VARIANTS OF THE FOCUSING NLS EQUATION. DERIVATION,JUSTIFICATION AND OPEN PROBLEMS RELATED TO
FILAMENTATIONÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL
Abstract. The focusing cubic NLS is a canonical model for the propagationof laser beams. In dimensions 2 and 3, it is known that a large class of initial
data leads to finite time blow-up. Now, physical experiments suggest that thisblow-up does not always occur. This might be explained by the fact that somephysical phenomena neglected by the standard NLS model become relevant
at large intensities of the beam. Many ad hoc variants of the focusing NLS
equation have been proposed to capture such effects. In this paper, we derivesome of these variants from Maxwell’s equations and propose some new ones.
We also provide rigorous error estimates for all the models considered. Finally,
we discuss some open problems related to these modified NLS equations.

1. Introduction
The cubic, focusing, nonlinear Schrödinger equation in space dimension d is given
by
(1)

(

i∂τ v + ∆v + |v|2 v = 0, τ > 0,v(0, x) = v0 (x), x ∈ Rd .

x ∈ Rd ,It is a canonical model for the propagation of laser beams.
From a result of Ginibre and Velo [15], equation (1) is locally well-posed in
H 1 = H 1 (Rd ) for d = 1, 2, 3, and thus, for v0 ∈ H 1 , there exists 0 < T ≤ +∞and a unique solution v(τ ) ∈ C([0, T ), H 1 ) to (1) and either T = +∞, we say
the solution is global, or T < +∞ and then limτ ↑T k∇v(t)kL2 = +∞, we say thesolution blows up in finite time.
The NLS equation (1) also admits the following (formal) conservation laws:
L2 − norm : kv(τ )k2L2 = kv0 k2L2 ;Z
Z
11Energy : E(v(τ )) =|∇v(τ, x)|2 dx −|v(τ, x)|4 dx = E(v0 );
2
4

Z

Z

Momentum : Im∇v(τ, x)v(τ, x)dx= Im

∇v0 (x)v0 (x)dx .

It is also known that a large group of symmetries leaves the equation invariant: ifddv(τ, x) solves (1), then ∀(λ0 , τ0 , x0 , β0 , γ0 ) ∈ R+
∗ × R × R × R × R, so does
(2)

β0

β0

u(τ, x) = λ0 v(λ20 τ + τ0 , λ0 x + x0 − β0 t)ei 2 ·(x− 2 τ ) eiγ0 .

The scaling symmetry u(τ, x) = λ0 v(λ20 τ, λ0 x) leaves the homogeneous Sobolev
space Ḣ sc (Rd ) invariant, where sc = d2 − 1.
D. L. acknowledges support from the ANR-13-BS01-0003-01 DYFICOLTI, the ANR BOND..
1

2

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Referring to conservation of the L2 norm by the flow, (1) is said to be L2 −subcriticalif sc < 0, L2 −critical if sc = 0 and L2 −supercritical if sc > 0. Thus, (1) is
L2 −subcritical if d = 1, L2 −critical if d = 2, and L2 −supercritical if d ≥ 3. In thesubcritical case, global existence (in C([0, ∞), L2 )) holds for arbitrarily large datain L2 . It turns out that in this case, global existence (in C([0, ∞), H 1 )) also holds
for arbitrarily large data in H 1 , due to the conservation of mass and energy. In
the critical and supercritical cases however, there exist stable finite time blow-up
dynamics. This has been known since the 60ies using global obstructive arguments
based on the virial identity (see e.g. [42]).1
There is however a discrepancy between the blow-up results predicted by (1) andphysical observations. Indeed, while the blow-up signifies a break-down of the solution v, physical observations show in many cases that lasers begin to focus according
to the scenarios associated to (1) but depart from this behavior slightly before thefocusing time. The reason advanced by physicists is that some physical phenomena that have been neglected to derive (1) become relevant at high intensities, andtherefore near focusing. This phenomenon is called filamentation: defocusing physical phenomena are triggered at high intensities and halt the collapse of the beam.
This interplay between diffraction, self-focusing, and defocusing mechanisms allow
for the beam to propagate along several times the focusing distance (called Rayleighlength in optics) and the resulting structure is called filament .
Many variants of (1) have been derived in optics to take into account these
additional physical phenomena and reproduce the filamentation mechanism. In
many cases, it is a mathematical open problem to prove whether these additionalterms prevent focusing or not, and a fortiori to understand the modification of the
dynamics induced by them.
Rather than adding as usual ad hoc modifications to (1) in order to take newphysical effects into account, we choose here to rigorously derive such modifications
from Maxwell’s equations. We then comment on some of the most physically relevant open mathematical problems that these modified equations raise and that are
natural milestones towards the understanding of filamentation. These variants canroughly be classified into two groups, depending on whether they take ionization
processes into account or not.
Notation 1.1. In the brief presentation below, we denote by z the direction of
propagation of the laser, by X⊥ = (x, y) the transverse variables, and by ∆⊥ =∂x2 + ∂y2 the transverse Laplace operator. In dimension d = 2, the variable y isomitted (and hence ∆⊥ = ∂x2 ), while in dimension d = 1, functions depend only onz (so that ∆⊥ = 0).
Models without ionization processes
We give below a family of variants to (1) that incorporate many physical phenomena
neglected by (1). It is of course possible to look at one or several of these additional
effects simultaneously. We state the equations in their most general form, startingwith a family of scalar NLS equations, and then give the corresponding vectorial –and more general – form of these equations. Let us therefore consider(3) iP2 (ε∇)∂τ v + (∆⊥ + α1 ∂z2 )v + iα2 v + (1 + iεα3 · ∇) 1 + f (εr |v|2 ) |v|2 v = 0,

1Much more is known about the finite time blow up dynamics for the focusing NLS and werefer the interested reader to [38] [26] [27] [28] [29] [30] [31] [32] and references therein.

VARIANT OF THE FOCUSING NLS EQUATION

3

where v is a complex-valued function. Here, ε > 0 is a (small) parameter; P2 (ε∇)
is a second (at most) order, self-adjoint, positive operator; α1 = 0, ±1; α2 ≥ 0;
α3 ∈ Rd ; f : R+ → R is a smooth mapping vanishing at the origin, and r > 0. The
physical meaning of these terms is commented below:(1) Nonlinearity. The cubic nonlinearity in (NLS) corresponds to a first orderapproximation of the nonlinear optical phenomena. At high intensities,
it is often worth including some next order terms captured here by theadditional term f (εr |v|2 ). We consider here three situations:(a) Cubic nonlinearity: f = 0.
(b) Cubic/quintic nonlinearity: f (r) = −r.
(c) Saturated nonlinearity: f is a smooth function on R+ vanishing at therorigin and such that (1 + f (r))r is bounded on R+ (e.g. f (r) = − 1+r).
(2) Group velocity dispersion (GVD). The coefficient α1 accounts for the dispersion of the group velocity and three different situations are possible:(a) No GVD: α1 = 0.
(b) Anomalous GVD: α1 = 1.
(c) Normal GVD: α1 = −1.
(3) Damping. The coefficient α2 accounts for damping phenomena:(a) No damping: α2 = 0.
(b) Damping: α2 > 0.
(4) Off-axis variations of the group velocity. The operator P2 (ε∇) is here toaccount for the fact that self-focusing pulses become asymmetric due tothe variation of the group velocity of off-axis rays2. The operator P2 (ε∇)
is a second order, self-adjoint, and positive operator in the sense that
(P2 (ε∇)u, u) ≥ C|u|2∗ , with |u|2∗ ≥ |u|22 . The norm | · |∗ may also control
derivatives of u; we consider three cases:(a) No off-axis dependence: P2 (ε∇) = 1, and therefore | · |∗ = | · |2 .
(b) Full off-axis dependence: the norm | · |∗ controls all first order derivatives, |u|2∗ ∼ |u|22 + ε2 |∇u|22 .
(c) Partial off-axis dependence: the norm | · |∗ controls some but not all
first order derivatives. More precisely, there exists j (j < d) linearlyPj
independent vectors vj ∈ Rd such that |u|2∗ ∼ |u|22 +ε2 k=1 |vk ·∇u|22 .
(5) Self-steepening of the pulse. The operator (1 + iεα3 · ∇) in front of thenonlinearity accounts for off-axis dependence of the nonlinearity, responsible for the possible formation of optical shocks. Various cases are considered
here:(a) No self-steepening. This corresponds to α3 = 0 and to the usual
situation where the nonlinearity does not contain any derivative.
(b) Longitudinal self-steepening. When α3 is colinear to ez , there is a
derivative in the nonlinearity along the direction z of propagation of
the laser.
(c) Transverse self-steepening. When α3 6= 0 and α3 · ez = 0, there isa derivative in the nonlinearity along a direction orthogonal to thedirection of propagation.
2This phenomenon is often referred to in optics as space-time focusing [35]; we do not use
this terminology here because this would be misleading. Indeed, physicists usually take z as theevolution variable and treat t as a space variable. This amounts to permuting t and z in (3) andelsewhere.

4

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL(d) Oblique self-steepening. When α3 is neither colinear nor orthogonalto ez .
Remark 1.1. The standard Schrödinger equation (1) is obtained with P2 (ε∇) = 1,
α2 = 0, α3 = 0, f = 0, and α1 = 1. Using the above terminology, it corresponds toa cubic nonlinearity, without damping terms, off-axis variation of the group velocity
and self-steepening, and with anomalous GVD.
As previously said, (3) stems from a more general vectorial equation. For thesake of simplicity, we give here the equation corresponding to the cubic case (orf = 0 in (3)):
1(3)vect
iP2 (ε∇)∂τ v+(∆⊥ +α1 ∂z2 )v+iα2 v+ (1+iεα3 ·∇) (v·v)v+2|v|2 v = 0,3
where v is now a C2 -valued function.
Remark 1.2. Equation (3) is in fact a particular case of (3)vect corresponding toinitial data living on a one-dimensional subspace of C2 . Indeed, if the initial condition to (3)vect has the form v|τ =0 = v 0 (x)v0 with v0 ∈ R2 and v 0 a scalar-valued
function, then the solution to (3)vect takes the form v(τ, x) = v(τ, x)v0 , where v
solves (3) with initial condition v 0 .
Models with ionization processes
In addition to the physical phenomena taken into account in (3), it is necessary athigh intensities to include ionization processes for a correct description of the laser
pulse. The reason why this phenomenon is singled out here is because a system of
two equations must be considered instead of the single equation (3). In the most
simple case (i.e. P2 = 1, f = 0, α2 = 0, α3 = 0; for a more general model, see(50)), this system is given by(i(∂t + cg ∂z )u + ε(∆⊥ +α1 ∂z2 )u + ε(|u|2 −ρ)u = −iεc(α4 |u|2K−2u + α5 ρu),(4)∂t ρ = εα4 |u|2K + εα5 ρ|u|2 ,with α4 , α5 ≥ 0, c > 0, and where ρ is the density of electrons created by ionization,
while cg = cg ez is the group velocity associated to the laser pulse.
The system (4) does not directly compare to (3) and (3)vect ; indeed, (4) is written
in the fixed frame of the laboratory, while (3) and (3)vect are written in a frame
moving at the group velocity cg = cg ez and with respect to a rescaled time τ = εt.
Rather than (4), the NLS equation with ionization used in the physics community
is its version written in the same variables as (3) and (3)vect . More precisely, if wesetu(t, X⊥ , z) = v(εt, X⊥ , z − cg t),ρ(t, X⊥ , z) = ρ̃(εt, X⊥ , z − cg t),

and τ = εt, the equations (4) are approximated 3 by the following ones,(i∂τ v + (∆⊥ + α1 ∂z2 )v + (|v|2 − ρ̃)v = −ic(α4 |v|2K−2 v + α5 ρ̃v),(5)−cg ∂z ρ̃ = εα4 |v|2K + εα5 ρ̃|v|2 .

The ionization processes taken into account by the systems (4) and (5) are:
3The approximation lies in the equation on ρ. In the new variables, the second equation of (4)is given byε∂τ ρ̃ − cg ∂z ρ̃ = εα4 |v|2K + εα5 ρ̃|v|2 .
In the physics literature, the term ε∂τ ρ̃ is neglected, and this corresponds to (5).

VARIANT OF THE FOCUSING NLS EQUATION

5(1) Photo-ionization. This corresponds to α4 > 0 and K > 0 (K is the numberof photons necessary to liberate one electron).
(2) Collisional ionization. When α5 > 0, a term corresponding to collisional
ionization is added to the evolution equation on ρ.
Remark 1.3.
(1) The coupling with ρ can of course be added to any equation of the family (3).
(2) A vectorial variant of (4) and (5) can also be derived in the same lines as (3)vect .
(2) When α4 = α5 = 0 and ρ|t=0 = 0 (respectively limz→−∞ ρ̃ = 0) one recovers (3)from (4) (respectively (5)).
The rest of the paper is as follows. In section 2, we recall the Maxwell equations,
we give an abstract formulation and we discuss the spaces of initial data used for
the Cauchy problem. In section 3, we prove our main result about the rigorous
derivation of general abstract versions of the models (3), (3)vect , and (4), (5). In
section 4, we analyze the role of the various parameters in (3) and (4) or (5) In
particular, we consider whether they indeed prevent the breakdown in finite timeor not. We also formulate a number of interesting open problems for these modified
NLS equations. Finally, an appendix contains explicit computations for a physically
relevant system of Maxwell equations which show that the abstract models derived
in Section 3 take indeed the form of (3), (3)vect , and (4).
1.1. Notations. We denote by- Fu(ξ) = ub(ξ), the Fourier transform of u with respect to the space variables
x ∈ Rd .
- Ft u(ω), the Fourier transform of u with respect to the time variable t.
- f (D), the Fourier multiplier with symbol f (ξ): f\(D)u(ξ) = f (ξ)bu(ξ).
- f (Dt ), Fourier multipliers with respect to time.
- Λ = (1 − ∆)1/2 , the Fourier multiplier with symbol (1 + |ξ|)1/2 .
2. The Maxwell equations and an abstract mathematicalformulation
2.1. The Maxwell equations. The Maxwell equations in a non magnetizable
medium are a set of two equations coupling the evolution of the magnetic field B
to the electric induction D, ∂t B + curl E = 0,1
(6)curl B = −J, ∂t D −µ0
where D is given in terms of the electric field E and a polarization P — modelingthe way the dipole moment per unit volume depends on the strength of the electricfield — by the relation(7)

D = ǫ0 E + P,

and where we used standard notations ǫ0 and µ0 for the electric permittivity andmagnetic permeability in vacuum. The evolution equations (6) go along with twoconstitutive laws,(8)∇ · D = ρ,∇ · B = 0,

6ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

where ρ is the electric charge density.
As a consequence of the relation ∇ · D = ρ and the second equation of (6) we get
the continuity equation coupling ρ to the current density J,(9)∂t ρ + ∇ · J = 0.

