Hamiltonian partial differential equations and Painlevé transcendents

Winter School on PDEs St Etienne de Tinée February 2-6, 2015 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN SISSA, Trieste

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 u t = F (u; u x, u xx,...) (1) R.h.s. analytic in jets at (u;0, 0,...) and F (u;0, 0,...) 0 ) n-dimensional family of constant solutions (n=dim u) Slow-varying solutions u(x) =f( x), u x = O( ), u xx = O 2 Rescale x 7! x, t 7! t and expand u t = 1 F u; u x, 2 u xx,... = A(u)u x + B 1 (u)u xx + B 2 (u)u 2 x + 2 C 1 (u)u xxx + C 2 (u)u xx u x + C 3 (u)u 3 x +... -independent initial data u(x, t) =u 0 (x)

Question 1: existence of solutions to (1) with slow varying! initial data Idea: for small times solutions to (1) u t = A(u)u x + B 1 (u)u xx + B 2 (u)u 2 x + 2 C 1 (u)u xxx + C 2 (u)u xx u x + C 3 (u)u 3 x... (1) and u t = A(u)u x (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 shock waves; u t + uu x = u xx or Hamiltonian (i.e., conservative) perturbations, e.g., Korteweg - de Vries equation oscillatory behaviour u t + uu x + 2 u xxx =0

Examples of Hamiltonian PDEs 1) KdV u t + uu x + ϵ2 12 u xxx =0, u t + @ x H u(x) =0 H = In the zero dispersion limit =0 Z apple u 3 6 2 24 u2 x dx Hopf equation u t + uu x =0 2) Toda lattice ϵu t = v(x) v(x ϵ) ϵv t = e u(x+ϵ) e u(x) } (n = 2) Long wave limit u t = v x v t = e u u x

More general class of systems of the Fermi-Pasta-Ulam type H = N n=1 p 2 n 2 + V (q n q n 1 ) For large N the equations of motion can be replaced by u t = 1 [v(x) v(x )] v t = 1 [V (u(x + )) V (u(x)] p n = v(n ), q n q n 1 = u(n ), = 1 N In the leading term one obtains an integrable PDE u t = v x v t = V (u)u x

3) Nonlinear Schrödinger equation i t + 2 2 xx + 2 =0 In the real-valued variables u = 2 x, v = 2i can be recast into the form x u t +(uv) x =0 v t + vv x u x + 2 4 1 2 u 2 x u 2 u xx u x =0

The Hamiltonian formulation u t + x v t + x H v(x) =0 H u(x) =0 H = 1 2 uv2 u 2 + 2 8u u2 x dx Hamiltonian operator 0 1 1 0 @ @x

General setting Class of systems of PDEs depending on a small parameter u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... u =(u 1,...,u n ) Terms of order ϵ k are differential polynomals of degree k +1 deg u (m) = m, m =1, 2,...

ε -dependent dynamical systems on the space of vector-functions u(x) u(x, t; ) solution of the Cauchy problem initial point u 0 (x)

Hamiltonian formulation u t = F (u; u x, u xx,...)=p H u(x) Hamiltonian local functional H H = 1 2 Z 2 0 h (u; u x, u xx,...) dx (the case of 2π-periodic function) u(x) = Euler - Lagrange operator of h H u i (x) = @h @u i @ x @h @u i x + @ 2 x @h @u i xx...

Finally, in the representation u t = F (u; u x, u xx,...)=p H u(x) P = P ij 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 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 ji k =( 1)k+1 P ij k defines a Poisson bracket 0 1 For KdV P = @ x, for NLS P = 1 0 @ x More general class P ij = X k P ij k (u; u x,...)@ k x

After rescaling x 7! x obtain P ij = X k 0 k+1 k X s=0 P ij k,s (u; u x,...,u (s) )@ x k s+1

This class is invariant with respect of the group of Miura-type transformations of the form u ũ = F 0 (u)+ F 1 (u, u x )+ 2 F 2 (u, u x, u xx )+... deg F k (u, u x,...,u (k) )=k det DF 0 (u) Du =0

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 Local Hamiltonians H = H 0 + H 1 + 2 H 2 + = Z [h 0 (u)+ h 1 (u; u x )+ 2 h 2 (u; u x, u xx )+...]dx

Perturbative approach to integrability of PDEs (B.D., Youjin Zhang) Integrable Hamiltonian system Complete family of commuting first integrals Definition. Perturbed Hamiltonian is integrable if every can be deformed to a first integral of and

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 d 2 dx 2 + u(x) Then where

More general second order perturbations H = H 0 + 2 H 2 = Z apple 1 6 u3 + 2 12 c(u)u2 x dx Thm. (B.D., J.Ekstrand) Integrability iff Lax operator It gives only half of the first integrals, for odd

Back to general case The main goal: to compare the properties of solutions to the perturbed system u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... with solutions to the dispersionless limit 0 u t = A(u)u x

Back to general case The main goal: to compare the properties of solutions to the perturbed system u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... with solutions to the dispersionless limit 0 Hamiltonian u t = A(u)u x

Back to general case The main goal: to compare the properties of solutions to the perturbed system u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... with solutions to the dispersionless limit 0 Hamiltonian u t = A(u)u x completely integrable

Back to general case The main goal: to compare the properties of solutions to the perturbed system u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... with solutions to the dispersionless limit 0 Hamiltonian u t = A(u)u x completely integrable finite life span (nonlinearity!)

For the dispersionless system u t = A(u)u x a gradient catastrophe takes place: the solution exists for t<t 0, there exists the limit lim u(x, t) t t 0 but, for some x 0 u x (x, t), u t (x, t) for (x, t) (x 0,t 0 ) The problem: to describe the asymptotic behaviour of the generic solution u(x, t; ), u(x, 0; )=u 0 (x) to the perturbed system for 0 in a neighborhood of the point of catastrophe (x 0,t 0 )

Gradient catastrophe for Hopf equation u t + uu x =0

Perturbation: Burgers equation u t + uu x = ϵu xx (dissipative case)

Perturbation: KdV equation u t + uu x + ϵ 2 u xxx =0 (Hamiltonian case)

The smaller is, the faster are the oscillations 1!=10 1.5 1!=10 2 0.5 0.5 u 0 u 0 0.5 0.5 1 4 3.5 3 2.5 2 x 1 4 3.5 3 2.5 2 x 1!=10 2.5 1!=10 3 0.5 0.5 u 0 u 0 0.5 0.5 1 4 3.5 3 2.5 2 1 4 3.5 3 2.5 2

Nonlinear Schrödinger equation (the focusing case) i t + 2 2 xx + 2 =0 u t v t NB: the dispersionless limit is a PDE of elliptic type + v u 1 v ux v x =0 = v ± i u, eigenvalues

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) u t + a(u)u x = u xx u(x, t; ) =u 0 + 1/4 x x0 a 0 (t t 0 ) 3/4, t t 0 1/2 + O 1/2 Here (, ) is the logarithmic derivative of the Pearcey function (, ) = 2 log e 1 8(z 4 2z 2 +4z ) dz

What kind of special functions is needed for the Hamiltonian case? Painlevé-1 equation U 00 =6U 2 X (P I ) and its 4th order analogue X = TU [ 1 6 U 3 + 1 24 ( U 2 +2UU ) + 1 240 U IV ] (P 2 I ) Painlevé property: any solution is a meromorphic function in X 2 C

Isomonodromy representation for P 2 I U = 0 1 2U 2 0