hamiltonian partial differential equations and …stolo/conf/hiver15/dubrovin15-1.pdfhamiltonian...

Hamiltonian partial differential equations and Painlevé transcendents

Boris DUBROVINSISSA, Trieste

Winter School on PDEsSt Etienne de TinéeFebruary 2 - 6, 2015

Sunday 8 February 15

• Cauchy problem for evolutionary PDEs with slow varying initial data

•Introduction into Hamiltonian PDEs

•On perturbative approach to integrability

• Phase transitions from regular to oscillatory behavior. Universality conjecture and Painlevé transcendents

Sunday 8 February 15

System

R.h.s. analytic in jets at

F (u; 0, 0, . . . ) ⌘ 0

(u; 0, 0, . . . )

and

) n-dimensional family of constant solutions (n=dim u)

Slow-varying solutions u(x) = f(✏x), ux

= O(✏), uxx

= O �✏

2�

Rescale x 7! ✏x, t 7! ✏ t and expand

ut

= F (u;ux

,uxx

, . . . )

ut

= 1✏

F�u; ✏u

x

, ✏2uxx

, . . .�

= A(u)ux

+ ✏⇥B1(u)uxx

+B2(u)u2x

⇤+ ✏2

⇥C1(u)uxxx

+ C2(u)uxx

ux

+ C3(u)u3x

⇤+ . . .

(1)

✏ -independent initial data u(x, t) = u0(x)

Sunday 8 February 15

Question 1: existence of solutions to (1) with slow varying!initial data

Idea: for small times solutions to (1)

ut

= A(u)ux

+✏⇥B1(u)uxx

+B2(u)u2x

⇤+✏2

⇥C1(u)uxxx

+ C2(u)uxx

ux

+ C3(u)u3x

⇤. . .(1)

and

ut

= A(u)ux (2)

with the same initial date are close

before the time of gradient catastrophe of (2)

Sunday 8 February 15

Question 2: comparison of solutions to (1) (“perturbed”)and (2) (“unperturbed”) near the point of gradient catastrophe of (2) = point of phase transition of (1)

Two main classes: dissipative perturbations, e.g., Burgers equation

ut

+ uux

= ✏uxx

or Hamiltonian (i.e., conservative) perturbations,e.g., Korteweg - de Vries equation

ut

+ uux

+ ✏2uxxx

= 0

shock waves;

oscillatory behaviour

Sunday 8 February 15

ut + u ux +ϵ2

12uxxx = 0

� = 0 �

ut + uux = 0

ϵ ut = v(x) − v(x − ϵ)ϵ vt = e

u(x+ϵ)− e

u(x)

!

(n = 2)

ut = vxvt = euux

�

Examples of Hamiltonian PDEs1) KdV

In the zero dispersion limit Hopf equation

2) Toda lattice

Long wave limit

, ut

+ @x

�H

�u(x)= 0 H =

Z u3

6� ✏2

24u2x

�dx

Sunday 8 February 15

H =N�

n=1

p2n2

+ V (qn � qn�1)

N

ut = 1� [v(x)� v(x� �)]

vt = 1� [V

�(u(x+ �))� V �(u(x)]

�

pn = v(n�), qn � qn�1 = u(n�), � =1

N

ut = vxvt = V ��(u)ux

�

More general class of systems of the Fermi-Pasta-Ulam type

For large the equations of motion can be replaced by

In the leading term one obtains an integrable PDE

Sunday 8 February 15

i� ⇥t +�2

2⇥xx + |⇥|2⇥ = 0

u = |⇥|2, v =�

2i

�⇥x

⇥� ⇥x

⇥

⇥

ut + (u v)x = 0

vt + v vx � ux +�2

4

�1

2

u2x

u2� uxx

u

⇥

x

= 0

3) Nonlinear Schrödinger equation

In the real-valued variables

can be recast into the form

Sunday 8 February 15

ut +⇥

⇥x

�H

�v(x)= 0

vt +⇥

⇥x

�H

�u(x)= 0

H =

⇧ ⇤1

2

�u v2 � u2

⇥+

�2

8uu2x

⌅dx

The Hamiltonian formulation

✓0 11 0

◆@

@x

Hamiltonian operator

Sunday 8 February 15

Class of systems of PDEs depending on a small parameter �

ut = A(u)ux + �A2(u;ux,uxx) + �2A3(u;ux,uxx,uxxx) + . . .

u = (u1, . . . , un)

ϵk

k + 1

deg u(m) = m, m = 1, 2, . . .