Introducing the speed of light in vacuum

1
,
c= √
ǫ0 µ0
the equations (6) can also be rewritten as a set of two evolution equations on themagnetic field B and the electric field E, ∂t B + curl E = 0,1
1
(10)
 ∂t E − c2 curl B = − ∂t P − J.
ǫ0ǫ0In order to get a closed system of equations, we still need two physical informations:(1) A description of the polarization response to the electric field E.
(2) A description of the current density J.
We first address the description of the polarization response in absence of current
density and then proceed to describe the modification to be made when current
density is included.
2.1.1. The polarization response to the electric field. Throughout this section, we
assume that there is no charge nor current density (ρ = 0, J = 0). The general casewill be handled in §2.1.2 below.
There exist various ways to describe the polarization P; we use here a simple
and natural model called “nonlinear anharmonic oscillator”, according to which
the polarization is found by solving the second order ODE(11)∂t2 P + Ω1 ∂t P + Ω20 P − ∇VN L (P) = ǫ0 bE,

where b ∈ R is a coupling constant and Ω0 , Ω1 > 0 are frequencies, and where VN L
accounts for nonlinear effects. When such effects are neglected, the description (11)goes back to Lorentz [23] and expresses the fact that electrons are bound to the
nucleus by a spring force. Nonlinearities have been added to this description by
Bloembergen [5] and Owyyoung [34] and the mathematical investigation of theirinfluence was initiated by Donnat, Joly, Métivier and Rauch [11, 16] (see also [20]).
Remark 2.1. In physics books, the polarization P is often sought as an expansion
P = ǫ0 χ1 [E] + χ2 [E, E] + χ3 [E, E, E] + . . . ,where the operator χ1 is called the linear susceptibility of the material, while forj > 1, the operators χj are the j-th order nonlinear susceptibilities. It is easy tocheck that the linear susceptibility corresponding to (11) is given by the nonlocal(in time) operator
χ1 [E] = χ1 (Dt )Ewith

χ1 (ω) =where we used the Fourier multiplier notation,b,Ω20 − ω 2 + iΩ1 ωFt [χ1 (Dt )E](ω) = χ1 (ω)Ft E(ω).
Example 2.1. Typical examples for VN L (P) are

VARIANT OF THE FOCUSING NLS EQUATION

7(i) Cubic nonlinearity:
a3
VN L (P) = |P|4 and therefore ∇VN L (P) = a3 |P|2 P
4(ii) Cubic/quintic nonlinearity:a3
a5
VN L (P) = |P|4 − |P|6 and therefore ∇VN L (P) = a3 |P|2 P − a5 |P|4 P.
46′(iii) Saturated nonlinearity: there exists a function vsat : R+ → R, with vsat+′′
and vsat bounded on R and such that1′(|P|2 )P;VN L (P) = vsat (|P|2 ) and therefore ∇VN L (P) = vsat
2
for instance, one can takevsat (r) =

a3
r2
,
5
2 1 + 2a3a r
3

2in which case ∇VN L (P) = a3 |P| P − a5 |P|4 P + h.o.t, and is therefore thesame at the origin as in (ii) above, up to higher order terms (seventh orderterms here).
We show in Appendix A.1 that Maxwell’s equations can be put under the following dimensionless form4 for all the nonlinearities considered in Example 2.1,
∂t B + curl E = 0,





1√ ♯


γQ = 0, ∂t E − curl B +

ε

1 √
γ
(12)

∂t Q♯ + ε1+p ω1 Q♯ − ( γE − ω0 P♯ ) = ε 3 1 + f (εr |P♯ |2 ) |P♯ |2 P♯ ,


ε
ω0




1

♯
♯
 ∂t P − ω0 Q = 0,εwhere γ, ω0 , ω1 , r and p are constants, 0 < ε ≪ 1 is a small parameter (the ratio ofthe duration of an optical cycle over the duration of the pulse, see Appendix A.1),
while f is a smooth function vanishing at the origin.
2.1.2. The case with charge and current density. The main mechanism at stake inlaser filamentation is certainly the local ionization of the medium: once a powerful self-focusing laser beam reaches high enough intensities, it ionizes the medium
around itself. It leaves behind a narrow channel of plasma, hereby causing local
defocusing that prevents blowup.
Taking current density into account, we come back to the set of equations (10)-(11),and a physical description of the current density J is needed. This current density
has the form
(13)

J = Je + Ji ,where Je and Ji are respectively the free electron and ionization current densities.
- Free electron current density. Partial ionization of the material medium by thelaser generates free electrons, with charge qe (= −1.6 × 10−19 C). This induces afree electron current density Je = qe ρe ve , where ρe is the electron density, and ve is
4The constitutive laws (8) are omitted because they are propagated by the equations if they
are initially satisfied.

8

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

the electron velocity. A rough5, but standard model in nonlinear optics is to take(see [4] and references therein),(14)

E(t, X) ∼ E01 (t, X)ei(kl ·X−ωl t) + c.c.,where kl and ωl are the laser wave number and pulsation respectively, with |∂t E01 | ≪|ωl E01 | (slowly varying envelope approximation); the polarization current is thentaken under the form
(15)

Je ∼ J01 (t, X)ei(kl ·X−ωl t) + c.c.

with

J01 = iqe2
ρe E01 ,ω l mewhere me is the electron mass. The drawback of this model is that it assumes thatthe electric field and the current density field can be written at leading order aswave packets (i.e. are given under the form (14)-(15)). In particular, it does notprovide any relation between the current density J and the electric field E that could
be used in Maxwell’s equations (10). We therefore propose here such a relation,
namely,q2kl
Je = e H·
D(ρe E),
ω l mekl2where kl = |kl |, and H is the regularization of the Hilbert transform given by the
Fourier multiplier
√
2 iDz
.
(16)
H(Dz ) =(1 + Dz2 )1/2
Quite obviously, this is consistent with the usual model (15) since this latter isrecovered at leading order when the electric field is a wave packet under the form
(14).
Finally, the evolution of the electron density ρe is given by a source term S
representing external plasma sources. Taking into account photo-ionization andcollisional ionization, but neglecting electron recombination (see for instance [3] forricher models), we haveσ
S = W (I)(ρnt − ρe ) + ρe I,
Ui
where the intensity is I = |E|2 and ρnt is the constant density of neutral species.
In the regime considered here6, ρe is negligible compared to ρnt and the photoionization rate W (I) takes the form
W (I) = σK I 2K ,for some constant coefficient σK > 0 and with K > 1 the number of photons
needed to liberate one electron. The collisional ionization cross-section σ depends
5This approximation can be deduced formally by assuming that ions are at rest and thatelectron motion is described by the compressible Euler system (see for instance [3]). Neglecting
q2

electron collisions, such a model yields ∂t Je = me ρe E which formally yields (15) assuming that Ee
is as in (14) and that ρe is not oscillating at leading order. It would of course be interesting toprovide a rigorous justification to these approximations.
6For higher intensities, electrons can tunnel out the Coulomb barrier of atoms, and W (I) is
modified.

VARIANT OF THE FOCUSING NLS EQUATION

9on the laser frequency, and Ui is the ionization potential. Summing up, we get thefollowing expression for the free electron current Je and ρ = ρe ,



2

 Je = qe H kl · D (ρE),
ω l me
kl2
(17)
σ

2K
 ∂t ρ = σK ρnt |E| + ρ|E|2 .
Ui- Ionization current density. It is also necessary to take into account losses due tophoto-ionization. We therefore introduce a ionization current density Ji such that
Ji · E represents the energy lost by the laser to extract electrons (per time andvolume unit). More precisely, Ji · E is equal to the energy necessary to extract one
electron (given by the ionization potential Ui ) multiplied by the number of electrons
per time and volume unit (given by ∂t ρ). Using the second equation of (17), thisgives
Ji · E = Ui σK ρnt |E|2K + σρ|E|2 .
We therefore take(18)

Ji = Ui σK ρnt |E|2K−2 + σρ)E.

We show in Appendix A.2 that after nondimensionalization, the set of equations(10)-(13)-(11)-(17)-(18) (for the nonlinearities considered in Example 2.1) becomes,∂t B + curl E = 0,






1√ ♯
k


γQ = −εH ε 2 · Dx (ρE) − εc0 c1 |E|2K−2 + c2 ρ E,
 ∂t E − curl B +


ε
k



1 √
γ
♯
1+p
♯
∂t Q + ε ω1 Q − ( γE − ω0 P♯ ) = ε 3 1 + f (εr |P♯ |2 ) |P♯ |2 P♯ ,
(19)
ε
ω0




1

 ∂t P♯ − ω0 Q♯ = 0,



ε


∂t ρ = εc1 |E|2K + εc2 ρ|E|2 ,with the same notations as in (12) for the constants γ, ω0 , ω1 , r and p, the small
parameter ε, and the function f . In addition, we have here constants c0 , c1 , c2 ≥ 0,
and we also recall that the definition of the regularized Hilbert transform H is givenin (16).
2.2. Abstract formulations.

2.2.1. The case without charge nor current density. We show in Appendix A.1 that
the Maxwell equations can be put under the dimensionless form (12), which itself
has the form
1
(20)
∂t U + A(∂)U + EU + ε1+p A0 U = εF (ε, U),εnwhere U is a R (n ≥ 1) valued function depending on the time variable t and thespace variable x ∈ Rd (d ≥ 1),
U:(t, x) ∈ R × Rd → Rn .

The operator A(∂) is defined asA(∂) =dX
j=1

Aj ∂j ,

10

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

where ∂j is the differentiation operator with respect to the j-th space coordinate.
The matrix A0 has size n × n, p is a positive number, and the following assumption
is made on the matrices Aj and E, and on the nonlinearity F .
Assumption 2.1.
(i) The matrices Aj (j = 1, . . . , d) are constant coefficient n × n, real valued, symmetric matrices.
(ii) The matrix E is a constant coefficient n×n, real valued, skew symmetric matrix.
(iii) There exists a smooth mapping f : R+ → R vanishing at the origin, a real
number r > 0, a quadratic form Q : Cn 7→ R+ and a trilinear symmetric mapping
T : (Cn )3 → Cn (with T (Rn × Rn × Rn ) ⊂ Rn ) such that
∀U ∈ Cn ,F (ε, U ) = 1 + f εr Q(U )) T (U, Ū , U ).
Remark 2.2. There exist of course situations where the leading order of the nonlinearity is not cubic (it can be quadratic for non centro-symmetric crystals for
instance) or not of this form; since we are interested here in deriving variants ofthe standard cubic nonlinear Schrödinger equation, we restrict ourselves to thisframework for the sake of simplicity.
Example 2.2. As previously said, the dimensionless version (12) of the Maxwell
equations can be put under the form (20) and they satisfy Assumption 2.1 withn = 12, U = (B, E, Q♯ , P♯ )T . See Appendix B for more details.
2.2.2. The case with charge and current density. As shown in Appendix A.2, thesystem (19) of Maxwell’s equations with partial ionization can be put under thegeneral form

(21)


1

∂t U + A(∂)U + EU + ε1+p A0 U =


ε



k
εF (ε, U) − εH ε 2 · D (WC1T C1 U) − εc C1T G(C1 U, W),

k



∂t W = εG(C1 U, W) · C1 U,

where, as in § 2.2.1, U is a Rn -valued function, whereas W is an R-valued functionof (t, x) ∈ R × Rd . The matrices Aj and E, as well as the nonlinearity F , satisfy
Assumption 2.1. Concerning the other coefficients of the system, we assume thefollowing.
Assumption 2.2.
(i) The real, constant matrix C1 has size m × n (with m ∈ N).
(ii) The constant c is positive.
(iii) There exists two real, positive constants c1 and c2 and an integer K ≥ 1 suchthat
∀E ∈ Cm , ∀w ∈ C,

G(E, w) = c1 |E|2K−2 E + c2 wE.

Remark 2.3. As in Remark 2.2 for the case without ionization, we can put Maxwell’sequation with ionization terms (19) under the abstract form (21). Using the same
notations as in Remark 2.2, the matrix C1 is the projection matrix such that C1 U =E, and w = ρ.

VARIANT OF THE FOCUSING NLS EQUATION

11

2.3. The Cauchy problem. We are considering initial conditions that correspondto laser pulses. In the case without charge nor current density (equation (20)), they
are fast oscillating wave packets slowly modulated by an envelope,k·x

U|t=0 = u0 (x)ei ε + c.c.,(22)

where k ∈ Rd is the (spatial) wave-number of the oscillations. Taking charge andcurrent density into account (equation (21)), we need to provide initial conditions
for W; since we are interested here in the situation where this quantity is createdby the laser when it reaches high intensities near self-focusing, we take these initialconditions to be initially zero7 for the sake of clarity.
W|t=0 = 0.

(23)

The evolution equation (20) (as well as (21)) being of semilinear nature, it isnatural to work with Banach algebra in view of a resolution by Picard iterations.
Throughout this article, we assume that u0 ∈ B, with
B = H t0 (Rd )n ,(t0 > d/2)

or
B = W (Rd )n := {f ∈ S ′ (Rd )n , |f |B := |fb|L1 < ∞}(the so called Wiener algebra, which is better adapted than H t0 (Rd )n to handle
short pulses, see [9, 21]). In both cases, B is stable by translations in Fourier space(this ensures that if u0 ∈ B in (22) then U|t=0 ∈ B) and is a Banach algebra in thesense that∀f, g ∈ B,
f · g ∈ B and |f · g|B . |f |B |g|B .
For all k ∈ N, we also define

B (k) = {f ∈ B,

∀α ∈ Nd ,∀|α| ≤ k,∂ α f ∈ B},

endowed with its canonical norm.
We are interested in deriving asymptotics to the solution formed by (20)-(22),with initial envelope u0 ∈ B, and more generally (21)-(22)-(23), if we want tobe able to handle ionization processes in nonlinear optics. This requires a further
assumption on the nonlinearity F , namely that F acts on B and is locally Lipschitz.
Assumption 2.3. In addition to (iii) of Assumption 2.1, the mapping F satisfies,
uniformly with respect to ε ∈ [0, 1),(i) For all f ∈ B, one has F (f ) ∈ B and|F (ε, f )|B ≤ C(|f |B )|f |B(ii) For all f, g ∈ B, one has|df F (ε, ·)g|B ≤ C(|f |B )|g|B .
Example 2.3. When B = H t0 (Rd )n (t0 > d/2), Assumption 2.3 is always satisfied(by Moser’s inequality); when B = W (Rd )n , the assumption holds for analytic
nonlinearities.
7One could more generally and without supplementary difficulty consider non-oscillating initialconditions for the charge density W

12ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

3. Derivation of NLS-type equations
The Schrödinger approximation takes into account the diffractive effects thatmodify over large times the propagation along rays of standard geometrical optics.
These diffractive effects are of linear nature and are known [11, 17, 16, 36, 20] toappear for time scales of order O(1/ε) for the initial value problem formed by thelinear part of (20) and (22). This is the reason why we are interested in proving
the existence and describing the solutions to the (nonlinear) initial value problem(20)-(22) over such time scales.
For the sake of simplicity, the initial value problem (20)-(22) (no ionization) is
first considered. Up to minor modifications, the results of §§3.1, 3.2 and 3.3 are
known [9]; we reproduce them here because they are necessary steps to derive thefamily of NLS equations (3) and, for the sake of clarity, their proof is sketched ina few words. The general idea to derive the Schrödinger equations of §3.5 withimproved dispersion was introduced in [9] but the computations are carried further
here. We then derive in §3.6 a new class of models with derivative nonlinearity forwhich a local well-posedness result is proved. When applied to Maxwell’s equations,these derivative nonlinearities yield the so-called self-steepening operators; to ourknowledge, this is the first rigorous explanation of these terms.
The asymptotic description of (21)-(22) (i.e. ionization is now included) is thenaddressed in §3.7.
3.1. The profile equation. We show here that under reasonable assumptions on
F , solutions to the initial value problem (20)-(22) exist for times of order O(1/ε)
and that there can be written under a very convenient form using a profile U ,k · x − ωt,(24)
U(t, x) = U t, x,εwith U (t, x, θ) periodic with respect to θ and for any ω ∈ R, provided that U solves
the profile equation ∂ U + A(∂)U + i L(ωD , kD )U + ε1+p A U = εF (ε, U ),
θ
θ
0
t
ε
(25)

U|t=0 (x, θ) = u0 (x)eiθ + c.c..

