Numerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations

Transcription

1 Numerical solutions of the small dispersion limit of KdV, Whitham and Painlevé equations Tamara Grava (SISSA) joint work with Christian Klein (MPI Leipzig) Integrable Systems in Applied Mathematics Colmenarejo, September

2 Main topics Construction of a global asymptotic solution for the small dispersion limit of KdV u t + 6uu x + ɛ 2 u xxx = 0 by a quantitative numerical comparison of Small dispersion limit of KdV Zero order asymptotic approximation and Whitham equations Double scaling limits and Painlevé equations

3 Equivalent Picture: linear WKB theory ɛ 2 f xx = (E V (x))f, Let k(x) = E V (x). The WKB solution is obtained by the matching f W KB (x) = f W KB (x) = 4 A ± k(x) e ± 1 ɛ B cos k(x) 4 R x k(ξ)dξ, x > x 2, x < x 1 ( α + 1 ɛ x k(ξ)dξ ), x 1 < x < x 2 At x = x 1,2 V (x 1,2 ) = E, ɛ 2 f xx = ( V (x 1,2 ))(x x 1,2 )f, f W KB (x) = Ai(z), z = ( V (x 1,2 ) ɛ 2 ) 1 3 (x x1,2 )

4 Korteweg de Vries equation, dispersive shock waves u t + 6uu x + ɛ 2 u xxx = 0 u 0 (x) = 1/ cosh 2 x ɛ = 10 2

5 Dispersionless limit: Hopf equation u t + 6uu x = 0 u(x, t) = u 0 (ξ), x = 6tu 0 (ξ) = ξ t c = min ξ R [ 1/(6u (ξ)]

6 For t > t c SHOCK SOLUTION (weak solution) x s(t) x c For t > t c the limit u(x, t, ɛ) as ɛ 0 of the dissipative equation u t + 6uu x = ɛ 2 u xx, determines the shock solutions among the infinite number of weak solutions of the Hopf equation.

7 Multiphase oscillations of KdV

8 Remark The oscillatory behaviour of the solution of the Hamiltonian perturbations of the Hopf equation is generic. For arbitrary functions c(u) and p(u) u t + uu x + ɛ2 24 (2cu xxx + 4c u x u xx + c u 3 x) + ɛ 4 [2pu xxxxx + 2p (5u xx u xxx + 3u x u xxxx ) + p (7u x u 2 xx + 6u 2 xu xxx ) + 2p u 3 xu xx ] = 0 c(u)=1, p(u)=0, corresponds to KdV c(u)=8u, p(u)=u/3, corresponds to the Camassa-Holm equation, up to Miura transformation (Dubrovin)

9 Numerical solution of the Camassa-Holm equation

10 Recurrence coefficients of orthogonal polynomials P n (z)p m (z)e NV t(z) dz = δ nm, Recurrence relations for OP: V (λ) = 1 2 λ tλ λ6. zp n (z) = γ n+1 P n+1 (z) + γ n P n 1 (z), and γ n satisfies the string equation: n N = γ n[1 + (t + γ n )(γ n 1 + γ n + γ n+1 ) + γ n 1 γ n γ n 1 (γ n 2 + γ n 1 + γ n ) + γ n+1 (γ n + γ n+1 + γ n+2 )].

11 Numerical Simulation of γ n (Jurkiewicz1991, C. Klein, T.G.) N = 400 N = 800 t = 2 to 1.7

12 Small dispersion limit of KdV for different values of ɛ (Trefethen s code) &!*&'!&#$ &!*&'!% '#$ '#$ t = 0.4 ) ' ) '!'#$!'#$!&!!!"#$!"!%#$!% (!&!!!"#$!"!%#$!% ( &!*&'!%#$ &!*&'!" '#$ '#$ ) ' ) '!'#$!'#$!&!!!"#$!"!%#$!% (!&!!!"#$!"!%#$!% (

13 Small dispersion limit of KdV The limit u(x, t, ɛ) as ɛ 0 has been obtained by Gurevich and Pitaevskii (1973) using the Whitham averaging technique; Lax-Levermore (1983) and Venakides (1985, 1990) using a steepest descent type argument for the Kay-Moses determinant τ ɛ (x, t). Deift-Venakides-Zhou (1997) using the steepest descent method for oscillatory Riemann-Hilbert problems (Deift-Zhou 1992). They get precise information about the phase of the oscillations.

