we will numerically construct the tritronquée solution Ω0, i.e. the tritronquée
solution with asymptotic behavior (5.3). We will drop the index 0 in the following. The
solution will be first constructed on a straight line in the complex plane. In a second step
we will then explore global properties of these solutions within the limitations imposed by a
numerical approach5.
5Cf. [15] where a similar technique was applied to solve numerically the Painlevé-II equation in the complex
domain.
Let the straight line in the complex plane be given by ζ = ay + b with a, b ∈ C constant
(we choose a to have a non-negative imaginary part) and y ∈ R. The asymptotic conditions
Ω ∼ −
, (6.1)
for y → ±∞. The root is defined to have its cut along the negative real axis and to as-
sume positive values on the positive real axis. This choice of the root implies the following
symmetry for the solution:
Ω(ζ∗) = Ω∗(ζ). (6.2)
Thus Ω is real on the real axis, see [24].
Numerically it is not convenient to impose boundary conditions at infinity. We thus
assume that the wanted solution can be expanded in a Laurent series in
ζ around infinity.
Such an asymptotic expansion is possible for the considered tritronquée solution in the sector
| arg ζ| < 4π/5. The formal series can be written there (see [24]) in the form
Ωf = −
ζ5k/2
, (6.3)
where a0 = 1, and where the remaining coefficients follow from the recurrence relation for
k ≥ 0
ak+1 =
25k2 − 1
amak+1−m. (6.4)
This formal series is divergent, the coefficients ak behave asymptotically as ((k − 1)!)2, see
[24] for a detailed discussion.
It is known that divergent series can be used to get numerically acceptable approxima-
tions to the values of the sum by properly truncating the series. Generally the best approx-
imations for the sum result from truncating the series where the terms take the smallest
absolute values (see e.g. [18]). Since we work in Matlab with a precision of 16 digits and
with values of |ζ| ≥ 10, we typically consider up to 10 terms in the series. In this case the
terms corresponding to a10 are of the order of machine precision (10−14 and below).
Thus we have constructed approximations to the numerical values of the tritronquée
solution for large values of |ζ|. These can be used as in [24] to set up an initial value problem
for the P-I equation and to solve this with a standard ODE solver. In fact the approach works
well on the real axis starting from positive values until one reaches the first singularity on
the negative real axis. It is straightforward to check the results of [24] with e.g. ode45, the
Runge-Kutta solver in Matlab corresponding to the Maple solver used in [24]. If one solves
P-I on a line that avoids the sector | arg ζ| > 4π/5, one could integrate until one reaches once
more large values of |ζ| for which the asymptotic conditions are known. This would provide
a control on the numerical accuracy of this so-called shooting approach. Shooting methods
are problematic if the second solution to the initial value problem has poles as is the case for
P-I. In this case the numerical errors in the initial data (here due to the asymptotic conditions)
and in the time integration will lead to a large contribution of the unwanted solution close
to its poles which will make the numerical solution useless. It is obvious that P-I has such
poles from the numerical results in [24] and the property (5.5). In [24] the task was to locate
poles in the tritronquée solution, and in this case the shooting approach seems to be the only
available. Here we are studying, however, the solution on a line in the complex plane where
we know the asymptotic conditions for the affine parameter tending to ±∞.
Thus we use as in [15] the asymptotic conditions on lines avoiding the sector | arg ζ| >
4π/5 to set up a boundary value problem for y = ±y0, y0 ≥ 10. The solution in the
interval [−y0, y0] is numerically obtained with a finite difference code based on a collocation
method. The code bvp4c distributed with Matlab, see [39] for details, uses cubic polynomials
in between the collocation points. The P-I equation is rewritten in the form of a first order
system. With some initial guess (we use Ω = −
ζ/6 as the initial guess), the differential
equation is solved iteratively by linearization. The collocation points (we use up to 10000)
are dynamically adjusted during the iteration. The iteration is stopped when the equation is
satisfied at the collocation points with a prescribed relative accuracy, typically 10−10. The
solution for a = i and b = 0 is shown in Fig. 1. The values of Ω in between the collocation
!20 !15 !10 !5 0 5 10 15 20
Figure 1: Real (blue) and imaginary part (red) of the tritronquée solution to the Painlevé I
equation for ζ = iy.
points are obtained by interpolation via the cubic polynomials in terms of which the solution
has been constructed. This interpolation leads to a loss in accuracy of roughly one order of
magnitude with respect to the precision at the collocation points. To test this we determine
the numerical solution via bvp4c for P-I on Chebychev collocation points and check the
accuracy with which the equation is satisfied via Chebychev differentiation, see e.g. [45].