Here, we used the notation(26)

L(ωDθ , kDθ ) = −ωDθ + A(k)Dθ +with Dθ = −i∂θ and A(k) =

Pd

E
,
i

j=1 Aj kj .

Theorem 3.1. Let B = H t0 (Rd )n or B = W (Rd )n and u0 ∈ B. Under Assumptions 2.1 and 2.3, there exists T > 0 such that for all 0 < ε ≤ 1 there exists a
unique solution U ∈ C([0, T /ε]; B) to (20)-(22). Moreover, one can write U under
the form
k · x − ωt,U(t, x) = U t, x,εwhere U solves the profile equation (25).

Proof. The proof is a slight adaptation of the one given in [9] in the trilinear case;consequently, we just give the main steps of the proof. Quite obviously, a solution

VARIANT OF THE FOCUSING NLS EQUATION

13

U to (20)-(22) is given by (24) if (25) admits a solution U ∈ C([0, T /ε]; H k (T; B))(k ≥ 1) where)(X
fn einθ , |f |H k (T,B) < ∞(27)
H k (T; B) = f =n∈ZPand with |f |2H k (T,B)

2 k
2
k
=
n∈Z (1 + n ) |fn |B . For k ≥ 1, H (T, B) is a Banach
algebra; moreover the evolution operator S(t) associated to the linear part of (25),iS(t) = exp −tA(∂) − tL(ωDθ , kDθ )εis unitary on H k (T; B) (thanks to point (i) and (ii) of Assumption 2.1). One cantherefore construct a (unique) solution to (25) by a standard iterative schemeZ tS(t − t′ ) F (ε, U l )(t′ ) − εp A0 U l dt′ ,U l+1 (t) = S(t)U 0 + ε
00with U = Uinit . Indeed, one has thanks to Assumption 2.3,
Z t

 p
ε |A0 U l |H k (T;B) +C(|U l |H k (T;B) )|U l |H k (T;B) ;|U l+1 (t)|H k (T;B) ≤ |U 0 |H k (T;B) +ε0

thanks to the ε in front of the integral. An estimate of the same kind is valid for adifference of iterates, by point (ii) of Assumption 2.3. By a fixed point argument,
this ensures that the sequence converges to a solution on [0, T /ε] for some T > 0
independent of ε. Uniqueness then follows classically from an energy estimate on
the difference of two solutions.

3.2. The slowly varying envelope approximation. The slowly varying envelope approximation (SVEA) consists in writing the profile U under the form
U (t, x, θ) ∼ uenv (t, x)eiθ + c.c.;(28)

plugging this approximation into the profile equation (25) and keeping only the
first harmonic in the Fourier expansion yields easily (writing u = uenv ),i∂t u + A(∂)u + L(ω, k)u + ε1+p A0 u = εF env (ε, u),
ε
where
(29)

F

env

1
(ε, u) =2πZ 2πe−iθ F (ε, ueiθ + c.c.) dθ.

0

Example 3.1. With F (u) = |u|2 u, one gets F env (u) = (u · u)u + 2|u|2 u.
Denoting D = −i∇, we observe thatiA(∂) + L(ω, k)εiA(∂) + (−ωId + A(k))ε
i=(−ωId + A(k + εD))εi
L(ω, k + εD),:=ε
where the last notation is of course consistent with (26).
=14

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTELAs a consequence of these computations, we see that in order for (28) to hold,it is necessary that u = uenv satisfies the envelope equation ∂ u + i L(ω, k + εD)u + ε1+p A u = εF env (ε, u),
t
0
(30)
ε

0
u|t=0 = u .

As implicitly assumed by omitting the fast oscillating scale in the argument of
the envelope function uenv (t, x), the envelope must not contain any fast oscillation.
However,• The singular part of the linear term in (30) creates fast oscillations withfrequencies ω − ωj (k), where the ωj (k) stand for the eigenvalues of L(0, k).
• The nonlinearity creates other oscillations that may resonate with the linear
propagator.
There is one way to avoid the catastrophic effects of these two scenarios. Choosing ω = ωj (k) for some j and assuming that, up to O(ε) terms, the initial envelope
u0 is contained in the corresponding eigenspace prevents the creation of oscillations
by the linear propagator. This is the polarization condition. The nonlinearity will
however create harmonics of the main oscillation k · x − ωj (k)t and it is necessary
to make a non resonance assumption. What is called characteristic variety in theassumption below is the set CL ⊂ Rd+1 defined as
CL = {(ω ′ , k′ ) ∈ R1+d ,det L(ω ′ , k′ ) = 0}.

Let us also recall that we assumed that the nonlinearity is under the formF (ε, U ) = 1 + f εr Q(U )) T (U, Ū , U ),with f (0) = 0, Q a quadratic form and T a trilinear symmetric mapping. If U is amonochromatic oscillation, the nonlinearity εF (ε, U ) creates third harmonic withsize O(ε), a fifth harmonic (if f ′ (0) 6= 0) with size O(ε1+r ), etc. The non-resonancecondition stated below holds for the (2p + 3)-th harmonics, for all p ≥ 0 such thatpr < 1 (the contribution of higher harmonics is small enough to be controlled even
if it is resonant).
Assumption 3.1. The characteristic variety CL and the frequency/wave number
couple (ω, k) satisfy:(1) There exist m functions ωj ∈ C ∞ (Rd \{0}) (j = 1, . . . , m) such thatCL \{0} =

m
[


j=1

(ωj (k′ ), k′ ), k′ ∈ Rd \{0} ;up to a renumbering, we assume that (ω, k) = (ω1 (k), k).
(2) There exists a constant c0 > 0 such that
inf |ω − ωj (k′ )| ≥ c0 ,k′ ∈Rdj = 2, . . . m.

(3) (Non resonance assumption) One has ±(2p + 3)(ω, k) ∈/ CL , for all p ≥ 0such that pr < 1.
Notation 3.1. We denote by πj (k) (j = 1, . . . , m) the eigenprojectors of the eigenvalues ωj (k) of A(k) + 1i E; in particular, we havemX1
ωj (k)πj (k).
L(0, k) = A(k) + E =i
j=1VARIANT OF THE FOCUSING NLS EQUATION

15

Example 3.2. For Maxwell’s equations, it is shown in Appendix B that Assumption
3.1 is satisfied with m = 7 in dimension d = 3, for ω 6= 0. Explicit expressions forthe eigenprojectors πj (k) are also provided in Appendix B.
Theorem 3.2. Let B = H t0 (Rd )n or B = W (Rd )n and u0 ∈ B (1) , r ∈ B. Let
Assumptions 2.1, 2.3 and 3.1 be satisfied and assume moreover that u0 = π1 (k)u0 +εr. Then(i) There exist T > 0 and, for all ε ∈ (0, 1], a unique solution u ∈ C([0, T /ε]; B (1) )to (30) with initial condition u0 .
(ii) There exists ε0 > 0 such that for all 0 < ε < ε0 , the solution U to (20) provided
by Theorem 3.1 exists on [0, T /ε] and|U − USV EA |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |u0 |B )(1 + |∇u0 |B + |r|B ),where USV EA (t, x) = u(t, x)eik·x−ωtε+ c.c..

Proof. Here again, the proof is a small variation of the one given in [9] for thetrilinear case and B = W (Rd )n . We just indicate the main steps of the proof:
Step 1. Existence and bounds of the solution u to (30) is established by a fixed
point argument as in Theorem 3.1.
Step 2. We decompose u as
u = u1 + uII ,

with

uII =

m
X

uj ,

j=2

and where uj = πj (k + εD)u (see Notation 3.1).
Step 3. Thanks to the assumption that ω = ω1 (k) one gets from the equation
obtained by applying π1 (k + εD) to (30) that |∂t u1 (t)|B is uniformly bounded on[0, T /ε].
Step 4. Using a non-stationary phase argument (on the semigroup formulation)
relying on point (ii) of Assumption 3.1 and the bound on ∂t u1 established in Step 3,
and taking advantage of Assumption 2.3, we get that 1ε |uII (t)|B remains uniformly
bounded on [0, T /ε].
Step 5. Using the non-resonance condition (iii) of Assumption 3.1, one can showthat the third and higher harmonics created by the nonlinearity remain of orderO(ε). More precisely, the solution U ∈ H 1 (T; B) to (25) provided by Theorem 3.1
can be written as
U (t, x, θ) = Uapp (t, x, θ) + εV (t, x, θ),
where Uapp (t, x, θ) = u(t, x)eiθ + c.c., and V remains bounded (with respect to ε)in C([0, T /ε]; H 1 (T; B)n ).
Step 6. Since U (t) − Uapp (t) = εV (t), it follows from the above thatsupt∈[0,T /ε]|U (t) − Uapp (t)|H 1 (T;B) ≤ εC(T, |u0 |B )(1 + |∇u0 |B + |r|B ),and the theorem follows therefore from the observation that|U − USV EA |L∞ ([0,T /ε]×Rd ) ≤supt∈[0,T /ε]|U (t) − Uapp (t)|H 1 (T;B) .


16

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

3.3. The Full Dispersion model. The idea is to diagonalize (30) in order to workwith a scalar equation. We project therefore (30) onto the eigenspace corresponding
to the oscillating term. These naturally leads to introduce
uF D = π1 (k + εD)uenv .
which naturally leads to the the full dispersion scalar8 equation (writing u = uF D )
∂t u + εi (ω1 (k+εD)−ω)u + ε1+p π1 (k+εD)A0 u = επ1 (k+εD)F env (ε, u)(31)
u |t=0 (x) = u0 (x)and with ω1 (·) as in Assumption 3.1.
The following corollary shows that the full dispersion scalar equation yields anapproximation of the same precision as the envelope equation for times t ∈ [0, T /ε].

Corollary 3.1 (Full dispersion model). Under the assumptions of Theorem 3.2,(i) There exists T > 0 and, for all ε ∈ (0, 1], a unique solution u ∈ C([0, T /ε]; B (1) )to (31) with initial condition u0 .
(ii) There exists ε0 > 0 such that for all 0 < ε < ε0 , the solution U to (20) provided
by Theorem 3.1 exists on [0, T /ε] and|U − UF D |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |u0 |B )(1 + |∇u0 |B + |r|B ),where UF D (t, x) = u(t, x)eik·x−ωtε+ c.c..

Remark 3.1. Equation (31) does not correspond exactly to the “full dispersion”
model of [9, 21], where the r.h.s is επ1 (k)F env (ε, u) rather than επ1 (k+εD)F env (ε, u)(but it can be found as an “intermediate model” in [8]). This change does not affect the estimate given in the Corollary, but it is important to keep track of thefrequency dependence of the polarization of the nonlinear term to introduce the“self-steepening” operators in §3.6. Note also that the ”full dispersion” model isrelated to the so-called ”unidirectional pulse propagation equation” used in nonlinear optics [19, 4].
Proof. This is actually a byproduct of the proof of Theorem 3.2 since uF D coincides
with u1 in Step 2 of the proof of Theorem 3.2.

3.4. The nonlinear Schrödinger (NLS) equation. The NLS equation is easily
derived from (31) using Taylor expansionsii(ω1 (k + εD) − ω) = cg · ∇ − ∇ · Hk ∇ + O(ε2 ),ε2
π1 (k + εD) = π1 (k) + O(ε),
where cg = ∇ω1 (k) and Hk stands for the Hessian of ω1 at k. Neglecting the O(ε2 )terms in (31) we define the N LS approximation u = uN LS as the solution to∂t u + cg · ∇u − ε 2i ∇ · Hk ∇u + ε1+p π1 (k)A0 u = επ1 (k)F env (ε, u).
(32)
u |t=0 (x) = u0 (x)We then get easily (see [9, 21]) the following justification of the NLS approximation.

Corollary 3.2 (Schrödinger approximation). Under the assumptions of Theorem
3.2, one has for all u0 ∈ B (3) such that u0 = π1 (k)u0 ,8The linear propagator is a scalar operator but the equation remains vectorial because of thenonlinear term. Indeed, π1 (k) is in general not of rank 1 (i.e. ω1 (k) is in general not of multiplicity
one)

VARIANT OF THE FOCUSING NLS EQUATION

17(i) There exists T > 0 and, for all ε ∈ (0, 1], a unique solution u ∈ C([0, T /ε]; B (3) )to (32) with initial condition u0 .
(2) There exists ε0 > 0 and cN LS > 0 such that for all 0 < ε < ε0 , the solution U
to (20) provided by Theorem 3.1 exists on [0, T /ε] and|U − UN LS |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |u0 |B )(1 + |∇u0 |B + cN LS |u0 |B (3) ),where UN LS (t, x) = u(t, x)eik·x−ωtε+ c.c..