14 The fundamental quantity to describe asymptotics is the minimization problem F 0 (x, t) = 1 π inf ψ A {(a(x, t, u 0), ψ) (Lψ, ψ)} Lψ(η) = 1 π log η µ η + µ ψ(µ)dµ The number of intervals of the support of the minimizer ψ determines qualitatively the behavior of the solution: one interval: the solution u(x, t, ɛ) is asymptotic to the Hopf solution; two intervals: the solution is asymptotic to a periodic modulated wave; many intervals: the solution is asymptotic to a quasi-periodic modulated wave.

15 Asymptotic solutions Let λ = 1 η 2, then ψ (η) φ (λ). 1. Supp φ = [u H (x, t), 1], and t u H + 6u H x u H = 0, u H (x, 0) = u 0 (x). u(x, t, ɛ) = 2ɛ 2 2 x 2 log [ e F 0/ɛ 2] + O(ɛ 2 ) = u H (x, t) + O(ɛ 2 ) 2. Supp φ = [β 1 (x, t), β 2 (x, t)] [β 3 (x, t), 1], the β i (x, t) satisfy the Whitham equations t β i + v i (β 1, β 2, β 3 ) x β i = 0, i = 1, 2, 3, with β 1 > β 2 > β 3 and u(x, t, ɛ) 2ɛ 2 2 x 2 log [ e F 0(x,t)/ɛ 2 ϑ ( Ω ɛ ; T )].

16 Asymptotic description t < t c :, u(x, t, ɛ) solution of the Hopf equation; t > t c : oscillatory zone, roughly where the Hopf solution becomes multivalued; outside: Hopf solution inside: modulated elliptic solution of KdV The Hopf solution and the modulated elliptic solution are attached continuously but not C 1. The size of the oscillatory zone in the asymptotic description does not depend on ɛ

17 For numerical simulations we need concrete formulas: the phase Ω takes the form Ω = β1 β 3 2K(s) [x 2t(β 1 + β 2 + β 3 ) q(β 1, β 2, β 3 )], where the function q is given by (Fei-Ran Tian) q(β 1, β 2, β 3 ) = dµdν f( 1+µ 2 ( 1+ν 2 β ν 2 β 2) + 1 µ 2 β 3) 2 2π 1 µ 1 ν 2, where f is the inverse of the decreasing part of the initial data. The solution of the Whitham equations is obtained by the following generalization of the method of characteristics (Tsarev) x = v i (β 1, β 2, β 3 )t + w i (β 1, β 2, β 3 ), i = 1, 2, 3, and w i = 1 2 [v i 2(β 1 + β 2 + β 3 )] ui q + q.

18 Numerical solution of the variational problem: support of the minimizer D. W. McLaughlin and J.A. Strain solved the variational problem. We solve the Whitham equations. Numerical technique: multidimensional zero-finding algorithm. The starting point of the algorithm is obtained at the boundary of the Whitham zone.

19 Comparison of the small dispersion limit of KdV and the asymptotic solutions (Hopf and modulated elliptic wave)

20 Difference of KdV and asymptotic solutions at time t=0.4 "(&%!&"# "(&%!$ %"& %"& %"%# %"%# % %!!%"%#!!%"%#!%"&!%"&!%"&#!%"&#!%"$!!"#!!!$"#!$ '!%"$!!"#!!!$"#!$ ' "(&%!$"# "(&%!! %"& %"& %"%# %"%# % %!!%"%#!!%"%#!%"&!%"&!%"&#!%"&#!%"$!!"#!!!$"#!$ '!%"$!!"#!!!$"#!$ '

21 Numerical evaluation of the error In the interior blue zone the error decreases like ɛ 3 while on the left boundary it decreases like ɛ At the leading edge in 3 the green zone the error decreases like ɛ and the size of the zone decreases as a power law compatible with ɛ 2 3 At the trailing edge the green zone on the right the error decreases like ɛ but the size of the zone does not decrease like a power law.