It is found that the numerical solution with a relative tolerance of 10−10 on the collocation
points satisfies the ODE to roughly the same order except at the boundary points where it
is of the order 10−8, see Fig. 2 where we show the residual ∆ by plugging the numerical
solution into the differential equation for the above example. It is straightforward to achieve
a prescribed accuracy by requiring a certain value for the relative tolerance. Notice that we
are not interested in a high precision solution of P-I here, but in a comparison of solutions
to the NLS equation close to the point of gradient catastrophe of the semiclassical system
with an asymptotic solution in terms of P-I transcendents. For this purpose an accuracy of
the solution of the order of 10−4 will be sufficient in all studied cases.
!20 !15 !10 !5 0 5 10 15 20
Figure 2: Error in the solution of the Painlevé I equation.
The quality of the used boundary conditions via the asymptotic behavior can be checked
by computing the solution for different values of y0. One finds that the difference between
the asymptotic square root and the tritronquée solution is only visible near the origin, see
Fig. 3. For large x it can be seen that the difference between the square root asymptotics
and the tritronquée solution reaches quickly values below the aimed at threshold of 10−4.
It is interesting to note that this difference is actually smaller than the difference between
the tritronquée solution and the truncated formal asymptotic series except at the boundary,
where the latter condition is implemented (see Fig. 3).
The dominant behavior of the square root changes if one approaches the critical lines
a = exp(4πi/5), b = 0. As can be seen from Fig. 4, the solution shows oscillations on top
of the square root. The closer one comes to the critical lines, the slower is the fall off of the
amplitude of the oscillations. We conjecture that these oscillations will have on the critical
10 12 14 16 18 20
!20 !10 0 10 20
Figure 3: The plot on the left side shows the absolute value of the difference between the
tritronquée solution and the asymptotic condition −
ζ/6 for a = i and b = 0. The plot on
the right side shows in blue the same difference for x > 10 and in red the difference between
the tritronquée solution and the truncated asymptotic series.
lines only a slow algebraic fall off towards infinity.
The above approach thus allows the computation of the tritronquée solution for a line
avoiding the sector | arg ζ| > 4π/5 with high accuracy. The picture one obtains by com-
puting Ω along several such lines is that there are indeed no singularities in the sector
| arg ζ| < 4π/5, and that the square root behavior is followed for large |ζ|. To obtain a
more complete picture, we compute the tritronquée solution for | arg ζ| < 4π/5 − 0.05 and
|ζ| < R ( we choose R = 20). The boundary data for |ζ| = R follow as before from the
truncated asymptotic series, the data for arg ζ = ±4π/5 − 0.05 are obtained by computing
the tritronquée solution on the respective lines as above.
To solve the resulting boundary value problem for the P-I equation is, however, computa-
tionally expensive since we have to solve an equation in 2 real dimensions iteratively. Since
the solution we want to construct is holomorphic there, we can instead solve the harmonicity
condition (the two dimensional Laplace equation) for the given boundary conditions. To this
end we introduce polar coordinates r, φ and use a spectral grid as described in [45]: the main
point is a doubling of the interval r ∈ [0, R] to [−R,R] to allow for a better distribution of
the Chebychev collocation points. Since we work with values of φ < φ0, we cannot use the
usual Fourier series approach for the azimuthal coordinate. Instead we use again a Cheby-
chev collocation method. The found solution in the considered domain is shown in Fig. 5
!20 !15 !10 !5 0 5 10 15 20
Figure 4: Real (blue) and imaginary part (red) of the tritronquée solution close to the critical
line (for a = exp(i(4π/5− .05))) with oscillations of slowly decreasing amplitude.
and Fig. 6. The quality of the solution can be tested by plugging the found solution to the
Laplace equation into the P-I equation. Due to the low resolution and problems at the bound-
ary, the accuracy is considerably lower in the two dimensional case than on the lines. This
is, however, not a problem since we need only the one dimensional solutions for quantitative
comparisons with NLS solutions. The two dimensional solutions give nonetheless strong
numerical evidence for the conjecture that the tritronquée solution has globally no poles in
the sector | arg ζ| < 4π/5.