Remark 3.2. The component cN LS |u0 |B (3) in the error estimate of the Corollary is
due to the bad frequency behavior of the Schrödinger equation when the envelopeof the oscillations ceases to be well localized in frequency. This is for instance thecase for short pulses, chirped pulses ([8, 9, 21]), and near the focusing point. To
describe such extreme situations, the standard NLS approximation does a poor job,
and this is why various variants have been derived in physics (e.g. [3]).
Remark 3.3. We assumed here that the polarization of the initial condition is exact(i.e. r = 0 in Theorem 3.2) for the sake of simplicity; indeed, the solution remains
polarized along π1 (k) for all times and computations on real physical models aremuch easier.
Example 3.3. In the frequent case where ω1 (·) has a radial symmetry, and writing
with a slight abuse of notation ω1 (k) = ω1 (k), with k = |k|, we can write,
cg = ω1′ (k)ez ,Hk =ω1′ (k)(I − ez ⊗ ez ) + ez ⊗ ez ω1′′ (k),|k|where we assumed without loss of generality that (0z) is the direction of the wave
number k, k = kez . In particular, (32) readsi ω1′ (k)ε∆⊥ u − i ω1′′ (k)∂z2 u + ε1+p π1 (k)A0 u = επ1 (k)F env (ε, u),2 k2where ∆⊥ = ∂x2 + ∂y2 is the Laplace operator in transverse variables. If we writev(t, x, z) = u(εt, x, z − ω1′ (k)t), we get∂t u + ω1′ (k)∂z u − εi ω1′ (k)1∆⊥ v − i ω1′′ (k)∂z2 v + εp π1 (k)A0 v = π1 (k)F env (ε, v).
2 k2
3.5. The nonlinear Schrödinger equation with improved dispersion relation. We propose here to investigate further the Schrödinger equation with improved dispersion relation derived in [9]. As said in Remark 3.2, the NLS approximation has too bad dispersive properties to capture correctly the envelope ofoscillating solutions to Maxwell’s equations in extreme situations, where high frequencies are released. Indeed, the dispersion relation ω1 (·) of (20) is approximatedby the second order polynomial
1(33)
ωN LS (k′ ) = ω1 (k) + cg · (k′ − k) + (k′ − k) · Hk (k′ − k).
2
For Maxwell’s equations and in dimension d = 1, Figure 1 shows that this dispersion
relation is very poor when k′ is not close to k. The idea introduced in [9] is to replace
the linear part of the Schrödinger approximation by a linear operator that differsfrom the linear part of the Schrödinger approximation by O(ε2 ) terms only, butwhose dispersion relation is far better.
More precisely, we consider an approximation under the form∂t v −(34)

Uimp (t, x) = uimp (t, x)eik·x−ωtε+ c.c.,

18

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Figure 1. The ω+,− component (see Appendix B) of the characteristic variety of Maxwell’s equation (solid) and the dispersion
relation corresponding to the Schrödinger approximation (left) and
improved Schrödinger (right). Here k = 2.
where u = uimp solves the nonlinear Schrödinger equation with improved dispersion
relation
1 − iεb · ∇ − ε2 ∇ · B∇ ∂t u




i

+ cg · ∇u − ε ∇ · Hk + 2cg ⊗ b ∇u + ε2 C3 (∇)u
(35)
2


+ ε1+p π1 (k)A0 u = επ1 (k)F env (ε, u)u |t=0 (x) = u0 (x),where b ∈ Cd , B ∈ Md×d (R) and C3 (∇) is a third order homogeneous differential
operator. We assume moreover that(36)

B is symmetric positive,b ∈ Range(B),and

4 − b · (B −1 b) > 0(note that even though B −1 b is not unique when B is not definite, the scalarb · (B −1 b) is uniquely defined). These assumptions ensure that the operator (1 −iεb · ∇ − ε2 ∇ · B∇) is invertible. This new model allows one to replace (33) byωimp (k′ ) = ω1 (k)+cg · (k′ − k) + 21 (k − k′ ) · (Hk + 2cg ⊗ b)(k′ − k) − C3 (k′ − k).
1 + b · (k′ − k) + (k′ − k) · B(k′ − k)A good choice of b, B and C3 allows a much better approximation of ω1 (·), as
shown in Figure 1 for Maxwell’s equation in dimension d = 1.
Exactly as for Corollary 3.2 we get the following result, where the only differencein the error estimate with respect to Corollary 3.2 is the constant cimp (with is muchsmaller than cN LS for good choices of b, B and C). We refer to [9] for the proof
and numerical validations of this model for the approximation of short pulses andchirped pulses.

Corollary 3.3 (Schrödinger approximation with improved dispersion). Under theassumptions of Theorem 3.2, one has, for all u0 ∈ B (3) such that π1 (k)u0 = u0 ,(i) There exists T > 0 and, for all ε ∈ (0, 1], a unique solution u ∈ C([0, T /ε]; B (3) )to (35) with initial condition u0 .

VARIANT OF THE FOCUSING NLS EQUATION

19(2) There exists ε0 > 0 and cimp > 0 such that for all 0 < ε < ε0 , the solution U to(20) provided by Theorem 3.1 exists on [0, T /ε] and|U − Uimp |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |u0 |B )(1 + |∇u0 |B + cimp |u0 |B (3) ),where Uimp (t, x) = u(t, x)eik·x−ωtε+ c.c..

Example 3.4. In the framework of Example 3.3, i.e. if ω1 (k) = ω1 (k) with k = |k|and k = kez , (35) can be writteni ω1′ (k)(1 − iεb · ∇ − ε2 ∇ · B∇)∂t u + ω1′ (k)∂z u − ε∆⊥ + ω1′′ (k)∂z2 u2k= −εiω1′ (k)b · ∇∂z u + ε2 C3 (∇)u + ε1+p π1 (k)A0 = επ1 (k)F env (ε, u).

If we write v(t, x, z) = u(εt, x, z − ω1′ (k)t) and choose C3 (∇) = −ω1′ (k)∇ · B∇∂z ,we geti ω1′ (k)∆⊥ + ω1′′ (k)∂z2 v + εp π1 (k)A0 v(1 − iεb · ∇ − ε2 ∇ · B∇)∂t v −2
k(37)= π1 (k)F env (ε, v).
A similar equation has been proposed in [22] §8.5.3 in the framework of water waves
equations.
3.6. The NLS equation with frequency dependent polarization. The idea
here is to improve the rough approximation π1 (k + εD) ∼ π1 (k) + O(ε) used toderive the NLS approximation (see §3.4). Indeed, when the description of theenvelope of the laser pulse requires a broad band of frequencies as in the situations
mentioned in §3.5, the variations of the polarization term π1 (k + εD) in front of
the nonlinearity in (31) become important and should be taken into account. We
therefore make the following approximation,
π1 (k + εD) ∼ (1 − iεb · ∇ − ε2 ∇ · B∇)−1 π1 (k) + επ1′ (k) · D − iε(b · ∇)π1 (k) ,where b and B are the same as in the NLS approximation with improved dispersion (35). In particular, if b = 0 and B = 0 (standard NLS), then the above
approximation coincides with the first order Taylor expansion
π1 (k + εD) = π1 (k) + επ1′ (k) · D.

The general formula has the same accuracy as this Taylor expansion as ε → 0. The
corresponding approximation is given by(38)

Upol (t, x) = upol (t, x)eik·x−ωtε+ c.c.,

where u = upol solves the nonlinear Schrödinger equation with frequency dependent
polarization
1 − iεb · ∇ − ε2 ∇ · B∇ ∂t u




i

+ cg · ∇u − ε ∇ · Hk + 2cg ⊗ b ∇u + ε2 C3 (∇)u + ε1+p π1 (k)A0 u
(39)
2




= ε π1 (k) + επ1 (k)π1′ (k) · D − iε(b · ∇)π1 (k) F env (ε, π1 (k)u)
u |t=0 (x) = u0 (x),where b, B and C3 (∇) are the same as in (31).
Contrary to all the previous models, the nonlinearity in (39) seems to be ofquasilinear rather than nonlinear nature.
It turns out however that the presence ofthe operator 1−iεb·∇−ε2 ∇·B∇ in front of the time derivative plays a smoothing

20

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

role allowing the control of one or several first order derivatives (see the discussionin point (4) in p. 3). If the first order derivatives involved in the nonlinearity are allcontrolled by this smoothing operator, then the nonlinearity remains semilinear innature. As shown in the proof below, the component −iε(b·∇)π1 (k)F env (ε, π1 (k)u)
of the nonlinearity is always semilinear in this sense. This is not the case for thecomponent επ1 (k)π1′ (k) · DF env (ε, π1 (k)u) that may be of quasilinear nature, inwhich case a symmetry assumption is needed on the nonlinearity to ensure local
well-posedness. In order to state this assumption, it is convenient to introduce thenorm | · |∗ defined as|u|2∗ = u, (1 − iεb · ∇ − ε2 ∇ · B∇)u .

Assumption 3.2. For all v ∈ W 1,∞ (Rd )n and u ∈ L2 (Rd )n such that π1 (k)u = u,
one has
∀1 ≤ j ≤ d,
ℜ π1 (k)π1′ (k) · ej dv F env Dj u, u ≤ Cst|v|W 1,∞ |u|2∗ ,where ej is the unit vector in the j-th direction of Rd and dv F env is the derivativeat v of the mapping u 7→ F env (ε, u).

Example 3.5. The computations performed in Appendix B show that this assumption is satisfied by the dimensionless Maxwell equations (73).
The approximation furnished by (39) is justified by the following corollary (thedifference in the estimate with respect to Corollary 3.3 is just a better nonlinear
constant, denoted Cpol to insist on this point). For the sake of simplicity, we take
B = H t0 (Rd )n (t0 > d/2) here, but adaptation to Wiener spaces are possible.
Corollary 3.4 (Schrödinger approximation with frequency dependent polarization). Let the assumptions of Theorem 3.2 be satisfied and assume moreover that
Assumption 3.2 holds with B = H t0 (Rd ) (t0 > d/2). Then for all u0 ∈ B (3) suchthat π1 (k)u0 = u0 , one has(i) There exists T > 0 and, for all ε ∈ (0, 1], a unique solution u ∈ C([0, T /ε]; B (3) )to (39) with initial condition u0 .
(2) There exists ε0 > 0 and cimp > 0 such that for all 0 < ε < ε0 , the solution U to(20) provided by Theorem 3.1 exists on [0, T /ε] and|U − Upol |L∞ ([0,T /ε]×Rd ) ≤ εCpol (T, |u0 |B )(1 + |∇u0 |B + cimp |u0 |B (3) ),where Upol (t, x) = u(t, x)eik·x−ωtε+ c.c..

Remark 3.4. We have introduced the variation of the polarization on the NLS
equation with improved dispersion (35), but the two steps are independent (i.e.,
one can take b = 0, B = 0 and C3 (∇) = 0 in (39)).
Remark 3.5. Note that in (39), we have applied π1 (k) to the full nonlinearity(hence the term π1 (k)π1′ (k) · D instead of π1′ (k) · D). This means that we keep
the effect of the frequency dependent polarization on the main polarized space
Range(π1 (k)) only. Similarly, we have replaced F env (ε, u) by F env (ε, π1 (k)u). Thislatter substitution would not change anything to the previous NLS models since wehave seen that polarized initial conditions remain polarized. Its purpose in (39) is
to make Assumption 3.2 much easier to check.

VARIANT OF THE FOCUSING NLS EQUATION

21

Proof. The justification of (39) is performed as for the other models. The only
difference here is that local well-posedness for a time scale of order O(1/ε) must be
established. We therefore show here that such a local well-posedness result holds if
u0 ∈ H s+1 (Rd )n with s > t0 + 1. We just prove a priori energy estimates for (39);existence, uniqueness and stability with respect to perturbations can be deduced
classically.
The natural energy associated to (39) is given for all s ≥ 0 by
 1
1
E s (u) = (1 − iεb · ∇ − ε2 ∇ · B∇)Λs u, Λs u = |Λs u|2∗ .
2
2
Under the assumptions (36) on B and b, E s (u)1/2 defines a norm that controls
uniformly the H s -norm. It may also control a certain number of first order derivatives. The important point for the local well-posedness of (39) is that it always
controls first order derivatives in the direction b · ∇; more precisely, we claim that
there exists c > 0 independent of ε such that for all u,(40)
E s (u) ≥ c |u|2H s + ε2 |b · ∇u|2H s .
A quick look on the Fourier side shows that it is equivalent to prove that∀X ∈ Rd ,1 + b · X + X · BX ≥ c(1 + b · X)2 ,which is a consequence of (36).
Multiplying Λs (39) by Λs ū and integrating by parts, we getd s
E (u) = −ε1+p ℜ π1 (k)A0 Λs u, Λs u + ε π1 (k)Λs F (u), Λs u
dt
d
X

2
+ε
ℜ π1 (k)π1 (k)′ · ej Λs du F Dj u, Λs u
j=1

2

:=

−ε ℜ iπ1 (k)Λs (b · ∇)F (u), Λs u
I1 + I2 + I3 + I4 .



where we denoted F (u) = F env (ε, π1 (k)u). It is straightforward to control I1 , andMoser’s estimate gives a control of I2 ,I1 . ε1+p |u|2H s ,I2 ≤ εC(|u|∞ )|u|2H s .

In order to control I3 , we must split it into two parts,
I3

=

ε2

d
X
j=1

+ε2

ℜ π1 (k)π1 (k)′ · ej du F Dj Λs u, Λs u

d
X
j=1




ℜ π1 (k)π1 (k)′ · ej [Λs , du F ]Dj u, Λs u ;the first component is controlled using Assumption 3.2 while the Kato-Ponce andMoser estimate give a control of the second one,
I3 ≤ ε2 Cst|u|W 1,∞ |u|2H s + ε2 C(|u|W 1,∞ )|u|2H s .

Remarking that Λs (b · ∇)F (u) = Λs du F (b · ∇)u and using the tame productestimate |f g|H s . |f |L∞ |g|H s + |f |H s |g|L∞ and Moser’s inequality, we get
I4 ≤ εC(|u|W 1,∞ )(|u|H s + ε|b · ∇u|H s )|u|H s .

22

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Gathering all the above estimates and using (40), we obtaind s
E (u) ≤ εC(|u|W 1,∞ )E s (u).
dt
Since moreover |u|2W 1,∞ . E s (u) for s > d/2 + 1, we deduce from Gronwall’s
estimate that for such s, the energy E s (u) remains bounded for times of orderO(1/ε).

Example 3.6. In the framework of Examples 3.3 and 3.4, we can check that v(t, x, z) =u(εt, x, z − ω1′ (k)t) solvesi ω1′ (k)∆⊥ + ω1′′ (k)∂z2 v + εp π1 (k)A0 v(1 − iεb · ∇ − ε2 ∇ · B∇)∂t v −
2
k


(41)
= π1 (k) + επ1′ (k) · D − iε(b · ∇)π1 (k) F env (ε, π1 (k)v).
3.7. Including ionization processes.