22 Difference in the interior of the Whitham zone (ϵ=0.001) Error in the phase Ω Determination of higher order corrections of the phase: Haberman (1995), Maltsev (2006) ' *+()!! &! # (! )!(!#!!!&

23 Comparison of KdV and asymptotic solutions at breakup time for ϵ= t= t= t=0.218 u 0.5 u 0.5 u x t= x t= x t= u 0.5 u 0.5 u x t= x t= x t= u 0.5 u 0.5 u 0.5

24 Error in the different parts of the (x,t) plane ɛ α ɛ 2 3 Whitham ɛ ɛ t ɛ 2 ɛ Hopf ɛ x

25 Comments Linear WKB theory: matching of inner and outer solutions is obtained by Airy function. Nonlinear WKB theory: matching of asymptotic solutions is obtained through special solutions of Painlevé equations.

26 Asymptotic solution near the point of gradient catastrophe The solution near this singular point can be obtained by Rigorous Riemann-Hilbert analysis of the singular point as obtained in random matrices (DKMVZ). Work in progress. Intuitive considerations. The Hopf solution near the point of gradient catastrophe takes the form x x c 6t(u u c ) + 1 3! f (u c )(u u c ) ! f (IV ) (u c )(u u c ) ! f (V ) (u c )(u u c ) 5.

27 Let h k be the KdV Hamiltonians of the KdV flow such that h k = u k+2 /(k + 2)! + O(ɛ 2 ) Then h 0 = u2 2 + ɛ2 u xx 6, h 1 = 1 6 (u3 + ɛ2 2 (u2 x + 2uu xx ) + ɛ4 10 u xxxx). x = 6tu + a 0 h 0 + a 1 h a k h k, is a symmetry of the KdV equation. Setting a 0 = 0, a 1 = f (u c ) /6 and a k>2 = 0, x x x c and u u u c x x c = 6t(u u c ) f (u c ) 6 [(u u c ) 3 + ɛ2 2 (u2 x+2(u u c )u xx )+ ɛ4 10 u xxxx], which is an ODE of Painlevé type (PI 2, Brezin, Marinari Parisi). For the fifth order approximation the corresponding ODE is of 8-th order.

28 Properties of the solution of the ODE P 2 I X = 6T U (U U 2 X + UU XX U XXXX). There is a real analytic solution which is pole free and which has asymptotic (T.Claeys, M.Vanlessen) U(X) = ±X 1 3 X. Conjecture: the solution of KdV near the point of gradient catastrophe is approximated by ( ) 2/7 ( ) ɛ x xc 6u c (t t c ) t t c u(x, t, ɛ) = u c + U, +O(ɛ 4 (ɛ 6 a 1 ) 1/7 (ɛ 4 a 1 ) 3/7 7 ). a 1 The same ansatz, up to different rescalings, is established for any Hamiltonian perturbations up to order ɛ 4 of the Hopf equations u t + 6uu x = 0 (Dubrovin).