7 Critical behavior in NLS and the tritronquée solution of
P-I: numerical results
In this section we will compare the numerical solution of the focusing NLS equation for
two examples of initial data for values of � between 0.1 and 0.025 with the asymptotic so-
lutions discussed in the previous sections, the semiclassical solution up to the breakup and
the tritronquée solution to the Painlevé I equation. The numerical approach to solve the NLS
equation is discussed in detail in [29]. For values of � below 0.04 we have to use Krasny
filtering [30] (Fourier coefficients with an absolute value below 10−13 are put equal to zero
to avoid the excitation of unstable modes). With double precision arithmetic we could thus
reach � = 0.025, but could not go below.
Figure 5: Real part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5− 0.05.
7.1 Initial data
We consider initial data where u(x, 0) has a single positive hump, and where v(x, 0) is
monotonously decreasing. For initial data of the form u(x, 0) = A2(x) and v(x, 0) = 0
where the function A(x) is analytic with a single positive hump with maximum value A0,
the semiclassical solution of NLS follows from (2.11) with f(u, v) given by
f(u, v) = 2=
dη ρ(η)
v)2 + u
 (7.1)
where
ρ(η) =
∫ x+(η)
x−(η)
A2(x) + η2
and where x±(η) are defined by A(x±(η)) = iη. The formula (7.1) follows from results by
S.Kamvissis, K.McLaughlin and P.Miller in [26].
From f(u, v) it is straightforward to recover the initial data from the equations
x = fu, fv = 0. (7.2)
Numerically we study the critical behavior of two classes of initial data, one symmetric with
respect to x which were used in [34], and initial data without symmetry with respect to x
Figure 6: Imaginary part of the tritronquée solution in the sector r < 20 and |φ| < 4π/5 −
0.05.
which are built from the initial data studied in [43]. For the former class the corresponding
exact solution of focusing NLS is known in terms of a determinant. Nonetheless we integrate
the NLS equation for these initial data numerically since this approach is not limited to
special cases, but can be used for general smooth Schwartzian initial data as in the latter
case.
7.1.1 Symmetric initial data
We consider the particular class of initial data data
u(x, t = 0) = A20sech
2 x, v(x, t = 0) = −µ tanhx, µ ≥ 0. (7.3)
Introducing the quantity
− A20,
we find that the semiclassical solution for these initial data follows from (2.11) with
f(u, v) =
(v − 2M)∆+ −
(v + 2M)∆− −
u log u
u log
v +M + ∆+)(−
v −M + ∆−)
] (7.4)
where
v ±M)2 + u
For µ = 0 we recover the Satsuma-Yajima [37] initial data that were studied numerically in
[34]. The function f(u, v) takes the form
f(u, v) = <
+ i A0
+ i A0
+ u log
+ i A0)
+ iA0)2 + u√
(7.5)
which can also be recovered from (7.1) by setting ρ = i. The critical point is given by
u0 = 2A
0, v0 = 0, x0 = 0, t0 =
. (7.6)
Furthermore we have
f 0uuu = 0, f
uuv =
, r = 4A30, ψ = 0, (7.7)
where r, ψ are defined in (2.13). For A0 = 1 the initial data (7.3) coincides with the one
studied in [43] by A.Tovbis, S.Venakides and X.Zhou. In the particular case µ = 2, A0 = 1
the function f(u, v) in (7.1) simplifies to
f(u, v) = v −
v2 + u+ u log
−12v +
v2 + u
 . (7.8)
In this case the critical point is given by
v0 = 0, u0 = 2 + µ, t0 =
2 + µ
, x0 = 0.