3.7.1. The profile equation. As in § 3.1, we solve the initial value problem (21)(22)-(23) for times of order O(1/ε), writing the solution under a profile form,
(42)



k · x − ωt
(U, W)(t, x) = (U, W ) t, x,.
εAgain, U (t, x, θ) and W (t, x, θ) are periodic with respect to θ, and we use anyω ∈ R. The action of the Fourier multiplier H ε kk2 · D from (19) (and (16)) istransferred at the profile level into the operator H Dθ + ε kk2 · D , with k = |k|,

 
 
 
 
k
k · x − ωt
k
,
t, x,
H ε 2 · D U (t, x) = H Dθ + ε 2 · D Ukkεwhere H Dθ + ε kk2 · D is the Fourier multiplier
X
 

X 
k
k
inθ
H n + ε 2 · D un (x)einθ ,un (x)e=
H Dθ + ε 2 · D
k
k
n∈Z

n∈Z

k

which acts continuously on H (T, B), for any k ∈ N. In order to get a solution tothe original problem, it is sufficient that (U, W ) solvesi∂t U + A(∂)U + L(ωDθ , kDθ )U + ε1+p A0 U =


ε


(43)
εF (ε, U ) − εH Dθ + ε kk2 · D (W C1T C1 U ) − εc C1T G(C1 U, W ), ∂ W − i ωD W = εG(C U, W ) · C U,
θ
1
1
t
ε
with initial conditions
(44)(U, W )|t=0 (x, θ) = u0 (x)eiθ + c.c., 0 .

Theorem 3.3. Let B = An × A, with A = H t0 (Rd ) or A = W (Rd ), and u0 ∈ An .
Under Assumptions 2.1, 2.2 and 2.3, there exists T > 0 such that for all 0 < ε ≤1 there exists a unique solution Z = (U, W) ∈ C ([0, T /ε]; B) to (21)-(22)-(23).
Moreover, one can write Z under the form
k · x − ωtZ(t, x) = Z t, x,,εwhere Z = (U, W ) solves the profile equation (43), with the initial condition (44).

VARIANT OF THE FOCUSING NLS EQUATION

23

Proof. Similar to the one of Theorem 3.1, by an iterative scheme in H k (T; B), withk ≥ 1.

3.7.2. The slowly varying envelope approximation. In this section, as in in section 3.2, we shall assume that ω = ω1 (k) is some characteristic frequency for L(·, k).
Postulating the Ansatz
(45)
Z(t, x, θ) ∼ uenv (t, x)eiθ + c.c., wenv (t, x) ,we obtain formally the following system for (uenv , wenv ) (denoted (u, w)),

i
1+p


 ∂t u + ε L(ω, k + εD)u + ε A0 u =(46)
εF env (ε, u) − ε iw C1T C1 u − εc C1T Genv (C1 u, w),

∂t w = 2εGenv (C1 u, w) · C1 u,where we used that H 1 + ε kk2 · D = i + O(ε). Here, F env is given by (29) andGenv is defined in the same way, filtering oscillations,
Z 2π
1Genv (u, w) =e−iθ G(ueiθ + c.c., w) dθ
2π 0Remark 3.6. The equation one obtains for w from direct computations is actually
Z 2πε∂t w =
G(C1 ueiθ + c.c., w) · (C1 ueiθ + c.c.) dθ
2π 0⌊K/2⌋ 

X K − 1  K − k (2|C1 u|2 )K−2k |C1 u · C1 u|2k
= c1 ε 2K |C1 u|2K + 2
kk
k=1+ 2c2 εw|C1 u|2= 2εGenv (C1 u, w) · C1 u,

in view of Assumption 2.2.

The approximation (45) is justified in the following theorem.
Theorem 3.4. Let B = An × A, with A = H t0 (Rd ) or A = W (Rd ). Let u0 ∈ An ,with ∇u0 ∈ And , and r ∈ An . Let Assumptions 2.1, 2.3 and 3.1 be satisfied andassume moreover that ω 6= 0 and that u0 = π1 (k)u0 + εr. Then(i) There exist T > 0 and, for all ε ∈ (0, 1], a unique solution (u, w) ∈ C([0, T /ε]; B (1) )to (46) with initial condition (u0 , 0).
(ii) There exists ε0 > 0 such that for all 0 < ε < ε0 , the solution Z to (43) provided
by Theorem 3.3 exists on [0, T /ε] and|Z − ZSV EA |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |u0 |A )(1 + |∇u0 |A + |r|A ),k·x−ωtwhere ZSV EA (t, x) = u(t, x)ei ε + c.c., w(t, x) .

Proof. The arguments are similar to the ones in the proof of Theorem 3.2.
Step 1. As in the proof of Theorem 3.1 and Theorem 3.3, we have local in time
T /ε (with T independent of ε) existence of (u, w), solution to (46), together with
(uniform w.r.t. ε) bounds.
Step 2. Decomposing u = u1 + uII as in Step 2 of the proof of Theorem 3.2,
one obtains in the same way that |∂t u1 |L∞ ([0,T /ε],An ) is bounded w.r.t. ε. This isalso the case for |∂t w|L∞ ([0,T /ε],A) (even of order O(ε)), as shows directly the third

24

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

equation in (46).
Step 3. As in Step 4 of the proof of Theorem 3.2, we deduce that
1|uII |L∞ ([0,T /ε],A) ≤ C T, |u0 |A 1 + |∇u0 |A + |r|A .
εThis is obtained by integration by parts in the integral formulation giving the u′j sof uII , using Step 2 and Assumption 3.1 to have non-stationary phase; we conclude
by a Gronwall argument.
Step 4. Approximation of Z by ZSV EA . Compared to Theorem 3.2, the new pointis the component w; for the sake of simplicity, we therefore set F = 0 throughout
this proof. Denote
Zapp (t, x, θ) = (Uapp , Wapp )(t, x, θ) := u(t, x)eiθ + c.c., w(t, x) , εZe = Z − Zapp ,where Z = (U, W ) is the solution to the profile equation (43) provided by Theoe, Wf ) in H k (T; B) (defined in (27)), k ≥ 1.
rem 3.3. We estimate Ze = (Uff
• For W . We have W|t=0 = 0 and
f + i ωDθ W
f = G(C1 U, W ) · C1 U − 2Genv (C1 u, w) · C1 u∂t WεX  1 Z 2π−iθe G(C1 U, W ) · C1 U dθ eikθ
= c1
2π 0
k6=0

+ c2 W |C1 U |2 − 2c2 w|C1 u|2 .

The terms in c1 and c2 can be treated similarly. For the sake of clarity, we therefore
set c1 = 0 and c2 = 1 in this proof, so that the right-hand side is given by
W |C1 U |2 − Wapp |C1 Uapp |2 + 2w (C1 u) · (C1 u)e2iθ + c.c. .

Since

e
W |C1 U |2 − Wapp |C1 Uapp |2 ≤ εC(T, |u0 |A )|Z|,

we easily deduce thatf (t)|H k (T;A) ≤ εC(T, |u0 |A )
|W

Z t
0

e ′ )|H k (T;A) dt′ + 2
|Z(t

Z t
0

′ω

e−2it ε w(C1 u) · (C1 u) dt′

Splitting u = u1 + uII as in Theorem 3.2, we have a uniform (in ε) bound for1|w(C1 u) · (C1 uII )|L∞ ([0,T /ε],An ) by Step 1 and Step 3. The only component leftεto control is therefore the one involving the product (C1 u1 ) · (C1 u1 ), for which we
write
Z t
Z t


′ω
ε
−2it′ ωεee−2it ε ∂t w(C1 u1 ) · (C1 u1 ) dt′
w(C1 u1 ) · (C1 u1 ) = − i2ω00ε −2it ω ε w(C u ) · (C u ) (t).
+i e
1 1
1 1
2ω

Using the equation satisfied by w to control ∂t w and Step 2 to control ∂t u1 , one
readily deduces that


Z t

′
′
0
0
e
f
|Z(t )|H k (T;A) dt .
(47) |W (t)|H k (T;A) ≤ C T, |u |A 1 + |∇u |A + |r|A + ε
0A.

VARIANT OF THE FOCUSING NLS EQUATION

25

e . We have Ue|= 0 and• For Ut=0
e + ε1+p A0 U
e = − H Dθ + ε k · D − i (W C T C1 U )e + i L(ωDθ , kDθ + εD)U
∂t U
1
ε
k2
T
T
f )C C1 (Uapp + εUe ) + i Wapp C C1 Uapp ,−i (Wapp + εW
11

where, for the sake of simplicity, we also have taken c = 0 becausethe corresponding
terms do not raise any difficulty. Since the Fourier multiplier H Dθ + ε kk2 · D − i
has a norm of order ε when acting on A and because the difference of the last twoe (t)|H k (T;A) satisfies the sameterms in easily bounded in terms of εZ̃, we get that |Ufupper bound as |W (t)|H k (T;A) in (47).
f and Ue we therefore get
Gathering the upper bounds for We ′ )|H k (T;A) ≤ C T, |u0 |A 1 + |∇u0 |A + |r|A
sup |Z(tt′ ∈[0,t]0+ ε C T, |u |AZ te ′′ )|H k (T;A) dt′ .
sup |Z(t0 t′′ ∈[0,t′ ]eBy a Gronwall estimate, we finally obtain a bound on Z,e H k (T;B) ≤ C T, |u0 |A 1 + |∇u0 |A + |r|A ,sup |Z(t)|t∈[0,T /ε]which, since the A− norm controls the L∞ − norm, immediately leads to the desired
estimate on |Z − ZSV EA |L∞ ([0,T /ε]×Rd ) .

3.7.3. The nonlinear Schrödinger equation with ionization. As in § 3.4, a Schrödinger
type equation can be derived in presence of ionization,

i

1+p

 ∂t u + cg · ∇u − ε ∇ · Hk ∇u + ε π1 (k)A0 u =2(48)
επ1 (k) F env (ε, u) − iwC1T C1 u − c C1T Genv (C1 u, w) ,∂t w = 2εGenv (C1 u, w) · C1 u.
Using Theorem 3.4, this model can be justified with a straightforward adaptationof the case without ionization.

Corollary 3.5 (Schrödinger approximation). Under the assumptions of Theorem
3.4, one has for all u0 ∈ A(3) such that u0 = π1 (k)u0 :(i) There exists T > 0 and, for all ε ∈ (0, 1], a unique solution (u, w) ∈ C([0, T /ε]; B (3) )to (48) with initial condition (u0 , 0).
(2) There exists ε0 > 0 and cN LS > 0 such that for all 0 < ε < ε0 , the solution Z to(43) provided by Theorem 3.3 exists on [0, T /ε] and|Z − ZN LS |L∞ ([0,T /ε]×Rd ) ≤ εC T, |u0 |A 1 + |∇u0 |A + cN LS |u0 |A(3) ,k·x−ωtwhere ZN LS (t, x) = u(t, x)ei ε + c.c., w(t, x) .
3.7.4. The most general model. In (48), ionization effects have been added to thestandard cubic NLS equation. It is of course possible to add them to a mostsophisticated model that takes into account more general nonlinearities, improved
frequency dispersion and frequency dependent polarization. We then obtain the

26

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

following generalization of (48) (for which the same justification as in Corollary 3.5
holds),


2
1
−
iεb
·
∇
−
ε
∇
·
B∇
∂t u




i



+ cg · ∇u − ε ∇ · Hk + 2cg ⊗ b ∇u + ε2 C3 (∇)u + ε1+p π1 (k)A0 u


2


(49)
= ε π1 (k) + επ1 (k)π1′ (k) · D − iε(b · ∇)π1 (k) F env (ε, π1 (k)u)






− iwC1T C1 u + c C1T Genv (C1 u, w) ,∂t w = 2εGenv (C1 u, w) · C1 u.

As shown in Appendix B.2, this system of equations takes the following form corresponding to (4) in the case of Maxwell’s equations,2i(P2 (ε∇)∂t + cg ∂z )u + ε(∆⊥ +α1 ∂z )u + iεα2 u+ε(1 + iεc3 · ∇)[(|u|2 −ρ)u + ic(α4 |u|2K−2u + α5 ρu)] = 0,(50)∂t ρ = εα4 |u|2K + εα5 ρ|u|2 .
4. Analysis of the equations (3) and (4), and open problems

In this section, we analyze and formulate open problems for the NLS-type equations (3), as well as the NLS-type equations (4) taking the ionization processes into
account. We first consider in section 4.1 the equations (3) in the case of respectively no and anomalous GVD (resp. α1 = 0 and α1 = 1), and also briefly discuss(4). Then, we discuss (3) in the case of normal GVD (i.e. α1 = −1) in section
4.2. Finally, we formulate additional open problems in section 4.3, section 4.4 andsection 4.5.
4.1. The case of no and anomalous GVD (resp. α1 = 0 and α1 = 1). Let
us first consider the case where P2 , α2 , α3 and f take the following values,
P2 = I,α2 = 0,α3 = 0andf = 0,

in which case equation (3) corresponds to the focusing cubic NLS (1) in dimensiond = 2 in the case of no GVD, and in dimension d = 3 in the case of anomalous GVD.
As recalled in the introduction, (1) is critical in dimension 2, and supercritical indimension 3, and there exist finite time blow-up solutions in both cases. Let us also
recall that some of these blow-up dynamics are stable (see for example [26] [27] [28][29] [30] in the critical case, and [32] in a slightly supercritical case).
Now, let us recall that the breakdown of solutions is not always observed in physical experiments on the propagation of laser beams and that instead a filamentation
phenomenon occurs. In this section, we would like to analyze the possibility thatthe modified model (3) in the case of no or anomalous GVD prevents the formationof singularities. Below, we analyze the role of each parameter of (3) separately
starting with the nonlinearity f .
4.1.1. The nonlinearity. In the case where P2 = I, α2 = 0, and α3 = 0, (3) takes
the following form,
(51)
i∂t v + ∆v + 1 + f (εr |v|2 ) |v|2 v = 0, t > 0, x ∈ Rd ,where the dimension is d = 2 in the case of no GVD, and d = 3 in the case ofanomalous GVD.

VARIANT OF THE FOCUSING NLS EQUATION27

As recalled in the introduction, standard modifications of the cubic nonlinearity
consist either of the cubic/quintic nonlinearity, i.e. f (s) = −s, or a saturated
nonlinearity, i.e. f is a smooth function on R+ vanishing at the origin and suchs). Let us first consider thethat (1 + f (s))s is bounded on R+ (e.g. f (s) = − 1+scase of a saturated nonlinearity. In that case, the fact that (1 + f (s))s is bounded
on R+ implies the following control of the nonlinear term in (51):
kvkL2
.
εr
Thus, running a fixed point argument yields the fact that this equation is locallywell posed in L2 (Rd ) for any integer d ≥ 1, with a time of existence only dependingon the size of ku0 kL2 . Since equation (51) still satisfies the conservation of mass,
this immediately implies global existence for any initial data in L2 (Rd ) and for any
d ≥ 1. Therefore, modifying the cubic nonlinearity in the standard NLS equation bya saturated nonlinearity does indeed prevent finite-time breakdown of the solutions.
Next, we consider the modification by a cubic/quintic nonlinearity. In that case,(51) becomes,k(1 + f (εr |v|2 ))|v|2 vkL2 .

(52)i∂t v + ∆v + |v|2 v − εr |v|4 v = 0, t > 0, x ∈ Rd .

