On Hamiltonian perturbations of hyperbolic PDEs

Transcription

1 Bologna, September 24, 2004 On Hamiltonian perturbations of hyperbolic PDEs Boris DUBROVIN SISSA (Trieste) Class of 1+1 evolutionary systems w i t +Ai j (w)wj x +ε (B i j (w)wj xx Ci jk (w)wj x wk x i = 1,..., n ε small parameter ) +O(ε 2 ) = 0 ε-expansion: Coefficient of ε m is a polynomial in w x, w xx,..., w (m+1) of the degree m + 1 deg w (k) = k, k 1 1

2 Perturbations of hyperbolic system v i t + A i j (v)vj x = 0, i = 1,..., n eigenvalues of ( A i j (v)) are real and distinct for any v = (v 1,..., v n ) ball R n. Particular class: systems of conservation laws v i t + x φ i (v) = 0, i = 1,.... Main goal: study of Hamiltonian perturbations of hyperbolic systems 2

3 Example 1 (Weakly dispersive) KdV w t + w w x + ε2 12 w xxx = 0 Example 2 Toda lattice Continuous version: q n = e q n+1 q n e q n q n 1. u n := q n+1 q n = u(nε), v n := q n = v(nε), t ε t u t = v(x + ε) v(x) ε v t = eu(x+ε) e u(x) ε = v x ε v xx + O(ε 2 ) = e u u x ε (eu ) xx + O(ε 2 ) Example 3 Camassa - Holm equation w t = ( { [ ) 1 ε 2 x w w x ε 2 w x w xx + 1 ]} 2 w w xxx = 3 ( 2 w w x + ε 2 w w xxx + 7 ) 2 w xw xx + O(ε 4 ) 3

4 Equivalencies: w i w i = f i 0 (w) + k 1 ε k f i k (w; w x,..., w (k) ) deg f i k (w; w x,..., w (k) ) = k det ( f i 0 (w) w j ) 0. f i k (w; w x,..., w (k) ) polynomials in derivatives 4

5 Questions: structure of solutions for t < tc t tc t > tc 5

6 Step 1: small t, quasitriviality. Locality of perturbative expansion Example 1 Riemann wave KdV v t + v v x = 0 w t + w w x + ε2 12 w xxx = 0 The substitution w = v + ɛ x (log v x ) ( v IV +ɛ 4 2 x 1152 v 2 x 7 v xxv xxx 1920 v 3 x ) + v3 xx + O(ɛ 6 ). 360 vx 4 Baikov, Gazizov, Ibragimov, 1989 So, for small t the solution to the Cauchy problem with smooth monotone initial data behaves like w(x, t) = v(x, t) + O(ε 2 ) v(x, t) defined by x = v t f(v) Universality for Riemann wave equation: near the point of gradient catastrophe any solution locally behaves as (A 2 singularity at (x, t) = (0, 0)) x = v t v3 6 6

7 Example 3 Riemann wave Camassa-Holm ]) v t = 3 2 v v x w t = (1 ε 2 2 x ) 1 ( 3 2 w w x ε 2 [w x w xx w w xxx ( v w = v + ɛ 2 vxx x v ) x 3 v x 6 +ɛ 4 x ( 7 v 2 xx 45 v x + 45 v v3 xx 16 v 3 x + 37 v2 vxx 2 v xxx 30 vx 4 + v2 v V 18 vx 2 31 v2 v xx v IV 90 vx 3 Lorenzoni, v2 vxxx 2 30 vx ) 3 45 v2 v 4 xx 32 v 5 x + O(ε 6 ) + 5 v viv 18 v x v xxx 8 59 v v xx v xxx 90 v 2 x 7

8 General results: (B.D., S.-Q.Liu, Y.Zhang) Any Hamiltonian PDE system of conservation laws w i t + x ψ i (w; w x,... ; ε) = 0, i = 1,... n Any bihamiltonian PDE is quasitrivial. The solution can be reduced to solving linear systems 8

