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 Lecture 2 1

2 Recall: the main goal is 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 )+... u(x, 0; )= (x) with solutions to the dispersionless limit 0 v t = A(v)v x v(x, 0) = (x) 2

3 Recall: the main goal is 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 )+... = 1 x H[u] u(x) u(x, 0; )= (x) with solutions to the dispersionless limit 0 v t = A(v)v x v(x, 0) = (x) 2

4 Recall: the main goal is 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 )+... = 1 x H[u] u(x) u(x, 0; )= (x) with solutions to the dispersionless limit 0 v t = A(v)v x = 1 H 0 [v] x v(x) v(x, 0) = (x) 2

5 Recall: the main goal is 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 )+... = 1 x H[u] u(x) u(x, 0; )= (x) with solutions to the dispersionless limit 0 v t = A(v)v x = 1 H 0 [v] x v(x) v(x, 0) = (x) H = H 0 + H H

6 Of particular interest is the comparison near the point of gradient catastrophe of the dispersionless system v t = A(v)v x i.e., such a point (x 0,t 0 ) that the solution exists for t<t 0, there exists the limit lim v(x, t) t t 0 but, for some x 0 v x (x, t), v 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 ) 3

7 Phase transition in KdV equation u t + uu x + ɛ 2 u xxx =0 4

8 Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour 5

9 Main Conjecture (B.D., 2005): a finite list of types of the critical behaviour (Universality) 5

10 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) 6

11 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) 6

12 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) 6

13 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) 6

14 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution 6

15 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

16 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

17 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

18 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function small oscillations Painlevé-II (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

19 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function small oscillations Painlevé-II (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

20 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function small oscillations Painlevé-II (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 solitonic asymptotics u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

21 KdV asymptotics u t + uu x u xxx =0 u(x, t =0, )= (x) S(x, t) u(x, t, ) U ell ; r1 (x, t),r 2 (x, t),r 3 (x, t) A.Gurevich, L.Pitayevski 73 P.Lax, D.Levermore 83 U ell (z; r 1,r 2,r 3 ) Weierstrass elliptic function small oscillations Painlevé-II (U ell ) 2 + 4(U ell r 1 )(U ell r 2 )(U ell r 3 )=0 solitonic asymptotics T.Claeys, T.Grava u(x, t, )=v(x, t)+o( ) where v t + vv x =0 v(x, 0) = (x) point of gradient catastrophe for the Hopf solution v(x, t) 6

22 Universality Conjecture, case of scalar PDEs u t + a(u)u x + 2 c(u)u xxx + b(u)u xx u x + d(u)u 3 x + O 3 =0 u(x, 0; )= (x) Compare with solutions to the unperturbed equation with the same initial condition v t + a(v) v x =0 v(x, 0) = (x) Suppose v(x, t) has a generic gradient catastrophe at (x 0,t 0 ) v 0 := v(x 0,t 0 ) 7

23 Part 1. For t<t 0 u(x, t; ) v(x, t) as 0 Part 2. There exists = ( ) > 0 such that the solution u(x, t; ) can be extended till t = t 0 + Part 3. The limit lim 0 2/7 [u(x, t; ) v 0 ] x = x 0 + a 0 4/7 T + 6/7 X t = t 0 + 4/7 T exists and depends neither on the choice of generic solution nor on the choice of generic Hamiltonian perturbation = 2/7 1/7, = 1/7 3/7, = 3/7 2/7 = (a 0 t 0 + f 0 ) where f ( (x)) x, = 12c 0 a 0 8

24 Part 1. For t<t 0 u(x, t; ) v(x, t) as 0 Part 2. There exists = ( ) > 0 such that the solution u(x, t; ) can be extended till t = t 0 + Part 3. The limit lim 0 2/7 [u(x, t; ) v 0 ] =: U(X, T) x = x 0 + a 0 4/7 T + 6/7 X t = t 0 + 4/7 T exists and depends neither on the choice of generic solution nor on the choice of generic Hamiltonian perturbation = 2/7 1/7, = 1/7 3/7, = 3/7 2/7 = (a 0 t 0 + f 0 ) where f ( (x)) x, = 12c 0 a 0 8