Terms of order are differential polynomals

of degree

General setting

Sunday 8 February 15

on the space of vector-functions

ε -dependent dynamical systems

u(x)

initial point

solution of the Cauchy problemu(x, t; ✏)

u0(x)

Sunday 8 February 15

Hamiltonian formulation

ut

= F (u;ux

,uxx

, . . . ) = P�H

�u(x)

Hamiltonian local functional

H =1

2⇡

Z 2⇡

0h (u;u

x

,uxx

, . . . ) dx

�H

�u(x)= Euler - Lagrange operator of

(the case of 2π-periodic function)

h

�H

�ui(x)=

@h

@ui

� @x

@h

@ui

x

+ @2x

@h

@ui

xx

� . . .

Sunday 8 February 15

Finally, in the representation

ut

= F (u;ux

,uxx

, . . . ) = P�H

�u(x)

is the matrix-valued operator of Poisson bracket

{F,G} =1

2⇡

Z 2⇡

0

�F

�u

i(x)P

ij �G

�u

j(x)dx

P =�P ij

�

• bilinearity

• skew symmetry {G,F} = � {F,G}

• Jacobi identity

{{F,G} , H}+ {{H,F} , G}+ {{G,H} , F} = 0

Sunday 8 February 15

For example, any linear skew-symmetric matrix operator with constant coefficients

P ij =X

k

P ij

k

@k

x

P jik = (�1)k+1P ij

k

defines a Poisson bracket

For KdV P = @x

, P =

✓0 11 0

◆@x

for NLS

More general classP ij =

X

k

P ij

k

(u;ux

, . . . )@k

x

Sunday 8 February 15

After rescalingx 7! ✏x obtain

P ij =X

k�0

✏kk+1X

s=0

P ij

k,s

(u;ux

, . . . ,u(s))@k�s+1x

Sunday 8 February 15

This class is invariant with respect of the group of

Miura-type transformations of the form

u ⇥� u = F0(u) + �F1(u,ux) + �2F2(u,ux,uxx) + . . .

degFk(u,ux, . . . ,u(k)) = k

det

�DF0(u)

Du

⇥�= 0

Sunday 8 February 15

Local Hamiltonians

Thm. Assuming any Hamiltonian operator

is equivalent to

Proof uses the theory of Poisson brackets of hydrodynamic type (B.D., S.Novikov, 1983) and triviality of Poisson cohomology (E. Getzler; F.Magri et al., 2001)

So, any Hamiltonian PDEs can be written in the form

H = H0 + ✏H1 + ✏2H2 + · · · =Z

[h0(u) + ✏h1(u;ux

) + ✏2h2(u;ux

,uxx

) + . . . ]dx

Sunday 8 February 15

Perturbative approach to integrability of PDEs(B.D., Youjin Zhang)

Integrable Hamiltonian system

Complete family of commuting first integrals

Definition. Perturbed Hamiltonianis integrable if every can be deformed

to a first integral of andSunday 8 February 15

Homological equation etc.

Remark. Regularity of commutativity of the centralizer

Example. Hopf equation

is integrable: commuting first integrals are

for an arbitrary function

Sunday 8 February 15

Proof.

where

An example of integrable deformation: KdV

Sunday 8 February 15

Claim. For any function f(u) there exists a first integral

of KdV equation

Sunday 8 February 15

An explicit formula in terms of Lax operator

L =�2

2

d2

dx2+ u(x)

Then

where

Sunday 8 February 15

More general second order perturbations

Thm. (B.D., J.Ekstrand) Integrability iff

Lax operator

It gives only half of the first integrals , for odd

H = H0 + ✏2H2 =

Z 1

6u3 +

✏2

12c(u)u2

x

�dx

Sunday 8 February 15

� � 0

The main goal: to compare the properties of solutions

with solutions to the “dispersionless limit

to the perturbed system

ut = A(u)ux + �A2(u;ux,uxx) + �2A3(u;ux,uxx,uxxx) + . . .

