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


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)


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)


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


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


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


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


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


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


on the space of vector-functions

ε -dependent dynamical systems

u(x)

initial point

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

u0(x)


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

� . . .


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


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


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


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


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


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


Proof.

where

An example of integrable deformation: KdV


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

of KdV equation


An explicit formula in terms of Lax operator

L =�2

2

d2

dx2+ u(x)

Then

where


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


� � 0

The main goal: to compare the properties of solutions

with solutions to the “dispersionless limit

to the perturbed system


ut = A(u)ux

Back to general case


� � 0



• Hamiltonian



ut = A(u)ux



� � 0



• Hamiltonian

• completely integrable



ut = A(u)ux



� � 0



• Hamiltonian

• completely integrable

• finite life span (nonlinearity!)



ut = A(u)ux



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)


ut + u ux = 0

Gradient catastrophe for Hopf equation


ut + u ux = ϵ uxxPerturbation: Burgers equation

(dissipative case)


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

(Hamiltonian case)


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


�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

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


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

(Universality)


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


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


Isomonodromy representation for P 2I

U =

✓0 �1

2U � 2� 0

◆


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