25 Equivalently, for the the case of scalar Hamiltonian PDEs at the point of phase transition x = x 0,t= t 0,u= u 0 the generic solution has the following asymptotics u(x, t) =v 0 + 2/7 U x a0 (t t 0 ) x 0 6/7, t t 0 4/7 + O 4/7 where U(X, T) is a particular solution to the ODE X = TU [ 1 6 U ( U 2 +2UU ) U IV ] P 2 I (a differential equation in X depending on the parameter T) 9

26 Proof of Part 1 (small time behaviour): D.Masoero and A.Raimondo u t = a(u)u x + k i=1 iu (2i+1) u(x, 0; )= (x) H s, s 2k +1 Thm. The solution is continuous in =( 1,..., k) R k 10

27 The smooth solution X = TU U(X, T) [ 1 6 U to P 2 I ( U 2 +2UU ) U IV ] Proof of existence: T.Claeys, M.Vanlessen,

28 Comparison of KdV with P 2 I 0 t= t= t=0.218 u!0.5 u!0.5 u!0.5!1!1.7!1.65!1.6!1.55!1.5 x t= !1!1.7!1.65!1.6!1.55!1.5 x t= !1!1.7!1.65!1.6!1.55!1.5 x t= u!0.5 u!0.5 u!0.5!1!1.7!1.65!1.6!1.55!1.5 x t= !1!1.7!1.65!1.6!1.55!1.5 x t= !1!1.7!1.65!1.6!1.55!1.5 x t= u!0.5 u!0.5 u!0.5!1!1.7!1.65!1.6!1.55!1.5 x!1!1.7!1.65!1.6!1.55!1.5 x!1!1.7!1.65!1.6!1.55!1.5 x 12

29 A non-integrable case: the Kawahara equation u t +6uu x + 2 u xxx 4 u xxxxx =0 versus P 2 I 13

30 Motivations (for a scalar Hamiltonian equation) u t + a(u)u x + 2 c(u)u xxx + b(u)u xx u x + d(u)u 3 x + O 3 =0 The solution to a Cauchy problem for the leading term v t + a(v) v x =0 v(x, 0) = (x) can be written in the implicit form x = a(v) t + f(v), f ( (x)) x Point of gradient x 0 = a(v 0 ) t 0 + f(v 0 ) catastrophe (x 0,t 0,v 0 ) such that 0=a (v 0 ) t 0 + f (v 0 ) 0=a (v 0 ) t 0 + f (v 0 ) 14

31 Local structure near the point of catastrophe v(x, t 0 ) v (x 0,v 0 ) x 15

32 Step 1: near the generic point of gradient catastrophe the solution to the dispersionless equation can be approximated by where x + a 0 v t 1 6 f 0 v 3 x = x x 0 a 0 (t t 0 ) t = t t 0 v = v v 0 a 0 = a(v 0 ), a 0 = a (v 0 ) etc. After rescaling x x t 2/3 t v 1/3 v one obtains the above cubic equation modulo O 1/3, 0 16

33 Step 2: replacing the original dispersive equation by KdV near the point (x 0,t 0,v 0 ) u t + a(u)u x + 2 c(u)u xxx + b(u)u xx u x + d(u)u 3 x + O 3 =0 Galilean transformation + rescaling yields x = x x 0 a 0 (t t 0 ) x t = t t 0 2/3 t ū = u v 0 1/3 ū 7/6 ū t + a 0 ū ū x + 2 c 0 ū x x x 0, c 0 = c(v 0 ) modulo O 1/3, 0 17

34 The calculation: after rescaling u t 1/3 u t A monomial in the rhs m u (i 1)...u (i k) i i k = m +1 So m u (i 1)...u (i k) 7 6 m+ 1 3 k (i 1+ +i k ) m u (i 1)...u (i k) = m 6 + k 3 1 m u (i 1)...u (i k) Must have m 6 + k or m +2k 4 Since m 2 m =2, k =1 Only one monomial m u (i 1)...u (i k) = 2 u xxx 18