Furthermore,
f 0uuu = 0, f
uuv =
(µ+ 2)3
, r =
(µ+ 2)3
, ψ = 0.
7.1.2 Non-symmetric initial data
Recall that we are interested here in Cauchy data in the Schwartz class of rapidly decreasing
functions. The above initial data are symmetric with respect to x, u is an even and v an odd
function in x. To obtain a situation which is manifestly not symmetric, we use the fact that if
f is a solution to (2.5), the same holds for derivatives and anti-derivatives of f with respect
to v and for any linear combination of those. If fv is an even function in v, this will obviously
not be the case for a linear combination of f and fv.
As a specific example, we consider the linear combination
f = f1 + αf2, α = const,
where f1 coincides with (7.8) and
f2 = 2u
v2 + u−
v2 + u
+ u v log
−12v +
v2 + u
 . (7.9)
The function f2 is obtained from the integration f2,v = f1 − v. The critical point is given in
this case by
u0 = 4(1− 16α2), v0 = −16α, x0 =
1 + 4α
1− 4α
, t0 =
1 + 4α
1− 4α
thus we have |α| < 1/4. Furthermore,
f 0uuv = −
4α2 − 1/8
1− 16α2
, f 0uuu =
1− 16α2
such that
r = 8u0, ψ = − arctan
1− 16α2
1/8− 4α2
We determine the initial data corresponding to f for a given value of |α| < 1/4 by
solving (7.2) for u, v in dependence of x. This is done numerically by using the algorithm of
[31] which is implemented as the function fminsearch in Matlab. The algorithm provides an
iterative approach which converges in our case rapidly if the starting values are close enough
to the solution, which is achieved by choosing u and v corresponding to f0 as an initial guess.
For α close to 1/4 we observe numerically a steepening of the initial pulse which will lead
to a shock front in the limit α→ 1/4. For the computations presented here, we consider the
case α = 0.1 which leads to the initial data shown in Fig. 7.
The initial data are computed in the way described above to the order of the Krasny filter
on the interval x ∈ [−15, 11] on Chebychev collocation points. Standard interpolation via
Chebychev polynomials is then used to interpolate the resulting data to a Fourier grid. To
avoid a Gibbs phenomenon at the interval ends due to the non-periodicity of the data, we use
a Fourier grid on the interval [−10π, 10π] to ensure that the function u takes values of the
order of the Krasny filter. For x < −15 and x > 11, the function u is exponentially small
which implies the zero-finding algorithm will no longer provide the needed precision. Thus
we determine the exponential tails of the solution to leading order analytically. We find for
x→ −∞
u = v2+ exp
2(x− αv+)
αv+ + 1
v = v+ − v+ exp
2(x− αv+)
αv+ + 1
2 log(v+) +
2(x− αv+)
αv+ + 1
, (7.10)
!4 !3 !2 !1 0 1 2 3 4
!4 !3 !2 !1 0 1 2 3 4
Figure 7: Initial data for the NLS equations without symmetry with respect to x.
and for x→ +∞
u = v2− exp
2(x+ α)
αv− + 1
v = v− − v− exp
2(x+ α)
αv− + 1
2 log(−v−)−
2(x+ α)
αv− + 1
, (7.11)
where v± = (
1± α − 1)/α. The initial data for the NLS equation in the form Ψ =√
u exp(iS/�) are then found by integrating v on the Chebychev grid by standard integration
of Chebychev polynomials. The exponential tails for S follow from (7.10) and (7.11). The
matching of the tails to the Chebychev interpolant is not smooth and leads to a small Gibbs
phenomenon. The Fourier coefficients decrease, however, to the order of the Krasny filter
which is sufficient for our purposes. Thus we obtain the non-symmetric initial data with
roughly the same precision as the analytic symmetric data.
7.2 Semiclassical solution
For times t� tc, the semiclassical solution gives a very accurate asymptotic description for
the NLS solution. The situation is similar to the Hopf and the KdV equation [19]. We find
for the symmetric initial data for t = tc/2 that the L∞ norm of the difference between the
solutions decreases as �2. More precisely a linear regression analysis in the case of symmetric
initial data (for the values � = 0.03, 0.04, . . . , 0.1) for the logarithm of this norm leads to an
error proportional to �a with a = 1.94, a correlation coefficient r = 0.9995 and standard error
σa = 0.03. In the non-symmetric case, we find a = 1.98, r = 0.999996 and σa = 0.003.