9 Explicit construction: under assumptions of existence of a tau-function and linear action of the Virasoro symmetries onto the tau-function τ τ + δ L m τ + O(δ 2 ), m 1 All solutions regular in ε obtained from the vacuum solution L m τ = 0, m 1 by shifts along the times of the hierarchy (completeness needed!). Parametrized by Frobenius manifolds n(n 1)/2 parametric family of integrable hierarchies (integrable hierarchies of the topological type) 9

10 Example n = 2 One-dimensional polytropic gas equation of state p = u t + ( u 2 κ κ+1 ρκ+1 : 2 + ρκ ) ρ t + (ρ u) x = 0 x = 0 Bihamiltonian structure (Olver, 1980) {u(x), u(y)} [0] λ {u(x), ρ(y)} [0] λ {ρ(x), ρ(y)} [0] λ = 2ρκ 1 (x) δ (x y) + ( ρ κ 1) x δ(x y), = (u(x) λ) δ (x y) + 1 κ v (x) δ(x y), = 1 κ ( 2 ρ(x) δ (x y) + ρ (x) δ(x y) ) 10

11 Integrable bihamiltonian perturbation { [ u u 2 κ 2 t + x 2 + ρκ + ɛ 2 8 ρκ 3 ρ 2 x + κ ] 12 ρκ 2 ρ xx +ɛ 4 (κ 2)(κ 3) [ a 1 ρ 4 u 2 x ρ 2 x + a 2 ρ κ 6 ρ 4 x+ a 3 ρ 3 u xx u x ρ x + a 4 ρ 2 u 2 xx + a 5 ρ 3 u 2 x ρ xx + a 6 ρ κ 5 ρ 2 x ρ xx +a 7 ρ κ 4 ρ 2 xx + a 8 ρ 2 u x u xxx + a 9 ρ κ 4 ρ x ρ xxx ] } +ɛ 4κ (κ2 1) (κ 2 4) ρ κ 3 ρ xxxx = O(ɛ 6 ), 360 { ( ρ (2 κ)(κ 3) t + x ρ u + ɛ 2 u x ρ x + 1 ) 12 κ ρ 6 u xx +ɛ 4 (κ 2)(κ 3) [ b 1 ρ 4 u x ρ 3 x + b 2 ρ 3 ρ 2 x u xx +b 3 ρ 3 u x ρ x ρ xx + b 4 ρ 2 u xx ρ xx + b 5 ρ 2 u xxx ρ x +b 6 ρ 2 u x ρ xxx + b 7 ρ 1 u xxxx ]} = O(ɛ 6 ) 11

12 The coefficients are given by a 1 = κ 59 κ κ κ 3, a 2 = κ κ2 182 κ κ κ 3 a 3 = 6 19 κ 11 κ2 4 κ κ 3, a 4 = 6 5 κ + 13 κ κ 3, a 5 = κ 7 κ κ 2 a 6 = κ 245 κ2 61 κ κ κ 2, a 7 = κ + 15 κ2 + 5 κ κ κ 2 a 8 = κ, a 9 = κ 240 b 1 = κ 97 κ κ κ 3, b 2 = κ + 47 κ2 10 κ κ 3 b 3 = κ 5 κ2 + 2 κ κ 3, b 4 = 6 4 κ + κ2 180 κ 2, b 5 = 6 + κ + κ2 720 κ 2, b 6 = 6 + κ + κ2 720 κ 2, b 7 = κ 12