35 Step 3: choosing a particular solution to KdV The trick: to replace PDE (the KdV) + initial condition by an ODE ( string equation ) u t + uu x u xxx =0 u(x, 0; )= (x) First, rewrite the solution x = vt+ f(v), f ( (x)) x to Hopf equation v t + vv x =0 in the form v v2 F (v)+t 2 xv =0, F (v) =f(v) 19

36 The idea: to determine the solution to the KdV equation u t + uu x + with the same, modulo 2 12 u xxx =0 u(x, 0; )=u 0 (x) O 2 initial data from the Euler-Lagrange equation u(x) H F [u; ]+ t u2 2 xu dx =0 ( string equation ) 20

37 Lemma. Given a first integral H F [u] of the KdV equation then the space of solutions to the string equation x = tu+ H F [u] u(x) is invariant wrt the KdV flow Recall: Lax-Novikov lemma (1974): Lax version: given a first integral H[u] then the set of stationary points is invariant wrt the KdV flow of KdV H[u] u(x) =0 21

38 Novikov version: given a first integral H[u] of the KdV then the space of stationary points of the Hamiltonian flow u s = x is invariant wrt the KdV flow H[u] u(x) u s =0 u t = x H KdV [u] u(x) Indeed, from commutativity of Hamiltonians {H KdV,H} =0 it follows commutativity of the flows (u t ) s =(u s ) t So, if the KdV initial data satisfies u s =0, then so does the solution. Observe u s =0 where H[u] =H[u] const udx H[u] u(x) = const H[u] u(x) =0 22

39 Galilean symmetry for KdV equation x x ct t t u u + c Infinitesimal form u =1 tu x, (u ) t =(u t ) So, the string equation x = tu+ H F [u] u(x) describes the stationary points of the linear combination s u =0 Here u s = x H F [u] u(x) 23

40 Construction of the functional H F [u; ] uses the theory of deformations of the conservation laws (see the previous lecture) For the Hopf equation v t + vv x =0 the functional H 0 F = F (v) dx is a conservation law for an arbitrary function F 24

41 Theorem. For any function F there exists a deformed functional H F = F (u) 2 24 F (u)u2 x F (4) u 2 xx F (6) u 4 x +... dx being a conservation law for the KdV equation u t + uu x u xxx =0 25

42 An explicit formula in terms of Lax operator L = 2 2 d 2 dx 2 + u(x) Then H F = h F dx where h F = res F (1/2) (L) 26

43 So, one arrives at studying solutions to the Euler-Lagrange equation u(x) H F [u; ]+ t u2 2 xu dx =0 where H F = F (u) 2 24 F (u)u2 x F (4) u 2 xx F (6) u 4 x +... dx Explicitly x = ut+ f(u) f (u)u xx + f (u)u 2 x f (u)u xxxx

44 The last step: to apply to the string equation x = ut+ f(u) f (u)u xx + f (u)u 2 x f (u)u xxxx +... a rescaling near the point of phase transition x = x x 0 v 0 (t t 0 ) x t = t t 0 2/3 t ū = u v 0 1/3 ū 7/6 to arrive at x =ū t f 0 ū ū ū x x +ū 2 x ū x x x x + O 1/3 Choosing = 6/7 one obtains P 2 I 28

45 Solutions to P 2 I X = TU [ 1 6 U ( U 2 +2UU ) U IV ] satisfy KdV U T + UU X U XXX =0 Matching condition: for large X U(X, T) (unique) root of cubic equation X = UT 1 6 U 3 Matching + smoothness choice of a particular solution 29

46 Proof of the Universality Conjecture for analytic solutions to KdV in T.Claeys, T.Grava, 2008 Uses: Riemann-Hilbert formulation of inverse scattering asymptotics of the scattering data of Lax operator L = 2 2 Deift-Zhou asymptotic analysis of the Riemann- Hilbert problem d 2 dx 2 + u 30