Close to the critical time the semiclassical solution only provides a satisfactory descrip-
tion of the NLS solution for large values of |x − xc|. In the breakup region it fails to be
accurate since it develops a cusp at xc whereas the NLS solution stays smooth. This behavior
can be well seen in Fig. 8 for the symmetric initial data. The largest difference between the
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 8: The blue line is the function u of the solution to the focusing NLS equation for
the initial data u(x, 0) = 2 sechx and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
semiclassical and the NLS solution is always at the critical point. We find that the L∞ norm
of the difference scales roughly as �2/5 as suggested by the Main Conjecture. More precisely
we find a scaling proportional to �a with a = 0.38 and r = 0.999997 and σa = 4.2 ∗ 10−4.
For the non-symmetric initial data, we find a = 0.36, r = 0.9999 and σa = 0.002. The
corresponding plot for u can be seen in Fig. 9.
The function v for the same situation as in Fig. 8 is shown in Fig. 10. It can be seen that
the semiclassical solution is again a satisfactory description for |x − xc| large, but fails to
be accurate close to the breakup point. The phase for the non-symmetric initial data can be
seen in Fig. 11. In the following we will always study the scaling for the function u without
further notice.
0.2 0.3 0.4 0.5 0.6 0.7
Figure 9: The blue line is the function u of the solution to the focusing NLS equation for
the non-symmetric initial data and � = 0.04 at the critical time, and the red line is the corre-
sponding semiclassical solution given by formulas (2.4). The green line gives the multiscales
solution via the tritronquée solution of the Painlevé I equation.
7.3 Multiscales solution
It can be seen in Fig. 8 and Fig. 10 that the multiscales solution (5.21) in terms of the
tritronquée solution to the Painlevé I equation gives a much better asymptotic description
to the NLS solution at breakup close to the breakup point than the semiclassical solution
for the symmetric initial data. For larger values of |x − xc|, the semiclassical solution pro-
vides, however, the better approximation. The rescaling of the coordinates in (5.21) sug-
gests to consider the difference between the NLS and the multiscales solution in an interval
[−γ�4/5, γ�4/5] (we choose here γ = 1, but within numerical accuracy the result does not
depend on varying γ around this value). These intervals can be seen in Fig. 12. We find that
the L∞ norm of the difference between these solutions in this interval scales roughly like
�4/5. More precisely we have a scaling �a with a = 0.76 (r = 0.998 and σa = 0.019).
For the non-symmetric initial data, the situation at the critical point can be seen in Fig. 9
and Fig. 11. Again the multiscales solution (5.21) gives a much better description close to
the critical point than the semiclassical solution. However, the approximation is here much
better on the side with weak slope for u than on the side with strong slope. We consider
again the L∞-norm of the difference between the multiscales and the NLS solution in the
interval [−γ�4/5, γ�4/5]. The scaling behavior of the solution can be seen in Fig. 13. For
γ = 1 we find a = 0.71, r = 0.998 and σa = 0.02. These values do not change much for
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
Figure 10: The blue line is the function v of the solution to the focusing NLS equation
for the initial data u0(x) = 2 sechx and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
larger γ. For smaller γ there are not enough points to provide a valid statistics. The value
of a smaller than the predicted 4/5 is seemingly due to the strong asymmetry in the quality
of the approximation of NLS by the multiscales solution as can be seen from Fig. 9. In the
considered interval, the deviation is already so big that the scaling no longer holds as in the
symmetric case. To study the scaling with a reliable statistics would, however, require the
use of a considerably higher resolution which would be computationally too expensive.
Going beyond the critical time, one finds that the real part of the NLS solution continues
to grow before the central hump breaks up into several humps. Notice that the multiscales
solution always leads to a function u that is smaller than the corresponding function of the
NLS solution at breakup and before. This changes for times after the breakup as can be
inferred from Fig. 14 which shows the time dependence of the NLS and the corresponding
multiscales solution for the non-symmetric initial data. The approximation is always best at
the critical time.