13 Step 2: Critical behavior For KdV w = v + ɛ x (log v x ) ( v +ɛ 4 x 2 IV 7 v xxv xxx 1152 vx vx 3 Near critical point v x 1/ε, ) + v3 xx + O(ɛ 6 ). 360 vx 4 v xx 1/ε 2,..., v (m) ε m All terms of the same order. Resummation needed Problem 1: Prove that near critical point the solution behaves as (universality) u ε 2 7 U ( ) x t ε 6/7, ε 4/7 + O ( ε 4 ) 7 where U(X, T ) is the unique smooth solution to the ODE [ U 3 X = T U (U 2 + 2U U ) + U IV ] 240 depending on the parameter T. Prove existence (cf. Brezin, Marinari, Parisi 1992) of such a solution (see also Kudashev, Suleimanov) 13

14 Step 3: After phase transition: oscillatory behavior. Gurevich, Pitaevski 1973, Whitham asymptotics (leading term) Problem 2 Determine full asymptotic behavior of averaged quantities, ε 0 Example Hermitean matrix integrals Z N (λ; ɛ) = 1 Vol(U N ) N N e 1 ɛ Tr V (A) da V (A) = 1 2 A2 λ k A k k 3 as function of N = x/ɛ, λ is a tau-function of Toda lattice Tau-function u = log v = ɛ t 0 log τ(x + ɛ)τ(x ɛ) τ 2 (x) τ(x + ɛ). τ(x) 14

15 Large N small ɛ expansion of τ(x, t; ɛ) = Z N (λ; ɛ) x = N ɛ, t k = (k + 1)!λ k+1 has the form log τ = ɛ 2g 2 F g (x, t) g 0 so the solution u, v admits regular expansion u = ɛ k u k (x, t) k 0 v = ɛ k v k (x, t) k 0 16

16 For small λ the ɛ-expansion can be obtained by applying the saddle point method to Z N = 1 Vol(U N ) e 1 ɛ TrV (A) da F g (x, t) = generating function of numbers of fat graphs on genus g Riemann surfaces Corresponds to the one-cut asymptotic distribution of the eigenvalues of the large size Hermitean random matrix A 17

17 Multicut case: gaps in the asymptotic distribution of eigenvalues of random matrices singular behaviour of the correlation functions (terms e i a ɛ t arise) (from Jurkiewicz, Phys. Lett. B, 1991) Smoothed correlation functions: average out the singular terms Question: Which integrable PDEs describe the large N expansion of smoothed correlation functions? 18

18 Example 2 n = 2, F (u, v) = 1 2 u v2 + e u The Frobenius manifold M 2 = { λ(p) = e p + v + e u p} =symbol of the difference Lax operator L = Λ + v + e u Λ 1, Λ = e ɛ x Extended Toda hierarchy (G.Carlet, B.D., Y.Zhang) ɛ L t k = ɛ L s k = 2 k! 1 [ (L k+1 ) +, L ] (k + 1)! [ (L k (log L c k ) )+ ], L c k = k s 0 = x, other times s 1, s 2,... are new. Remark Interchanging time/space variables x = s 0 t 0 = x transforms Toda NLS 20

19 Back to matrix models (B.D., T.Grava, in progress) Claim Substituting τ vac Toda (t 0, t 1, t 2,... ; s 0, s 1, s 2,... ; ɛ) t 0 = 0, t 1 = 1, t k = (k + 1)!λ k+1,, k 2 s 0 = x, s k = 0, k 1 one obtains F := log τ vac Toda (0, 1, 3!λ 3, 4!λ 4,... ; x, 0,... ; ɛ) = x2 2ɛ 2 ( log x 3 ) log x+ ( ) ɛ 2g 2 B 2g g 2 x 2g(2g 2) + g 0 ɛ 2g 2 F g (x; λ 3, λ 4,...) 21

20 = n h = 2 2g and where F g (x; λ 3, λ 4,...) k 1,...,k n a g (k 1,..., k n )λ k1... λ kn x h, ( n k 2 ) a g (k 1,..., k n ) = Γ, k = k k n, 1 # Sym Γ Γ = a connected fat graph of genus g with n vertices of the valencies k 1,..., k n. E.g.: genus 1, one vertex, valency 4 22