Note that (52) still satisfies the conservation of mass, and the conserved energy is
now given by,
Z
Z
Z
1
εr
1
|∇v(t, x)|2 dx −|v(t, x)|4 dx +|v(t, x)|6 dx = E(v0 ).
E(v(t)) =2
4
6Note also that,1
3
E(v(t)) +kv(t)k2L2+
2 32εr

2

Z
1
εr
3
dx
kv(t)k2H 1 +|v(t)|2 |v(t)|2 − r
2
6
4ε
1
≥
kv(t)k2H 1
2
which together with the conservation of mass and energy yields,s
3
kv0 k2L2 .
(53)
kv(t)kH 1 ≤ 2E(v0 ) + 1 +16εr=

Let us first consider the case of dimension 2. In this case, both the cubic focusing
NLS and the quintic defocusing NLS are H 1 subcritical in the sense that the equation is locally well posed in H 1 (R2 ) with a time of existence only depending on thesize of ku0 kH 1 (see [15]). The proof extends to (52) which is thus locally well posedin H 1 (R2 ) with a time of existence only depending on the size of ku0 kH 1 . Together
with the bound (53), this immediately implies global existence for any initial datain H 1 (R2 ). The case of dimension 3 is more involved, since the quintic defocusing
NLS is H 1 critical in the sense that the equation is locally well posed in H 1 (R3 )with a time of existence depending on the shape of u0 (see [6]). Thus, one cannot
rely solely on the bound (53) to prove global existence. However, the case of combined nonlinearities is addressed in [41], where global existence of solutions with
H 1 initial data is proved, when the nonlinearity is a sum of two H 1 -subcritical powers, or the sum of an H 1 -subcritical power and of a defocusing H 1 -critical power.
This includes equation (52). Therefore, modifying the cubic nonlinearity in thestandard NLS equation by a quintic defocusing nonlinearity does indeed prevent

28

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

finite-time breakdown of the solutions both in dimension 2 and 3. Let us mention an interesting phenomenon regarding the qualitative behavior of the solutions.
Physical experiments suggest an oscillatory behavior of the solution which focuses,
then defocuses, refocuses,... in an almost periodic fashion (see [3] and references
therein). Such a behavior is also observed in numerical simulations and suggested
by heuristic arguments (see for example [38] and [13]).
Open Problem 1. Establish rigorously the “oscillatory” phenomenon of the solutions to (51) observed in physical experiments (see e.g. [3]).
Another open problem concerns the behavior of the solution as ε → 0. Pick an
initial data v0 ∈ H 1 (Rd ), d = 2, 3, leading to a finite time blow-up solution v attime T > 0 to the cubic NLS (1). Consider the solution vε of (51) with the same
initial data v0 . In the case of dimension 2, it is shown in [24] that vε (t) → v(t) in
H 1 (R2 ) as ε → 0 on [0, T ) and,lim kvε (T )kH 1 (R2 ) = +∞.

ε→0An interesting question is the understanding of the limit of vε (t) as ε → 0 for t > T .
Partial results have been obtained in this direction in [25]. There, it is proved that
only few scenarios are possible, but one would like to establish whether all scenarios
do occur or only some of them, and which scenarios are generic.
Open Problem 2. Let v0 ∈ H 1 (Rd ), d = 2, 3 be leading to a finite time blow-up
solution v at time T > 0 to the cubic NLS (1). Consider the solution vε of (51)with the same initial data v0 . Describe the behavior of vε (t) as ε → 0 for t > T .
4.1.2. Taking the ionization process into account. Let us discuss equations (4) and(5) in the case of no or anomalous GVD which take the ionization process into
account and also correspond to a modification of the nonlinearity in the focusing
cubic NLS (1). Ionization is certainly the most important phenomenon leading to
the formation of laser filaments. Recall that (4) is given by(i(∂t + cg ∂z )u + ε(∆⊥ +α1 ∂z2 )u + ε(|u|2 −ρ)u = −iεc(α4 |u|2K−2u + α5 ρu),(54)
∂t ρ = εα4 |u|2K + εα5 ρ|u|2 ,where α4 , α5 ≥ 0, c > 0, where ρ is the density of electron created by ionization,
while cg = cg ez is the group velocity associated to the laser pulse, and where d = 2
in the case of no GVD, and d = 3 in the case of anomalous GVD. Also, recall that(5) is given by(i∂τ v + (∆⊥ + α1 ∂z2 )v + (|v|2 − ρ̃)v = −ic(α4 |v|2K−2 v + α5 ρ̃v),(55)−cg ∂z ρ̃ = εα4 |v|2K + εα5 ρ̃|v|2 ,where ρ̃ corresponds to ρ written in a frame moving at the group velocity cg = cg ez
and with respect to a rescaled time τ = εt, i.e.
ρ(t, X⊥ , z) = ρ̃(εt, X⊥ , z − cg t).
In view of the second equation of (55), a boundary condition for ρ̃ has to beprescribed at z = z0 for some z0 in order to obtain a well-posed problem. A natural
choice, which ensures that ρ̃ ≥ 0, is to prescribe ρ̃ at +∞:(56)

lim (∇⊥ )l ρ̃(τ, X⊥ , z) = 0, ∀ l ≥ 0.

z→+∞

VARIANT OF THE FOCUSING NLS EQUATION

29

Let us first discuss the local well-posedness theory for equations (54) and (55),starting with the first one. One may obtain the existence of solutions to (54) overan interval of time with size O(ε−1 ). Indeed, for an integer N > d/2, differentiating
N times with respect to space variables the equations both for v and ρ, multiplying(54) respectively by ∇N v and ∇N ρ, integrating on Rd , taking the imaginary partand integrating by parts for the first equation, and summing both equations yields:d[kvk2H N + kρk2H N ]dt.

ε(kvk2H N + kρk2H N )(kvkL∞ + kvk2K−2+ kρkL∞ )L∞.

ε(kvk2H N + kρk2H N )(1 + kvk2H N + kρk2H N )K−1

where we used the fact that N > d/2 together with simple product rules and the
Sobolev embedding. Integrating this differential inequality, we obtain a time ofexistence with size O(ε−1 ).
Next, we discuss the local well-posedness theory for equation (55) supplemented
with the boundary condition (56). For any integer N , we introduce the space HN ,


N


X
.
<
+∞
k∇j f kL2X L∞
HN = f : Rd → R,
⊥ z


j=0

For an integer N > d, differentiating N times with respect to space variables theequations for v, multiplying the first equation of (55) by ∇N v, integrating on Rd ,taking the imaginary part and integrating by parts yields:d
kvk2H Ndt.

2
kvk4H N + kvk2K
H N + kvkH N kρ̃kHN

where we used the fact that N > d together with the Sobolev embedding in thelast inequality. Next, we estimate kρ̃kHN . In view of the second equation of (55)and the boundary condition (56), we have


Z
Z +∞
α5 ε σ 2|v| (t, x, y, s)ds dσ.
α4 ε|v|2K (t, x, y, σ) exp
ρ̃(t, x, y, z) =c zz
We infer|ρ̃(t, x, y, z)| .

Z +∞
−∞

|v|

2K



(t, x, y, σ)dσ exp(kvk2H N ),which yields
2kρ̃kH0 . kvk2K
H N exp(kvkH N ),where we used the fact that N > d together with the Sobolev embedding. Next,
differentiating the second equation of (55) N times, we obtain−c∂z ∇N ρ̃ = α4 ε∇N (|v|2K ) + α5 ε∇N (ρ̃|v|2 ).
In view of the boundary condition (56), this yields
Z +∞N
α4 ε∇N (|v|2K )(t, x, y, σ) + α5 ε(∇N (|v|2 ρ̃) − |v|2 ∇N ρ̃)(t, x, y, σ)∇ ρ̃(t, x, y, z) =zZ
α5 ε σ 2
× exp|v| (t, x, y, s)ds dσ.
c z

30

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Hence, we deduce.

|∇N ρ̃(t, x, y, z)|
Z +∞


N
2K
N
2
2 N
|∇ (|v| )(t, x, y, σ) + |∇ (|v| ρ̃) − |v| ∇ ρ̃|(t, x, y, σ) dσ exp(kvk2H N ),−∞which yields

2kρ̃kHN . (kρ̃kHN −1 kvk2H N + kvk2K
H N ) exp(kvkH N ).

By induction, we obtain
2K
2kρ̃kHN . (1 + kvk2N
H N )kvkH N exp((N + 1)kvkH N ).

This yieldsd
kvk2H N
dt

.

2
2N
2K
2
kvk4H N + kvk2K
H N + kvkH N (1 + kvkH N )kvkH N exp((N + 1)kvkH N ).

Integrating this differential inequality, we obtain local existence for (v, ρ̃) in H N ×
HN for N > d.
Remark 4.1. Note that the factor ε in the second equation of (55), while present
in the model, is not needed for the well-posedness theory.
Let us now come back to the issue of global well-posedness/finite time singularity
formation. For equations (54) and (55), note that the L2 norm of u (resp. v) isdissipated. Indeed, in the case of (55), multiplying by v, integrating on Rd , taking
the imaginary part and integrating by parts yields:


Z
d
2
+
α
kvk2L2 = −c α4 kvk2K
|v|
ρ̃dx
,
5
L2K
dt

from which we deduce that the L2 norm is dissipated since ρ̃ ≥ 0. On the other
hand, the energy is not conserved, nor dissipated. We thus can not carry out
the analysis of section 4.1.1 for equation (52), even in the case of dimension 29.
An interesting problem would then be to prove that taking the ionization process
into account (i.e. replacing the cubic focusing NLS (1) by either equation (54)or equation (55)) does indeed prevent finite-time breakdown of the solutions indimensions 2 and 3. This is formulated in the following open problem.
Open Problem 3. Prove that the solutions to equations (54) and (55) are global
in dimensions 2 and 3.
4.1.3. The damping. In the case where P2 = I, α3 = 0 and f = 0, (3) takes thefollowing form,(57)

i∂t v + ∆v + iα2 v + |v|2 v = 0,

where d = 2 in the case of no GVD, and d = 3 in the case of anomalous GVD. The
mass and energy are not conserved quantities anymore. The mass decreases,(58)

kv(t)kL2 = e−α2 t kv0 kL2 ,9Note that a simpler model of damped NLS where the ρ or ρ̃ term are not present has been
investigated in [2]. The authors obtain global existence for a certain range of parameters bycontrolling a modified energy even if it is not conserved.

VARIANT OF THE FOCUSING NLS EQUATION

31

while for the energy we have,




Z
Z
d 1
1
4
2
4
2
(59)
|v| dx = −α2 k∇vkL2 − |v| dx ,k∇vkL2 −dt 2
4so that the energy is neither increasing nor decreasing. Equation (57) has been analyzed in several works (see e.g. [40], [38], [12] and [33]). In particular, from standard
arguments using Strichartz estimates, there is in dimensions 2 and 3 a continuous
and increasing function θ such that global existence holds for α2 > θ(ku0 kH 1 ) (see[38] and [33]). Also, in the 3-dimensional case, there is a function θ such that stableblow-up solutions exist for 0 ≤ α2 ≤ θ(u0 ) (see [40] and [33]). Finally, the same
behavior has been observed in numerical simulations and suggested by heuristic
arguments in the 2-dimensional case (see [12]). Thus, a small α2 does not prevent
finite time blow up. Since the constant α2 obtained in (4) is usually small, it appears that the damping cannot by itself explain the physical observation according
to which the breakdown of solutions does not occur.
4.1.4. Off-axis variation of the group velocity. In the case where α2 = 0, α3 = 0
and f = 0, (3) takes the following form,(60)

iP2 (ε∇)∂t v + ∆v + |v|2 v = 0,

where d = 2 in the case of no GVD, and d = 3 in the case of anomalous GVD. Theenergy remains unchanged and is still conserved, while the mass is replaced by thefollowing conserved quantity,(61)(P2 (ε∇)v(t), v(t)) = (P2 (ε∇)v0 , v0 ).

Let us first consider the case of full off-axis dependence, i.e. the case where,(62)(P2 (ε∇)v, v) & kvk2L2 + ε2 k∇vk2L2 .

The operator P2 (ε∇) is a second order self-adjoint operator which is invertible inview of (62). We denote by P2 (ε∇)−1 its inverse. We rewrite (60) in the following
form,
(63)

i∂t v + P2 (ε∇)−1 ∆v + P2 (ε∇)−1 (|v|2 v) = 0.
−1

Using Duhamel’s formula, and the semi-group eitP2 (ε∇) ∆ , we obtain,
Z t
−1
−1
ei(t−s)P2 (ε∇) ∆ P2 (ε∇)−1 (|v|2 v)(s)ds.
(64)
v = eitP2 (ε∇) ∆ v0 +0

As a simple consequence of (62) and the Sobolev embedding in dimensions 2 and3, we have,
11(65)
kP2 (ε∇)−1 (|v|2 v)kH 1 (Rd ) . k|v|2 vkL2 (Rd ) . kvk3H 1 (Rd ) , d = 2, 3.
εεNow, a fixed point argument based on the formulation (64) together with the esti−1
mate (65) and the fact that the semi-group eitP2 (ε∇) ∆ is unitary on H 1 , implies
that (60) is locally well posed in H 1 for dimensions d = 2, 3, with a time of existence only depending on the size of ku0 kH 1 . Together with the lower bound (62)and the conserved quantity (61) which yield a uniform in time bound on the H 1
norm of the solution, this immediately implies global existence for any initial datain H 1 when d = 2, 3. Therefore, modifying the focusing cubic NLS equation byadding a full off-axis dependence does indeed prevent the finite-time breakdown of
the solutions.

32

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Next, let us consider the case of partial off-axis dependence, i.e. the case where,(P2 (ε∇)v, v) & kvk2L2 + ε2

(66)
d

j
X

k=1

kvk · ∇vk2L2 ,for vectors vk in R , k = 1, . . . , j, where j = 1 if d = 2, and j = 1 or j = 2 if d = 3
so that the right-hand side of (66) does not control the full H 1 norm. Notice thatthe well-posedness theory has yet to be investigated in this case. Indeed, consider
the nonlinear term in the formulation (64),(67)

eitP2 (ε∇)−1∆P2 (ε∇)−1 (|v|2 v).