To study the quality of the approximation (5.21), we use rescaled times. The scaling of
the coordinates in (5.21) suggests to consider the NLS solution close to breakup at the times
t±(�) with
t±(�) = tc + u0/r −
(u0/r)2 ± �4/5β, (7.12)
0.2 0.3 0.4 0.5 0.6 0.7
Figure 11: The blue line is the function v of the solution to the focusing NLS equation
for the non-symmetric initial data and � = 0.04 at the critical time, and the red line is
the corresponding semiclassical solution given by formulas (2.4). The green line gives the
multiscales solution via the tritronquée solution of the Painlevé I equation.
where β is a constant (we consider β = 0.1). We will only study the symmetric initial data in
this context. Before breakup we obtain the situation shown in Fig. 15. It can be seen that the
multiscales solution always provides a better description close to xc than the semiclassical
solution, and that the quality improves in this respect with decreasing �. We find that the L∞
norm of the difference scales in this case as �a with a = 0.55 (r = 0.994 and σa = 0.03).
The situation for times after breakup can be inferred from Fig. 16. Close to the central
region the multiscales solution shows a clear difference to the NLS solution. But it is inter-
esting to note that the ripples next to the central hump are well approximated by the Painlevé
I solution. The L∞ norm of the difference between the two solutions scales roughly like �.
More precisely we find a scaling �a with a = 1.02 (r = 0.9999 and σa = 7.7 ∗ 10−3).
8 Concluding remarks
In this paper we have started the study of the critical behavior of generic solutions of the fo-
cusing nonlinear Schrödinger equation. We have formulated the conjectural analytic descrip-
tion of this behavior in terms of the tritronquée solution to the Painlevé-I equation restricted
to certain lines in the complex plane. We provided analytical as well as numerical evidence
−0.05 0 0.05
ε=0.03
−0.1 0 0.1
ε=0.1
Figure 12: The blue line is the solution to the focusing NLS equation for the initial data
u0(x) = 2 sechx at the critical time, and the green line gives the multiscales solution via the
tritronquée solution of the Painlevé I equation. The plots are shown for two values of � at the
critical time.
supporting our conjecture. In subsequent publications we plan to further study the Main
Conjecture of the present paper by applying techniques based, first of all, on the Riemann -
Hilbert problem method [26, 43, 44] and the theory of Whitham equations (see [19] for the
numerical implementation of the Whitham procedure in the analysis of oscillatory behavior
of solutions to the KdV equations). The latter will also be applied to the asymptotic descrip-
tion of solutions inside the oscillatory zone. Furthermore we plan to study the possibility
of extending the Main Conjecture to the critical behavior of solutions to the Hamiltonian
perturbations of more general first order quasilinear systems of elliptic type. Last but not
least, it would be of interest to study the distribution of poles of the tritronquée solution in
the sector | arg ζ| > 4π
and to compare these poles with the peaks of solutions to NLS inside
the oscillatory zone. The elliptic asymptotics obtained by Kitaev [28] might be useful for
studying these poles for large |ζ|.
In this paper we did not study the behaviour of solutions to NLS near the boundary u = 0.
Such a study is postponed for a subsequent publication.
0.35 0.4 0.45 0.5
!=0.03
0.2 0.3 0.4 0.5 0.6 0.7
!=0.1
Figure 13: The blue line is the solution to the focusing NLS equation for the non-symmetric
initial data at the critical time, and the green line gives the multiscales solution via the
tritronquée solution of the Painlevé I equation. The plots are shown for two values of �
at the critical time.
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	Introduction
	Dispersionless NLS, its solutions and critical behaviour
	First integrals and solutions of the NLS/Toda equations
	Critical behaviour and elliptic umbilic catastrophe
	The tritronquée solution to the Painlevé-I equation and the Main Conjecture
	Numerical analysis of the tritronquée solution of P-I
	Critical behavior in NLS and the tritronquée solution of P-I: numerical results
	Initial data
	Symmetric initial data
	Non-symmetric initial data
	Semiclassical solution
	Multiscales solution
	Concluding remarks