21 [ ( 1 F = ɛ 2 2 x2 log x 3 ) + 6x 3 λ x 3 λ x 4 λ 2 3 λ x 4 2 λ x 5 λ x 4 λ 3 λ x 5 λ 3 λ 4 λ 5 +90x 5 λ x 6 λ 4 λ x 4 λ x 5 λ 3 2 λ x 5 λ 4 λ x 6 λ 4 2 λ x 6 λ 3 λ 5 λ x 7 λ 5 2 λ x 6 λ x 7 λ 4 λ x 8 λ 6 3 ] 1 12 log x+3 2 xλ 3 2 +xλ x 2 λ 3 2 λ 4 +30x 2 λ x 3 λ x 2 λ 3 λ x 3 λ 3 λ 4 λ x 3 λ x 4 λ 4 λ x 2 λ x 3 λ 3 2 λ x 3 λ 4 λ x 4 λ 4 2 λ x 4 λ 3 λ 5 λ x 5 λ 5 2 λ x 4 λ x 5 λ 4 λ x 6 λ 6 3 +ɛ 2 [ 1 240x xλ xλ 3 λ 4 λ xλ x 2 λ 4 λ xλ 3 2 λ xλ 4 λ x 2 λ 4 2 λ x 2 λ 3 λ 5 λ x 3 λ 5 2 λ x 2 λ x 3 λ 4 λ x 4 λ 6 3 ]

22 Proof uses Toda equations and the large N expansion for the Hermitean matrix integral ( t Hooft; D.Bessis, C.Itzykson, J.-B.Zuber) Z N (λ; ɛ) = 1 Vol(U N ) N N e 1 ɛ Tr V (A) da V (A) = 1 2 A2 λ k A k k 3 where one has to replace N x ɛ Remark This is the topological solution for the (extended) nonlinear Schrödinger hierarchy 24

23 Multicut case: G gaps in the spectrum of random matrices Claim: The full large N expansion of the smoothed correlation functions is given via the topological tau function associated with the Frobenius structure M n, n = 2G + 2 on the Hurwitz space of hyperelliptic curves µ 2 = 2G+2 i=1 (λ u i ) 25

24 Recall the general construction: Frobenius structure on the Hurwitz space M n = moduli of branched coverings λ : Σ G P 1 fixed degree, genus G, ramification type at infinity, basis of a- and b-cycles (n = number of branch points λ = u i for generic covering). Must choose a primary differential dp (say, holomorphic differential with constant a- periods) Then, for any two vector fields 1, 2 on M n the inner product 1, 2 = n i=1 res λ=ui 1 (λdp) 2 (λdp) dλ for any three vector fields 1, 2, 3 on M n 1 2, 3 = n i=1 res λ=ui 1 (λdp) 2 (λdp) 3 (λdp) dλ dp 26

25 Example G = 1 (two-cut case). Here n = 4. Flat coordinates on the Hurwitz space of elliptic double coverings with 4 branch points are u, v, w, τ. Can describe by the superpotential (= symbol of Lax operator) λ(p) = v + u ( log θ 1(p w τ) θ 1 (p + w τ) The Frobenius structure given by F = i 4π τ v2 2 u v w + u 2 log [ 1 π u ) θ 1 (2w τ) θ 1 (0 τ) ] Recall [ ] θ1 (x τ) log π θ 1 (0 τ) = log sin π x + 4 m=1 q 2m sin 2 π m x 1 q 2m m q = e i π τ 27

26 Corresponding integrable hierarchy of the topological type for the functions u, v, w, τ, four infinite chains of times t u,p, t v,p, t w,p, t τ,p. Then Z τ vac with t w,1 t w,1 1, t w,0 = 0, t w,k = (k+1)!λ k+1 other couplings = 0. t u,0 = x 28

27 Problem 3 Critical and after critical behaviour for Camassa - Holm? 29