47 Second order systems u t = A(u)u x + A 2 (u; u x, u xx )+ 2 A 3 (u; u x, u xx, u xxx )+... 1) Hyperbolic case: eigenvalues of A(u) are real and distinct. Riemann invariants diagonalize the leading term r t (r)r x + + O( )=0 r t + (r)r x + O( )=0 For the leading order system only one of Riemann invariants breaks down. Critical behaviour described by the same solution to P 2 I 2) Elliptic case. The Riemann invariants are complex conjugate. They break down simultaneously 31

48 Dispersionless case, characteristic directions x +, x r + x =0, r =0 x + At the critical point: Whitney fold catastrophe (hyperbolic case) x + = r + x = r + r 1 6 r3 In the elliptic case x = x +, hence the Cauchy-Riemann eqs. r + x + =0 At the critical point x + = r 2 + (elliptic umbilic singularity) 32

49 How to solve dispersionless equations? Example of (focusing) NLS H 0 = 1 Z (u 2 uv 2 ) dx 2 u t +(uv) x =0 v t + vv x u x =0 u t x H 0 v(x) 9 >= First integrals F 0 = Z f(u, v) dx v t x H 0 u(x) >; 0={H 0,F 0 } = @ 0 dx, f vv + uf uu =0 33

50 Lemma. Given a first integral f, the system x = f u (u, v) t = f v = 0 = 0 defines a solution to dispersionles NLS. Any generic solution can be obtained in this way. 34

51 Dispersive deformations of first integrals (NLS case) 2 h f = D NLS f = f f uuu u f uu u 2 x +2f uuv u x v x uf uuu vx f uuuu u f uuu u 2 xx +2f uuuv u xx v xx uf uuuu vxx f uuuuu xx vx u f uuuvv xx u 2 1 x 3456u 3 30f uuu 9uf uuuu + 12u 2 f 5u +4u 3 f 6u u 4 x 1 432u 2 3f uuuv +6uf uuuuv +2u 2 f 5uv u 3 x v x (9f uuuuv + 10uf 5uv ) u x v 3 x 288u 9f uuuu +9uf 5u +2u 2 f 6u u 2 x v 2 x u 4320 (18f 5u +5uf 6u ) v 4 x + O( 6 ) where f vv + uf uu =0 Remark. Using dispersionless limit of the standard NLS conservation laws one obtains only half of solutions f(u,v). Another half from construction of Extended Toda/ Extended NLS hierarchy, G.Carlet, B.D., Youjin Zhang

52 Adding dispersive corrections Hyperbolic case r + = r+ 0 + cx /7 U a 6/7 x ; b 4/7 x + + O 6/7 r = r 0 + 2/7 U a 6/7 x ; b 4/7 x + + O 4/7 where U(X,T) is the solution to P 2 I 36

53 Elliptic case r + = r /5 w 4 5 (x+ x 0 +) + O 4/5 where w = w(z) is the tritronquée solution to the Painlevé-I eq. w =6w 2 z 37

54 Thm. (O.Costin et al. 2012) analyticity of the tritronquée solution in the sector arg z <

55 Thm. (O.Costin et al. 2012) analyticity of the tritronquée solution in the sector arg z <

56 Thm. (O.Costin et al. 2012) analyticity of the tritronquée solution in the sector arg z <

57 NLS, dispersionless NLS, tritronquée solution to PI u x u = 2 39

58 NLS, dispersionless NLS, tritronquée solution to PI !0.5 u!1!1.5!2! x v = Im(log ) x 40

59 About tritronquée solution to Painlevé-1 equation w =6w 2 z Any solution is meromorphic on the entire complex plane Distribution of poles: Theorem of Boutroux (1913) Poles of a generic solution to P-1 accumulate along the rays arg z = 2πn 5, n =0, ±1, ±2 41

60 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z 42

61 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z Hint: do a substitution 42

62 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z Hint: do a substitution w = z 1/2 W Z = 4 5 z5/4 42

63 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z Hint: do a substitution w = z 1/2 W Z = 4 5 z5/4 to arrive at 42

64 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z Hint: do a substitution w = z 1/2 W Z = 4 5 z5/4 to arrive at W =6W Z W W Z 2 42