In the directions vk of (66), the operator P2 (ε∇)−1 gains two derivatives, while−1
the semi-group eitP2 (ε∇) ∆ does not disperse (and thus does not satisfy a useful Strichartz estimate). On the other hand, in the directions orthogonal to thevectors vk , the operator P2 (ε∇)−1 does not gain any derivative, while the semi−1group eitP2 (ε∇) ∆ should satisfy a useful Strichartz estimate. Thus, in order toinvestigate the well-posedness theory, one should try to combine the regularization
provided by the operator P2 (ε∇)−1 in the directions vk of (66) with the dispersive−1
properties of the semi-group eitP2 (ε∇) ∆ in the direction orthogonal to the vectors
vk . Now, it would be interesting to investigate whether a suitable well-posedness
theory together with the conserved quantities given by the energy and (61) yield
global existence. This suggests the following open problem.
Open Problem 4. Investigate both the local and global well-posedness for equation(60) in the case of partial off-axis dependence. In particular, does the modificationof the focusing cubic NLS (1) by the addition of a partial off-axis dependence preventthe finite-time breakdown of the solutions?
4.1.5. Self-steepening of the pulse. In the case where P2 = I, α2 = 0 and f = 0, (3)takes the following form,(68)i∂t v + ∆v + (1 + iεα3 · ∇)|v|2 v = 0,

where d = 2 in the case of no GVD, and d = 3 in the case of anomalous GVD.
Adding the operator (1 + iεα3 · ∇) in front of the nonlinearity of the focusing cubic
NLS does certainly not prevent finite-time breakdown of the solutions. Indeed, one
expects to obtain even more blow up solutions since the formation of optical shocks
is expected in this case. The term shock is used in view of the similarity with the
Burgers equation - both in terms of the nonlinearity of the equation and the profilesof the solutions observed in some numerical simulations (see [1] for more details
on optical shocks). The reason we included this modification in the discussion is
because it may account for the physical observation of the self-steepening of thepulse: an initial pulse which is symmetric becomes asymmetric after propagatingover a large distance and its profile seems to develop a shock. This phenomenon has
been widely observed and we refer the reader to [3] and references therein. Thus,
an interesting open problem would be to exhibit solutions of (68) for which thecorresponding profile develops a shock. We formulate a slightly more general open
problem below.
Open Problem 5. Describe the blow-up scenarios for the finite time blow-up solutions to equation (68).

VARIANT OF THE FOCUSING NLS EQUATION

33

4.2. The case of normal GVD (i.e. α1 = −1). Let us consider the case where
P2 , α1 , α2 , α3 and f take the following value,
P2 = I, α1 = −1, α2 = 0, α3 = 0 and f = 0,in which case equation (3) corresponds to the following equation in dimension 2 or
3,(69)i∂t v + (∆⊥ − ∂z2 )v + |v|2 v = 0.

This equation is sometime referred to as hyperbolic cubic NLS, or non elliptic cubic
NLS. Since Strichartz estimates only depend on the curvature of the corresponding characteristic manifold (see [37]), equation (69) satisfies the same Strichartz
estimates as the cubic focusing NLS. Thus, the result of Ginibre and Velo [15]immediately extends to (69) which is locally well-posed in H 1 = H 1 (Rd ) withd = 2, 3. Therefore, for v0 ∈ H 1 , there exists 0 < T ≤ +∞ and a unique solutionv(t) ∈ C([0, T ), H 1 ) to (69) and either T = +∞, and the solution is global, or thesolution blows up in finite time T < +∞ and then limt↑T k∇u(t)kL2 = +∞.
Equation (69) admits the following conservation laws in the energy space H 1 ,L2 − norm : kv(t)k2L2 = kv0 k2L2 ;
Z
Z
Z
1
1
1
Energy : E(v(t)) =|∇⊥ v(t, x)|2 dx −|∂z v|2 dx −|v(t, x)|4 dx = E(v0 );
2
2
4

Z

Z
∇v0 (x)v0 (x)dx .
Momentum : Im∇v(t, x)v(t, x)dx = Im

A large group of H 1 symmetries leaves the equation invariant: if u(t, x) solves (69),3then ∀(λ0 , τ0 , x0 , γ0 ) ∈ R+
∗ × R × R × R, so does(70)u(t, x) = λ0 v(λ20 t + τ0 , λ0 x + x0 )eiγ0 .

Note that (69) is not invariant under the usual Galilean transform. However, it isinvariant under a twisted Galilean transform. For β = (β1 , β2 , β3 ) ∈ R3 , we define,β̂ = (β1 , β2 , −β3 ) ∈ R3 .

Then (69) is invariant under the following twisted Galilean transform,
(71)

β̂

β

vβ (t, x) = v(t, x − βt)ei 2 ·(x− 2 t) .

Note also that the scaling symmetry u(t, x) = λ0 v(λ20 t, λ0 x) leaves the space L2 (R2 )invariant so that (69) is critical with respect to the conservation of mass in dimen1
sion 2, while it leaves the homogeneous Sobolev space Ḣ 2 (R3 ) invariant so that(69) is supercritical with respect to the conservation of mass in dimension 3.
In contrast to the cubic focusing NLS, the existence or absence of finite time
blow up solutions for (69) is widely open. While there exists a counterpart tothe virial for (69), it is too weak to provide the existence of finite-time blow up
dynamics (see the discussion in [38]). Also, there are no standing waves in the formv(t, x) = Q(x)eiωt , ω ∈ R in the energy space for the equation (69) (see [14]), while
for the focusing cubic NLS, such an object exists and is fundamental in the analysisof the blow up dynamics (see e.g. [26] [27] [28] [29] [30] [31] in dimension 2, and[32] in dimension 3). Now, numerical simulations suggest that the solutions to (69)do not break down in finite time (see e.g. [38], [13] and references therein). In
particular, the simulations exhibit a phenomenon called pulse splitting where thepulse focuses until it reaches a certain threshold. Once this threshold is attained,

34

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

the pulse splits in two pulses of less amplitude moving away from each other. Thisphenomenon might repeat itself (multiple splitting) and ultimately prevent finite
time blow up, but this has not been clearly backed up by numerics so far. Thus
an interesting problem would be to prove that the cubic NLS equation (69) (i.e. in
the case of normal GVD) does not have finite time blow up solutions, whether thisis due to pulse splitting, or to some other phenomenon. This is formulated in thetwo open problems below.
Open Problem 6. Prove that the solutions to equation (69) are global in dimension 2 and 3.
Open Problem 7. Describe rigorously the phenomenon of pulse splitting for equation (69).
Remark 4.2. We only considered in this section the case where,
P2 = I, α2 = 0, α3 = 0 and f = 0
which is (69). While it is of interest to investigate the qualitative role of theparameters P2 , α2 , α3 and f of equation (3) in the case of normal GVD (i.e. thecase α1 = −1), we chose not to investigate their effects since we are interested inthis paper primary on the non existence of focusing dynamics, and equation (69)is expected to have only global solutions.
Remark 4.3. In a recent preprint [18], standing waves and selfsimilar solutions havebeen exhibited for equation (69) in dimension 2. However, these solutions do notbelong to the energy space. In particular, they are not in C 2 and do not decay
along the two diagonals in R2 .
4.3. Mixing several phenomena. In section 4.1 and section 4.2, we have investigated the effect of each of the parameters P2 , α1 , α2 , α3 and f on the solutions
to equation (3), and in particular whether these modifications of the cubic NLS(1) prevent the existence of finite time blow-up solutions. We have also formulated
several open problems. Now, instead of studying all these effects separately, one
may investigate the case when all these phenomenon are present at the same time,and ask which are the dominant ones? Consider for instance the case where thereis partial off-axis variation of the group velocity in the direction z, and at the sametime a potential self-steepening of the pulse in the same direction, i.e. P2 = 1 − ∂z2and α3 = ez ,(72)i(1 − ε2 ∂z2 )∂t v + ∆v + (1 + iε∂z )(|v|2 v) = 0.

Then, one may wonder whether the partial off-axis variation of the group velocity
prevents the self-steepening from taking place. This is formulated in the open
problem below.
Open Problem 8. Do the solutions to equation (72) exhibit the phenomenon ofself-steepening?
4.4. The vectorial case. Recall that equation (3) is a particular case of the vectorial equation (3)vect ,1iP2 (ε∇)∂t v + (∆⊥ + α1 ∂z2 )v + iα2 v + (1 + iεα3 · ∇) (v · v)v + 2|v|2 v = 0,3
where v is now a C2 -valued function, and where we consider for simplicity theequation corresponding to the cubic case (i.e. f = 0 in (3)). In fact, (3) is a

VARIANT OF THE FOCUSING NLS EQUATION

35

particular case of (3)vect corresponding to initial data living on a one-dimensional
subspace of C2 (see Remark 1.2). Now, one may of course consider the previous
questions formulated for equation (3), and investigate the same problems for thevectorial case. This is formulated in the open problem below.
Open Problem 9. Investigate the vectorial counterparts for equation (3)vect of thevarious open problems formulated for the scalar equation (3) in section 4.1, section
4.2, section 4.3 and section 4.5.
4.5. The approximation of the Maxwell equations over longer times. We
have provided in §3 a rigorous justification for all the NLS-type models derived inthis paper. More precisely, we have proved that there exists T > 0 such that
(1) The exact solution U of (20)-(22) exists on [0, T /ε].
(2) The NLS-type model under consideration (e.g. (39)) admits a unique solution uapp with initial condition u0 on the same time interval.
k·x−ωt(3) The approximation Uapp (t, x) = uapp (t, x)ei ε + c.c. remains close to
U on this time interval,10|U − Uapp |L∞ ([0,T /ε]×Rd ) ≤ εC(T, |uapp |L∞ ([0,T /ε];B) )(1 + |∇u0 |B + |u0 |B (3) ).

Such a justification is far from being sharp. In particular, for the standard cubic focusing NLS equation, one has T < Tfoc , where Tfoc is the time when focusing occurs. Indeed, the above justification process requires that uapp remains
bounded on [0, T /ε] (after rescaling, this is equivalent to require that the solution v to (3) is bounded on [0, T ]). In the error estimate above, the constantC(T, |uapp |L∞ ([0,T /ε];B) ) therefore blows up as T → Tfoc .
It follows that all the NLS-type equations derived here are justified far enoughfrom the focusing time. Now, relevant differences between the different models canonly be observed close enough to focusing (they all differ by formally O(ε2 ) terms
that become relevant only near the focusing point). So, roughly speaking, we have
only proved that all the models are justified on a time interval for which they arebasically identical.
As seen above, existence beyond the focusing time of the standard cubic NLS
equation is proved or expected for many of the variants considered here. For suchmodels, the above argument does not work, i.e., the constant C(T, |uapp |L∞ ([0,T /ε];B) )of the error estimate does not blow up as T → Tfoc . This is for instance the caseof the cubic/quintic NLS equation (52) that admits a global solution v ε . However(see Open Problem 2), we have
lim lim |v ε (t, ·)|H 1 = +∞,ε→0 T →Tfoc

so that there is no reason to expect the error term εC(T, |v ε |L∞ ([0,T /ε];B) ) to besmall near the focusing point and for small ε.
For the moment, the merits of the NLS-type models derived here to describe
correctly the mechanisms at stake during focusing can only by assessed numerically.
Hence the following interesting open problem,
Open Problem 10. Rigorously justify one of the NLS-type models derived here
on a time scale [0, T /ε] with T ≥ Tfoc .
10In the error estimates of §3, the constant on the r.h.s.
is C(T, |u0 |B ) rather thanC(T, |uapp |L∞ ([0,T /ε];B) ). Since |uapp |L∞ ([0,T /ε];B) = C(T, |u0 |B ), this is of course equivalent,but the first form is more convenient for the present discussion.

36

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

Acknowledgements. The autors warmly thank Christof Sparber for pointing reference [41] out.
Appendix A. Nondimensionalization of the equations
A.1. The case without charge nor current density. There are two characteristic times for the situation considered here. The first one, denoted t̄, is the inverseof the frequency of the laser pulse; the second one, denoted T̄ is the duration of thepulse. In the regimes considered here11, one has T̄ ≫ t̄, and the small parameter εis defined as
t̄
ε= .
T̄
The time variable is naturally nondimensionalized with T̄ , while the space variables
are nondimensionalized by L = cT̄ , namely,t = T̄ t̃,

x = Lx̃,

where dimensionless quantities are denoted with a tilde. We also nondimensionalize
the unknowns E, B and P with typical orders E0 , B0 and P0 . The typical scale forthe electric and magnetic fields are directly deduced from P0 ,1
1
E0 = P0 ,B0 = E0 .
ǫ0c
In order to choose the remaining P0 , some consideration on the polarization
equation is helpful. We recall that equation (11) is given by∂t2 P + Ω1 ∂t P + Ω20 P − ∇VN L (P) = ǫ0 bE.

For all the applications we have in mind, on has ∇VN L (P) = a3 |P|2 P + h.o.t., sothat we nondimensionlize the nonlinear term as∇P VN L (P) = a3 P30 ∇P̃ ṼN L (P̃),
where ṼN L is the dimensionless potential
ṼN L (P̃) =1
VN L (P).
a3 P40

Example A.1. For the three cases considered in Example 2.1, we find, denoting
a P2
ãεr = a5 3 0 , where r > 0 is chosen such that ã = O(1),(i) Cubic nonlinearity:∇P̃ ṼN L (P̃) = |P̃|2 P̃(ii) Cubic/quintic nonlinearity:∇P̃ ṼN L (P̃) = (1 − ãεr |P̃|2 )|P̃|2 P̃.

(iii) Saturated nonlinearity:r

1 + ãε3 |P̃|2|P̃|2 P̃.
∇P̃ ṼN L (P̃) =
r
2 )2
|
P̃|
(1 + 2ãε3All these examples can therefore be put under the form
∇ ṼN L (P̃) = 1 + f˜(ãεr |P̃|2 ) |P̃|2 P̃,P̃

where f˜ : R+ → R is a smooth function such that f˜(0) = 0.
11We refer to [10] for variants of this nondimensionalization.

VARIANT OF THE FOCUSING NLS EQUATION

37

In dimensionless form, the polarization equation becomes therefore∂t̃2 P̃ + T̄ Ω1 ∂t̃ P̃ + T̄ 2 Ω20 P̃ − a3 T̄ 2 P20 ∇P̃ ṼN L (P̃) = ǫ0 b

T̄ 2 E0
Ẽ
P0

We therefore choose P0 to be the typical nonlinear scale for which the nonlinear
terms in the above equations are of the same order as the second order time derivative, namely,
1
P0 = √ .
T̄ a3
Finally, we need some information on the size of Ω0 , Ω1 and b to give our final
dimensionless form of the equations. The resonance frequency of the harmonic
oscillator is typically of the same order as the frequency of the laser pulse,Ω0 = t̄−1 ω0 ;the damping frequency is written under the form
Ω1 = εα+1 t̄−1 ω1 ,where the dimensionless damping frequency ω1 has size O(1) while the coefficient α
depends on the medium in which the laser propagates. Typically, α = 1 or α = 2.
Writing the coupling constant b as
b=

γ
,
t̄2

the dimensionless polarization equations can therefore be written as∂t̃2 P̃ + εα ω1 ∂t̃ P̃ + ε−2 ω02 P̃ − ∇P̃ ṼN L (P̃) = ε−2 γ Ẽ.
Omitting the tildes for the sake of clarity, Maxwell’s equations (10)-(11) read therefore (without charge nor current density), ∂t B + curl E = 0,∂t E − curl B = −∂t P,
 2∂t P + εα ω1 ∂t P + ε−2 ω02 P − ∇VN L (P) = ε−2 γE.

