Hamiltonian partial differential equations and Painlevé transcendents

Transcription

1 The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 22-26, 2014 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN SISSA, Trieste 1

2 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 2

3 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

4 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) 4

5 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 5

6 Examples of Hamiltonian PDEs 1) KdV u t + uu x + ɛ2 12 u xxx =0, u t x H u(x) =0 H = In the zero dispersion limit =0 Z apple u u2 x dx Hopf equation u t + uu x =0 2) Toda lattice ɛu t = v(x) v(x ɛ) ɛv t = e u(x+ɛ) e u(x) } (n = 2) Long wave limit u t = v x v t = e u u x 6

7 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 7

8 3) Nonlinear Schrödinger equation i t 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 u 2 x u 2 u xx u x =0 8

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

10 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 ɛ k are differential polynomals of degree k +1 deg u (m) = m, m =1, 2,... 10

11 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) 11

12 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 i x i xx... 12

13 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 13

14 For example, any linear skew-symmetric matrix operator with constant coefficients P ij = X k P ij x P ji k =( 1)k+1 P ij k defines a Poisson bracket 0 1 For KdV P x, for NLS P = 1 x More general class P ij = X k P ij k (u; u x,...)@ k x 14

15 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 15

16 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 16

17 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 H 2 + = Z [h 0 (u)+ h 1 (u; u x )+ 2 h 2 (u; u x, u xx )+...]dx 17

18 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 18

19 Homological equation etc. Remark. Regularity of commutativity of the centralizor Example. Hopf equation is integrable: commuting first integrals are for an arbitrary function 19

20 Proof. where An example of integrable deformation: KdV 20

21 Claim. For any function f(u) there exists a first integral of KdV equation 21

22 An explicit formula in terms of Lax operator L = 2 2 d 2 dx 2 + u(x) Then where 22

23 More general second order perturbations H = H H 2 = Z apple 1 6 u c(u)u2 x dx Thm. (B.D., J.Ekstrand, Di Yang) Integrability iff Lax operator It gives only half of the first integrals, for odd 23

24 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 24

25 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 24

26 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 24

27 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!) 24

28 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 ) 25

29 Gradient catastrophe for Hopf equation u t + uu x =0 26

30 Perturbation: Burgers equation u t + uu x = ɛu xx (dissipative case) 27

31 Perturbation: KdV equation u t + uu x + ɛ 2 u xxx =0 (Hamiltonian case) 28

32 The smaller is, the faster are the oscillations 1!= != u 0 u x x 1!= != u 0 u x x 29

33 Nonlinear Schrödinger equation (the focusing case) i t 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 30

34 31

35 Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour 32

36 Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour (Universality) 32

37 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 33

38 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 ( U 2 +2UU ) U IV ] (P 2 I ) 34

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

40 apple d d W( ), d dx U( ) =0, P 2 I apple d dt V( ), d dx U( ) =0, KdV So, solutions to P 2 I satisfy KdV X = TU 1 6 U U 02 +2UU U IV U T + UU X U XXX =0 9 = ; U = U(X, T) 36

41 Digression. Consider an analogue of P 2 I with constant coefficients a, b, c a = bu + c 1 2 U apple 1 24 U 00 6 U U 02 +2UU U IV Thm. (S.P.Novikov; P.Lax, 1974) Any smooth solution to this ODE is a two-gap potential of Schrödinger operator E.g., take a = 9 L = 1 2 d 2 dx 2 + U(X) 40 g 3, b = 7 40 g 2, c =0, then U = 3} (X; g 2,g 3 ) is a solution 37

42 Solving KdV equation U T + UU X U XXX =0 with initial data U(X, 0) = 3}(X) one obtains a family of two-gap potentials For T=0 the three poles merge Only triple collisions! 38

43 Back to P 2 I X = TU [ 1 6 U ( U 2 +2UU ) U IV ] Looking for common solutions with KdV U T + UU X U XXX =0 Lemma (B.D., A.Kapaev) Poles of satisfy (cf. S.Shimomura 2001) 39

44 Proof by substitution of Laurent expansion Poles of second type of solutions to P 2 I equation are not compatible with KdV dynamics. They correspond to triple collisions 40

45 I.e., the function has a triple branch point at So the equation for pole dynamics does not satisfy the Painlevé property Triple branch points depends on the choice of a solution, hence movable critical singularities; to be determined from a spectral problem for Schrödinger equation with quintic potential 41

46 NB: the equation also admits an isomonodromy representation!, 42

47 Thm. (T.Claeys, M.Vanlessen, 2008) For any real T there exists unique solution U(X, T) smooth for all X 2 R satisfying to P 2 I U(X, T) ' (6X) 1/3 2 2/3 T, X!1 (3X) ( 1/3) 43

48 The Whitham approximation requires an improvement 0.8!= != u u x x KdV Whitham (T.Grava, C.Klein, 2005) 44