29 NUMERICAL SOLUTION OF KDV AND PI2 AT THE BREAK TIME (joint work with G.Carlet) & )*&"+!, & )*&"+!$ & )*&"+!- (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"++& &!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"+++ &!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"++, & (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"++$ &!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"++- &!!!!"#!!"$%!!"$!!"%%!!"% ' )*&"+.& & (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"% '!!!!"#!!"$%!!"$!!"%%!!"% '!!!!"#!!"$%!!"$!!"%%!!"% '

30 Asymptotic solution near the leading edge: oscillations go to zero, error O(ɛ 1 3 ) (Random Matrices opening of a gap in the spectrum: Bleher-Its, DKMVZ, Bleher-Eynard, Kuijlaars-Claeys-Vanlessen,... KdV: Kudryashev-Suleimanov ) Double scaling expansion: y = (x x (t))ɛ 2 3 u(x, t, ɛ) = U 0 + ɛ 1/3 U 1 + ɛ 2/3 U where U 0 = u H ( ) Φ(y, t) U 1 (y, t) = a(y, t) cos, ɛ ( 2Φ(y, t) U 2 (y, t) = b 1 (y, t) + b 2 (y, t) cos ɛ and the phase ) Φ(y, t) = Φ 0 (y, t)+ɛ 1/3 Φ 1 (y, t)+ɛ 2/3 Φ 2 (y, t)+ɛφ 3 (y, t)+ɛ 4/3 Φ 4 (y, t)

31 Consistency of the expansion with KdV produces differential equations for a, b 1, b 2 and Φ k, k = 0,..., 4. The function a = a(y, t) satisfies the Painlevè II equation (PII) 4(β 1 (t) β 3 (t))a yy 2ya 3 tβ 3 (t) = 1 2 a3. The relevant solution is the Hastings Mcleod solution of PII. Differential equations for Φ k give free constants of integrations. The integration constants for Φ 0, Φ 1 and Φ 2 are obtained analitycally from the matching with the elliptic modulated solution. The integration constants for Φ 3 and Φ 4 are obtained numerically.

32 Numerical solution of PII: Tracy-Widom using a standard differential solver. Praehofer and Spohn with in principle arbitrary precision using a Taylor series method. We use spectral method and asymptotic Taylor series A !10!8!6!4! z

33 Comparison of the KdV solution and the PII asymptotics

34 Difference between KdV and PII asymptotic solution 0.04 ε= ε= Δ 0 Δ x x 0.04 ε= ε= Δ 0 Δ

35 Combination of asymptotic solutions: difference between KdV and PII or elliptic solutions ε= ε= Δ x ε=10 3 Δ Δ x x

36 Error in the different parts of the (x,t) plane after the matching with Painleve ɛ ɛ 3 4 ɛ t PII ɛ 0.26 ɛ 2 ɛ P x

37 Open problems Matching of the asymptotic expansions: P12- elliptic solution, PI2-PII, elliptic solution-pii in all the relevant regions of the (x,t) plane. Understanding of the error of order the Whitham zone. ɛ 3 4 inside Understanding the asymptotic description at the trailing edge where the error decreases like. ɛ 1 2 The size of the zone does not decreases as a power law and double scaling asymptotics is not possible.

38 Conclusions: comparison with the analytical results obtained for the large N expansion in Hermitian random matrix models or asymptotic for orthogonal polynomials. 1. KdV: inside the oscillatory zone the error is of order ɛ. Orthogonal polynomials: asymptotics for γ N for regular external fields is determined by theta-functions up to terms of order 1/N (DKMVZ). 2. Leading edge. KdV: expansion of the solution in powers of ɛ 1 3 and the amplitude is determined by the Hasting-Mcleod solution of PII. Same scaling in random matrices for the opening of a gap in the spectrum, the size of the gap is determined by PII (Bleher-Its, Kuijlaars-Claeys-Vanlessen,Bleher-Eynard, DKMVZ,... )

39 1. Trailing edge: error ɛ. In random matrices, the opening of a band in the spectrum is characterized by an error term of order 1/ N (DKMVZ).The double scaling limit has been studied by Eynard (2006). 2. Point of gradient catastrophe: second term of the asymptotic expansion is of order ɛ 2 7. The equivalent situation for the asymptotic of γ N has been rigourously proved by Clayes and Vanlessen (2006) and it is described by the nonsingular solution of P12.

40