Introducing the auxiliary unknowns,
ω0P♯ = √ P,
γ

Q♯ =

ε
ε
∂t P♯ = √ ∂t P,
ω0γ

and working with nonlinearities as those considered in Example A.1, this system
can be written as a first order system,∂t B + curl E = 0,





1√ ♯



 ∂t E − curl B + ε γQ = 0,

1 √
(73)
γ

∂t Q♯ + εα ω1 Q♯ − ( γE − ω0 P♯ ) = ε 3 1 + f (εr |P♯ |2 ) |P♯ |2 P♯ ,


ε
ω0




1

♯
♯
 ∂t P − ω0 Q = 0,ε38ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

where we wrote f (x) = f˜(ã ωγ2 x).
0This system has the form (20) with n = 12, U = (B, E, Q♯ , P♯ )T and



0
0
0
0
curl 0 0√ 0 −curl

0
0
0
0
γI
,
E=
A(∂) =  0 −√γI

0
0
0 0 
0
0
0
0 0
0
0
−ω0 I


0
0 

ω0 I 
0while A0 is the block diagonal matrix A0 = diag(0, 0, ω1 I, 0), so that the firsttwo points of Assumption 2.1 are satisfied. For the third one, remark that thenonlinearity is given by

T

γ
(74)
F (ε, U) = 0, 0, 3 (1 + f (εr |P♯ |2 ) |P♯ |2 P♯ , 0,
ω0which is of the same form as in Assumption 2.1 with Q(U ) = |P♯ |2 and T (U, Ū , U ) =(0, 0, ωγ3 |P♯ |2 P♯ , 0)T . One can check that such nonlinearities also satisfy Assumption
0
2.3.

A.2. The case with charge and current density. Now, we deal with the nondimensionalization of equations (10)-(11)-(17). We proceed in the same way as in theprevious paragraph, simply adding the characteristic sizes J0 and ρ0 of the free electron current density Je and charge density ρ. Considerations on the polarization
equation are the same.
We choose ρ0 from the equation for the charge density,σE2K∂t̃ ρ̃ = σK ρnt 0 T̄ |Ẽ|2K + E20 T̄ ρ̃|Ẽ|2 .
ρ0Ui
Many configurations are possible, according to the numerical value of the various
coefficients involved (see for instance [3] for experimental data). We choose here aconfiguration that leads to the richest model, where the coupling constants of thetwo nonlinear terms are of the same order; more precisely∂t̃ ρ̃ = εc1 |Ẽ|2K + εc2 ρ̃|Ẽ|2 ,for non-negative constants c1 , c2 . This corresponds to the following choice for ρ0 ,
ρ0 =σK ρnt E02K.
εc1

Knowing ρ0 , we can then determine J0 from the equation for the free electron
current density,q2
Je = e ρ0 E0 ρ̃ H(εDz̃ )Ẽ,
ω l mewhich naturally leads toJ0 =q 2 T̄ ρ0 E0
qe2
ρ0 E0 = ε e.
ω l meme

Going back to Maxwell’s equations, we get ∂t̃ B̃ + curl Ẽ = 0,

1 J0
Ui ρ0
εc1 |Ẽ|2K−2 + εc2 ρ̃ Ẽ, ∂t̃ Ẽ − curl B̃ = −∂t̃ P̃ − T̄ J̃e −ǫ0 E0
ǫ 0 E0VARIANT OF THE FOCUSING NLS EQUATION

39

Here again, many configurations can be found, and we choose one that leads to therichest models, namely,
1 J0= εc3 ,
T̄
ǫ0 E0

Ui ρ0
= c0
ǫ 0 E0where c0 , c3 are dimensionless positive constants. Replacing J̃e by c3 J̃e , ρ̃ by c3 ρ̃and c1 by c1 c3 , we can assume that c3 = 1.
Finally, dropping the tildes, introducing the unknowns Q♯ and P♯ , and using thesame notations as in (73), we obtain∂t B + curl E = 0,





1√ ♯


∂t E − curl B +γQ = −ερH(εDz )E − εc0 c1 |E|2K−2 + c2 ρ E,



ε



γ
1 √
∂t Q♯ + εα ω1 Q♯ − ( γE − ω0 P♯ ) = ε 3 1 + f (εr |P♯ |2 ) |P♯ |2 P♯ ,
(75)
ε
ω0




1



∂t P♯ − ω0 Q♯ = 0,


ε


∂t ρ = εc1 |E|2K + εc2 ρ|E|2 .
Appendix B. Explicit computations for Maxwell equations

B.1. The case without charge nor current density. We derive here the variants of the NLS equations derived in this paper in the particular case of the Maxwell
equations (73). We derive all the versions of the NLS equations that do not take into
account the frequency dependence of the polarization (this corresponds to (37)).
This latter effect, which leads to the generalization (41) of the previous model, isexamined separately. For the sake of simplicity, we only consider cubic nonlinearities here; we show that in this context, (41) reduces to the vectorial family of NLS
equation (3)vect .
B.1.1. Without frequency dependent polarization. In order to check that Assumption 3.1 is satisfied by the dimensionless Maxwell equations (73), we need to compute the eigenvalues and eigenvectors of the matrix
1
L(0, k) = A(k) + E,iwhere A(k) and E are given in §A.2, which is equivalent to find all non trivial
solutions to the equation
L(ω, k)u = 0,with

1
L(ω, k) = −ωI + A(k) + E,iand ω ∈ R, u = (b, e, q♯ , p♯ ) ∈ C12 .
From the last two equations of (73), we can always write p♯ and q♯ in terms of e,
√
√
ω0 γ
ω γ
♯
(76)
p♯ = − 2
e,
q
=
i
e.
ω − ω02ω 2 − ω02
Let us now consider several cases:• If e k k then by the first equation, either ω = 0 and b k k or b = 0 andω 2 = ω02 + γ.

40

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL• If e 6k k then ω 6= 0 and combining the first two equations, we get 2 ω 2 − (ω02 + γ)ω
− k 2 e = −(k · e)k,ω 2 − ω02

where we denoted as usual by k the modulus of k, k = |k|; this in turns
implies (since k is not parallel to e) that k ⊥ e andω 4 − ω 2 (ω02 + γ) + k 2 + ω02 k 2 = 0;2denoting ∆ = (ω02 + γ) + k 2 − 4ω02 k 2 > 0, this equation admits four real
solutionsq√1
ω02 + γ + k 2 ± ∆.
ω±,± (k) = ± √2
There are m = 7 different branches of CL : three constant sheets given byqω = 0 and ω = ± γ + ω02(the first one being of multiplicity two, the two other ones of multiplicity one),and four curved sheets (each one being of multiplicity two) given by ω = ω±,± .
We denote by π±,± the associated eigenprojectors (which are therefore of rank 2).
Remarking that π±,± (k)u = u if and only if
1
e · k = 0,b = − k ∧ e,ωand p♯ and q♯ are given in terms of e by (76), one readily finds that π±,± (k) canbe written as a block matrix
1k2
ω(k)2 + ω022π±,± (k) =
(P
)
with
N
(k)
=
+
1
+
γ
ij
1≤i,j≤4
N (k)2
ω(k)2(ω(k)2 − ω02 )2(for the sake of clarity, we write ω(k) = ω±,± (k)), and where the 3 × 3 blocks P1j
are given by
k2
Π ⊥,
P11 =
ω(k) k√γk∧,P13 = iω(k)2 − ω02
P22 = Πk⊥ ,√ω0 γ
P24 = −Π ⊥,ω(k)2 − ω02 k
ω(k)ω0 γ
P34 = −iΠ ⊥,(ω(k)2 − ω02 )2 k

1
k∧,
ω(k)
√
γ
ω0
P14 =
k∧,
ω(k) ω(k)2 − ω02√
ω(k) γP23 = −iΠ ⊥,ω(k)2 − ω02 k
ω(k)2 γΠ ⊥,P33 =(ω(k)2 − ω02 )2 k
ω02 γ
Π ⊥,
P44 =
(ω(k)2 − ω02 )2 k
P12 = −

where Πk⊥ is the orthogonal projector onto the orthogonal plane to k; the remaining∗
blocks stem from the symmetry relations Pij = Pji.
It follows from these computations that in the case of a cubic nonlinearity12 (i.e.
f = 0 in (74)), F env (ε, u) is given for all u such that u = π±,± (k)u by
F env (ε, u) = 0, 0, −

T
γ 5/2
[(e · e)ē + 2|e|2 e], 0 ;(ω 2 − ω02 )3

12The treatment of other kinds of nonlinearities is absolutely similar

VARIANT OF THE FOCUSING NLS EQUATION

41

similarly,
π±,± (k)F env (ε, u) = ∗,
and

T
i
γ3ω
[(e · e)ē + 2|e|2 e], ∗, ∗ .
242
2
N (k) (ω − ω0 )Tγω 2e, ∗, ∗ .
222
(ω − ω0 )We can now write explicitly the general nonlinear Schrödinger equation (37) derived in Example 3.4 in the particular case of the Maxwell equations (73). Choosingω1 (·) = ω±,± (·) and therefore π1 (·) = π±,± (·) = ω(·), and remarking that the solution v to (37) remains polarized along π±,± (k) if it initially polarized, it is enoughto give an equation on its electric field component e (the components b, p♯ and q♯
being recovered as indicated above). Moreover, we can assume the k = kez , so thatthe electric field e only has transverse components e ∈ C2 (i.e. e = (eT , 0)T ) and(37) finally reduces to
π±,± A0 u = ∗, ω1


i ω ′ (k)∆⊥ + ω ′′ (k)∂z2 e
2
k
2
γω(k)
γ 3 ω(k)i+εp ω1e=[(e · e)ē + 2|e|2 e].
(ω(k)2 − ω02 )2
N (k)2 (ω(k)2 − ω02 )4(1 − iεb · ∇ − ε2 ∇ · B∇)∂t e −(77)

Proceeding to the following rescaling of the space variables1ω ′ (k) 1/2 ′ ′(x, y) =(x , y ) and z = | ω ′′ (k)|1/2 z ′2k2′′(the latter one only if ω (k) 6= 0), and rescaling e as
1/2
 1
γ 3 ω(k)e′ ,e=2N (k)2 3(ω(k)2 − ω0 )4
the above equation becomes
(78)


1
iP2 (ε∇)∂t e + ∆⊥ + α1 ∂z2 e + iα2 e + [(e · e)ē + 2|e|2 e] = 0
3
2

γω(k)
′′
where α1 = sgn(ω±,±(k)) ∈ {0, ±1}, α2 = εp ω1 (ω(k)
2 −ω 2 )2 and0

2P2 (ε∇) = 1 − iεM b · ∇ − ε ∇ · M BM ∇,′(k) −1/2 1 ′′(k) −1/2, ω2k, | 2 ω (k)|−1/2 . This equation is exactly
where M = diag ω2k
under the form (3)vect with α3 = 0 (i.e. without the terms modeling the frequency
dependence of the polarization).
′

B.1.2. With frequency dependent polarization. Let us now describe the modifications that have to be made in order to take into account a frequency dependent polarization, as in (41). Since initial data polarized along π±,± (k) conserve this polarization during the evolution in time, the only difference between′(41) and (37) is the presence of the terms επ±,± (k)π±,±(k) · DF env (ε, u) and−iε(b · ∇)π±,± (k)Fenv (ε, u). Only the first one requires a nontrivial computation. Thanks to the expression for π±,± (k) given above, we can write for all
F = (0, 0, f, 0),π±,± (k)F = (ig1 (k)k ∧ f, −ig2 (k)Πk⊥ f, g3 (k)Πk⊥ f, ig4 (k)Πk⊥ f )T ,

42

ÉRIC DUMAS, DAVID LANNES AND JÉRÉMIE SZEFTEL

with

√
γ
1
,
2
2
N (k) ω(k) − ω02√
ω(k) γg3 (k) =g2 (k),
ω(k)2 − ω02

g1 (k) =g2 (k) = ω(k)g1 (k),√ω0 γ
g4 (k) =g2 (k).
ω(k)2 − ω02

One has therefore, with h = kk · Df and l = dk (k′ 7→ Πk′ ⊥ ) · Df ,′π±,±(k) · DF=+(ig1′ (k)k ∧ h, −ig2′ (k)Πk⊥ h, g3′ (k)Πk⊥ h, ig4′ (k)Πk⊥ h)T(ig1 (k)D ∧ f, −ig2 (k)l, g3 (k)l, ig4 (k)l)T .

If f · k = 0, we can use the identity Πk⊥ dk (k′ 7→ Πk′ ⊥ ) · DΠk⊥ = 0 to deduce that
Tm(k) k′π±,± (k)π±,±(k) · DF = ∗, −· ∇f, ∗, ∗ ,2N (k) kwhere the ∗ can be easily computed, and wherem(k) =√
γk(ω(k)g3′ (k) − ω0 g4′ (k)).
(g1 (k) + kg1′ (k)) + g2′ (k) +2ω(k)ω(k) − ω02

The nonlinearity in the r.h.s. of (77) must therefore be replaced byγ 3 ω(k)i(1 − iεb · ∇ + iα3 ∂z )[(e · e)ē + 2|e|2 e],
N (k)2 (ω(k)2 − ω02 )4
ω 2 −ω 20with α3 = − √γω(k). The reduction to (3)vect then follows the same steps as for(78).

B.2. The case with charge and current density. According to (48) and moregenerally (49), ionizations effects are taken into account by addingI := −επ1 (k) iwC1T C1 u + c C1T Genv (C1 u, w)

to the right-hand side of the NLS equation, and by considering the following equation for w,∂t w = 2εGenv (C1 u, w) · C1 u.
In the particular case of the Maxwell equations (19), one has w = ρ andT
I = −επ1 (k) 0, iρe + c0 c1 |e|2K−2 + c2 ρ e, 0, 0 ;consequently, one must add to the right-hand side of (77) the ionization term−iρe − c0 c1 |e|2K−2 + c2 ρ e,so that after rescaling as for (78), the equation takes the form (54) (when written
in a fixed frame rather than the frame moving at the group velocity).
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Éric Dumas: Institut Fourier - Université Grenoble 1 - 100, rue des mathématiques BP 74 - 38402 Saint Martin d’Hères - France. Email address: Eric.Dumas@ujf-grenoble.fr
David Lannes: DMA - École Normale Supérieure - 45, rue d’Ulm - 75005 Paris - France.
Email address: David.Lannes@ens.fr
Jérémie Szeftel: Laboratoire Jacques-Louis Lions - Université Pierre et Marie Curie- 4, place Jussieu - 75005 Paris - France. Email address: Jeremie.Szeftel@upmc.fr