65 Boutroux ansatz (1913): solutions to P-1 w(z) z ( ) 4 5 z5/4 for large z Hint: do a substitution w = z 1/2 W Z = 4 5 z5/4 to arrive at W =6W Z W W Z 2 6 W

66 Z-plane z-plane 43

67 2) For any three consecutive rays there exists a unique tritronquée solution w(z) such that the lines of poles truncate along these three rays for large z Denote w 0 (z) the tritronquée solution with lines of poles truncated along the green rays w 0 (z) z 6, z, arg z <

68 Towards rigorous proof: the focusing NLS case: inverse spectral transform for the Zakharov - Shabat operator i d dx + semiclassical asymptotics for scattering data Deift - Zhou nonlinear steepest descent analysis of the Riemann - Hilbert problem 45

69 Towards rigorous proof: the focusing NLS case: inverse spectral transform for the Zakharov - Shabat operator i d dx + semiclassical asymptotics for scattering data Deift - Zhou nonlinear steepest descent analysis of the Riemann - Hilbert problem Bertola & Tovbis,

70 Towards rigorous proof: the focusing NLS case: inverse spectral transform for the Zakharov - Shabat operator i d dx + semiclassical asymptotics for scattering data??? Deift - Zhou nonlinear steepest descent analysis of the Riemann - Hilbert problem Bertola & Tovbis,

71 Remark. Universality and random matrices. Z N (t; ) = 1 Vol N Z N N e 1 Tr V (M) dm Integral over the space of N N Hermitean matricesm = M V (M) = 1 2 M 2 X k t k M k+1 (k + 1)! Vol N = Vol (U(N)/U(1) U(1)) N = x, x = t Hooft parameter ) Z N (t; )! Z(x, t; ) 46

72 Claim. = Z(x, t; ) is tau function of a solution to Toda hierarchy i.e., the functions u = log (x+ ) (x ) 2 (x) v log (x+ 0 k = 1 (k + 1)! (L k+1 ) +,L where L = + v(x)+e u(x) 1, = x difference Lax operator. 47

73 Proof uses representation Z N = h 0 h 1...h N 1 in terms of monic orthogonal polynomials p n (x) =x n +..., Z 1 1 p n (x)p m (x)e 1 V (x) dx = h n mn For t =0 p n (x) H n (x) (Hermite polynomials) ) initial conditions u(x, t =0, ) = log x, v(x, t =0, ) =0 Gradient catastrophies=various critical regimes 48

74 (from J.Jurkiewicz, Phys. Lett. B, 1991) 49

75 References B.Dubrovin, Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour, Comm. Math. Phys. 267 (2006) B.Dubrovin, T.Grava. C.Klein, On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonlinear Sci. 19 (2009) B.Dubrovin, On universality of critical behaviour in Hamiltonian PDEs, Amer. Math. Soc. Transl. 224 (2008) T.Claeys, M.Vanlessen, The existence of a real pole-free solution of the fourth order analogue of the Painlevé-I equation, Nonlinearity 20 (2007) T.Claeys, T.Grava, Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann - Hilbert problem, Comm. Math. Phys. 286 (2009) D.Masoero, Poles of integrales tritronquée and anharmonic oscillators. A WKB approach, J. Phys. A 43 (2010) 5201 B.Dubrovin, T.Grava, C.Klein, Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations, SIAM J. Appl. Math. 71 (2011) D.Masoero, A.Raimondo, Semiclassical limit for generalized KdV equation before the time of gradient catastrophe, arxiv: O.Costin, M.Huang, S.Tanveer, Proof of the Dubrovin conjecture and analysis of tritronquée solutions of PI, arxiv: B.Dubrovin, M.Elaeva, On critical behaviour in nonlinear evolutionary PDEs with small viscosity, Russ. J. Math. Phys. 19 (2012)

76 Thank you! 51

77 About viscous case 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]+... Critical behaviour (shock wave) 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 B.D., M.Elaeva, 2012 Particular case u t + a(u)u x = (A.Il in 1985) u xx 52