Hidden solutions ofdiscrete systems
Nalini Joshi@monsoon0

Supported by the Australian Research Council

The University of Sydney

Lustri 2015

Real part of tritronquée solution of first Painlevé equation
yxx = 6y 2 x!2Properties on Ry⇠r

1
Xx
ak,x!
1
5k/2
6x
k=0

• A one-parameter family of solution exists
with this behaviour on the positive realline, real for x 2 R• One unique solution is bounded, analytic
and monotonically decaying forx 2 R+

Joshi & Kitaev 2001!3

Coefficients of asymptotic series
ak = c✓25p8 6◆k((k1)!) (1 + o(1)) , k

1• The asymptotic series diverges.
• It provides information aboutboundaries of validity.
• Late terms have the formak ⇠↵ (2k + )2k+!4

Coefficients of asymptotic series
ak = c✓25p
8 6◆k((k1)!) (1 + o(1)) , k1

• The asymptotic series diverges.
• It provides information aboutboundaries of validity.
• Late terms have the formak ⇠↵ (2k + )2k+Singulant!4

In C

Image by H. Dullin• The unique solution extends to | arg(x)|  4⇡/5
• But the behaviour is completely different
for x 2 R!5

Do such solutions exist for discrete systems?

!6

The answer is yes*.

* dPI Joshi & Lustri 2015
* dPII Joshi, Lustri & Luu 2017
* a-dPI Joshi & Takei 2017
* qPI Joshi, Lustri & Luu 2018!7

q-discrete first Painlevé equation
1
ww =w

1
xw2w = w(qx), w = w(x), w = w(x/q)• For each x 2 C the compactified &regularised initial value space is a foliated(1)
Avector bundle with each fibre equivalent to 7
• But the foliation fails in in the limit x ! 1
when two base points merge.
!8Solutions?
• Nishioka (2010) proved that qPI has nosolution expressible in terms of solutions offirst-order difference equations.
• So it has no rational solutions or qhypergeometric-type solutions.
• The only explicit information about solutions
we have is their asymptotic behaviours.

9Asymptotic behaviours• qPI has an unstable (quicksilver*) solutionw ⇠ 1/x, |x| ! 1• Its asymptotic expansion is divergent,hiding exponentially small terms withfree coefficients.
*Joshi 2015!10

How does it behave in C ?

!11

Rescalingx = x0 q n , s = ✏ n, q = 1 + ✏, w(x) = W (s)2W (s + ✏)W (s) W (s✏) = W (s)1x0 (1 + ✏)s/✏

To leading order as ✏ ! 0 we have
W (s) ⇠ W0 (s)W04 = W0e s!12

Asymptotic expansion
W (s) ⇠1X

✏2r Wr (s) +r=01
X
r=1✏2r+1 Vr (s), ✏ ! 0

For simplicity, we focus on one of the 4
behaviours given by
W04 = W0e sJ., Luu and Lustri, arXiv:1807.00450 [math-ph] 2018!13

Late order terms
8>><Wr (s)>>:Vr (s)F (s) (2r + 1 )⇠,2r+1⇠(s)
G(s) (2r + 2 )⇠µ(s)2r+ 2r

1

It can be shown that the even coefficientsdominate the odd ones in the limit.

!14

The Singulant
Matching terms of order Wr (s)cosh(which implies⇠ ± (s) = ±withgives

1 2W03⇠s ) = 2W 30Z s

cosh 1(t) dt

s0(s) =12W02
2W03
!15

The Prefactor
Matching terms of order Wr 1/2 (s) gives
F (s) =⌥W0 (s)(s)2

1

and matching near singularity gives

1/4

1 =1/2 .

!16

Optimal truncation
For each s, there is a smallest term in the series,occurring at
1N ⇠ |⇠|/✏ +  ,  2 [0, 1)
2
We truncate at this term and write
2NX1
W (s) =✏r Wr (s) + RN (s)r=0

and matching near singularity gives.

RN (s) ⇠ S (s)F (s)e ⇠/✏Stokes multiplier!17

D0 = {s 2 C =(s) 2 ( ⇡, ⇡]},F (s) =3 (s)Im(s)¬e3 /✏: exp. smalle

3 /✏: exp. largee

3 /✏: exp. smallRe(s)e3 /✏: exp. smalle

3 /✏: exp. large

Stokes curvee

3 /✏: exp. large

anti-Stokes curve!18Region of validity
Im(s)Exp. Large Contribution
Exp. Small Contribution
ValidValidFor a generic value of S.

Re(s)

!19

Extended region of validity
Im(s)Exp. Large Contribution
Exp. Small Contribution
No Exp. Contribution
Re(s)
Valid

For a special value of S=0.

!20

Region of validity in x
Im(x)

Re(x)

For a generic value of S!21

Extended region of validity in x
Im(x)

Re(x)

For a special value of S=0.

!22

Type B solutions• There also exists another type of solution, which matches withthe quicksilver solution.
• Its Stokes structure (indicated below) is more complicated.
Im(s)Region I¡¬ Re(s)¿Region II
Region III
Region IV!23

Summary• Asymptotic behaviours suggest there exist“hidden” solutions of discrete systems.
• Their asymptotic expansions are divergent.
• Estimates of late-order terms show Stokes
phenomena in C .
• The results suggest the existence of uniquesolutions with interesting properties.
• These are analogous to tritronquée solutionsof the Painlevé equations.
!24