ut = A(u)ux

Back to general case

Sunday 8 February 15

� � 0

The main goal: to compare the properties of solutions

with solutions to the “dispersionless limit

• Hamiltonian

to the perturbed system

ut = A(u)ux + �A2(u;ux,uxx) + �2A3(u;ux,uxx,uxxx) + . . .

ut = A(u)ux

Back to general case

Sunday 8 February 15

� � 0

The main goal: to compare the properties of solutions

with solutions to the “dispersionless limit

• Hamiltonian

• completely integrable

to the perturbed system

ut = A(u)ux + �A2(u;ux,uxx) + �2A3(u;ux,uxx,uxxx) + . . .

ut = A(u)ux

Back to general case

Sunday 8 February 15

� � 0

The main goal: to compare the properties of solutions

with solutions to the “dispersionless limit

• Hamiltonian

• completely integrable

• finite life span (nonlinearity!)

to the perturbed system

ut = A(u)ux + �A2(u;ux,uxx) + �2A3(u;ux,uxx,uxxx) + . . .

ut = A(u)ux

Back to general case

Sunday 8 February 15

x0

� � 0

(x0, t0)

For the dispersionless system

a gradient catastrophe takes place:

for

the solution exists

, there exists the limit

but, for some

(x, t) � (x0, t0)for

The problem: to describe the asymptotic behaviour of the generic solution

to the perturbed system for

in a neighborhood of the point of catastrophe

t < t0

ut = A(u)ux

limt�t0

u(x, t)

ux(x, t), ut(x, t) � ⇥

u(x, t; �), u(x, 0; �) = u0(x)

Sunday 8 February 15

ut + u ux = 0

Gradient catastrophe for Hopf equation

Sunday 8 February 15

ut + u ux = ϵ uxxPerturbation: Burgers equation

(dissipative case)

Sunday 8 February 15

ut + u ux + ϵ2uxxx = 0Perturbation: KdV equation

(Hamiltonian case)

Sunday 8 February 15

The smaller is , the faster are the oscillations�

−4 −3.5 −3 −2.5 −2−1

−0.5

0

0.5

1

x

u

!=10−1.5

−4 −3.5 −3 −2.5 −2−1

−0.5

0

0.5

1

x

u

!=10−2

−4 −3.5 −3 −2.5 −2−1

−0.5

0

0.5

1

x

u

!=10−2.5

−4 −3.5 −3 −2.5 −2−1

−0.5

0

0.5

1

x

u

!=10−3

Sunday 8 February 15

�ut

vt

⇥+

�v u�1 v

⇥�ux

vx

⇥= 0 � = v ± i

�u

i� ⇥t +�2

2⇥xx + |⇥|2⇥ = 0

Nonlinear Schrödinger equation (the focusing case)

NB: the dispersionless limit is a PDE of elliptic type

, eigenvaluesSunday 8 February 15

Sunday 8 February 15

Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour

Sunday 8 February 15

Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour

(Universality)

Sunday 8 February 15

Universality for the generalized Burgers equation (A.Il’in 1985)

ut + a(u)ux = �uxx

u(x, t; ⌅) = u0 + ⇤ ⌅1/4�

⇤x� x0 � a0(t� t0)

� ⌅3/4,t� t0⇥ ⌅1/2

⌅+O

�⌅1/2

⇥

Here �(�, ⇥) is the logarithmic derivative of the Pearcey

function

�(�, ⇥) = �2⇤

⇤�log

� ⇥

�⇥e�

18 (z

4�2z2⇥+4z �)dz

Sunday 8 February 15

What kind of special functions is needed for the Hamiltonian case?

Painlevé-1 equation U 00 = 6U2 �X

and its 4th order analogue

X = T U −

!

1

6U

3+

1

24

"

U′2

+ 2U U′′#

+1

240U

IV

$

(P 2I )

(PI)

Painlevé property: any solution is a meromorphic function

in X 2 C

Sunday 8 February 15

Isomonodromy representation for P 2I

U =

✓0 �1

2U � 2� 0

◆

Sunday 8 February 15