Numerical Study of Oscillatory Regimes in the KP equation

Transcription

1 Numerical Study of Oscillatory Regimes in the KP equation C. Klein, MPI for Mathematics in the Sciences, Leipzig, with C. Sparber, P. Markowich, Vienna, math-ph/"#"$"%& C. Sparber (generalized KP), personal-homepages.mis.mpg.de/klein/

2 Outline Introduction, general features of the KP equation Low dispersion limit for small data, multi-scale expansion Numerical methods Low dispersion limit for general amplitudes Outlook

3 Kadomtsev-Petviashvili equation (KP) nonlinear dispersive waves on the surface of fluids, essentially one-dimensional propagation of the waves with weak transverse effects KP I (λ = 1): strong surface tension, KP II (λ = +1): weak surface tension x ( t u + u x u + ɛ 2 xxx u ) + λ yy u = 0, λ = ±1, evolutionary form t u + u x u + ɛ 2 xxx u + λ 1 x yy u = 0, u t=0 = u I (x, y). anti-derivative (Fourier multiplier) 1 x f(x) := 1 2 ( x f(ζ) dζ + x ) f(ζ) dζ,

4 Constraint on intial data constraint for initial data u I (x, y): R yyu I (x, y) dx = 0, - satisfied: smooth solution in time to Cauchy problem, - not satisfied: constraint satisfied for any t > t 0. %!$#! ( "$# "!"$#!" # " '!#!!"!!"!# & " #!"

5 (Line)Solitons (localized in one direction), 2-soliton

6 (Line)Solitons, 4-soliton

7 KP I: lump solitons (localized, algebraic fall off)

8 Hyperelliptic solutions (g=2)

9 Hyperelliptic solution (g=4)

10 Low-dispersion limit, small data, multi-scale expansion constraint for initial data u I (x, y): R yyu I (x, y) dx = 0 powerlaw tails for initial data with compact support example (λ = 1) u I = x sech 2 R, R = x 2 + y 2 small amplitude: u(t, x, y) ɛ u 0 (t, x, y) + ɛ 2 u 1 (t, x, y) + O(ɛ 3 ) ( ) u 0 (t, x, y) = 2R ψ 0 (ɛt, x + 3η 2 t, y) e i(ηx+η3 t)/ɛ, ɛ 1

11 Davey-Stewartson equation equation for the amplitude i τ ψ 0 3η ξξ ψ 0 + λ η yyψ 0 ( ) 1 6η ψ ηφ ψ 0 = 0, 3η 2 ξξ φ + λ yy φ + ξξ ψ 0 2 = 0. y-independent case: non-linear Schrödinger equation (NLS)

12 Davey-Stewartson solution, real part

13 KP II solution (ϵ=0.1)

14 KP solution (blue) and asymptotic solution (green) "&"' "&"( "&"# * "!"&"#!"&"( t = 4 ɛ = 0.1!"&"'!!"!#$!#"!%$!%"!$ " )

15 Difference of KP and asymptotic solution t = 4 ɛ = 0.1

16 L 2 norm of the difference difference decreases as result (Schneider) &"& ɛ 5/2, expected from KdV & %"& " # % $"& $ #"&!!"!!"#!"$!"%!"&!"'!"(!")!"* #!+,-!!. The data can be fitted by a straight line log 10 2 = a log 10 ɛ + b, with a = 2.48 and b = The correlation coefficient in this case is r = and the standard error for a is σ a = 0.17.

17 Numerical approach task: resolve steep gradients in rapid oscillations Fourier series for spatial coordinates fourth-order time stepping to avoid aliasing, integrating factor method (fourth-order Runge- Kutta) with dealiasing exponential time differencing more efficient

18 Fourier space: equation of the form U t = cu + N[U] here: U vector (1+1) or matrix (2+1), c array, N[U] convolution, steep gradients: high frequency terms in c lead to large absolute values despite small ɛ exponential time differencing: time discretization and integration with integrating factor U(t n + h) = e ch U(t n ) + h 0 dτe c(h τ) N[U(t n + τ)] fourth-order Runge-Kutta scheme (Cox-Matthews), coefficients via contour integrals (Kassam-Trefethen) integrating factor, fourth-order Runge-Kutta (e.g. Trefethen): ( e ct U ) t = e ct N[U]

19 Hyperelliptic solution (g=2, periodic in x,y)

20 Difference between numerical and exact solution (ETD, IF)!'!!!& N x = N y = 256!# t [0, 1]!+,-. '(!12!!$!*!%!)!'(!'' 1!!"!!"#!"$!"% & &"! &"# &"$ &"% #,-. / '( 0 linear regression: log 10 L 2 = a log 10 N t + b, with a = 4.32 and b = 5.90; correlation coefficient r = , standard error for a σ a = 0.14

21 Difference in dependence of CPU time!'!!!&!#!+,-. '(!34!!$!*!%!)!'(!''!!"!!"#!"$!"% & &"! &"# &"$,-. '( / 012

22 General amplitude, dispersionless KP (dkp), dissipative regularization u 0 = 6 x sech 2 (x)

23 Gradient of the dkp solution

24 dkp solution as asymptotic solution before breakup %"& % " # $"& $ #"&!!"!!"#!"$!"%!"&!"'!"(!")!"* #!+,-!.! L 2 norm of the difference for several values of ɛ. The data can be fitted by least square analysis with a straight line log 10 2 = a log 10 ɛ + b with a = 1.45 and b = The correlation coefficient is r = 0.999, the standard error for a is σ a =

25 KP I solution ϵ=0.1

26 KP II solution ϵ=0.1

27 KP I and II solution ϵ=0.1 #!*& % ) '!%!#!!!"!#!$!%!& ' & % $ # ( +!*!&! # ) % '!%!#!!!"!#!$!%!& ' & % $ # (

28 KP I solution (negative initial data)

29 KP I solution ϵ=0.01

30 Contour plot for t=0.4 # *%& " ) *%& *!*%&!) *!*%&!)!)%&!$!$%& #!$%&&!$%&!$%'&!$%'!$%(&!$%(!$%$&!$%$!$%)&!!( ) *%& # - + " * '!*%&!) # )%( )%' )%& )%+ )%, )%- )%. $ $%) $%$ $%(! $ *

31 Oscillatory zone shrinks with ϵ *!#$%!$ & % " %!%+' "!* & * %!* &!'!(!)!*!$ % $ * )!!#$%!$+'!'!(!)!*!$ % $ * )! &!$ %!%+'!$!$+' *!#$%!* & % " %!%+'!* &!'!(!)!*!$ % $ * )!!$

32 Dependence on the `weak coordinate - )!(!!!' )"* )"$ )"&, )"! )!!"#!!"$!!"% +!!"&!!"'!!"!!!"( - * $ &! ) )"* )"$ )"&, )"! ) ("$ ("* +!!"!!"&

33 y-dependence deformation of the KdV sector: R ν = x 2 + νy 2 oscillations increased by tail formation, more mass in the tails implies stronger gradients &$!)*)$+$& &$!)*)$+& % % ( ( $ $!%!!!"!# $ # " '!%!!!"!# $ # " ' &$!)*)$+, &$!)*)&$ % % ( ( $ $!%!!!"!# $ # " '!%!!!"!# $ # " '

34 KP II solution, ν = 10

35 Generalized KP equations t u + 6σu n x u + ɛ 2 xxx u + λ 1 x yy u = 0, σ = ±1 n = 2, σ = 1

36 Outlook Asymptotic solution for KP oscillations, Whitham equations Step-like solutions, simplified initial data generalized KP, blow up behavior other numerical methods

37 Focusing DS