Recurrence in the KdV Equation? A. David Trubatch United States Military Academy/Montclair State University

Transcription

1 Nonlinear Physics: Theory and Experiment IV Gallipoli, Lecce, Italy 27 June, 27 Recurrence in the KdV Equation? A. David Trubatch United States Military Academy/Montclair State University With: Ben Herbst J. Andre Weideman University of Stellenbosch

2 Korteweg-de Vries equation (KdV) u t + uu x + δ 2 u xxx = a particular continuum limit of an FPU lattice d 2 dt 2 y = (y n 2y n + y n+ ) [ α (y n y n+ )] Repeated near-recurrences are observed in FPU lattices with [FPU, 955] y n () = sin( nπ N ), d dt y n() =. 2

3 Look for (near) recurrence in KDV by simulation [Zabusky & Kruskal,965] u t + uu x + δ 2 u xxx = δ =.22 2 << u(x, ) = sin(πx) d dt u n = (u n + u n + u n+ ) (u n u n+ ) 3h +δ 2 u n 2 2u n + 2u n+ u n+2 2h 3 d dt u = (A 3u) (D c u) δ 2 D 3c u 3

4 Recurrence t =. t = 9.76 t = Simulation with N = 28 4

5 Recurrence t Simulation with N = 28 5

6 Emergence of Solitons t =.25 t= Simulation with N = 28 6

7 KdV solitary waves Emergence of Solitons u(x, t) = 2δ 2 k 2 sech 2 [ k(x 4δ 2 k 2 t) ] are the limit of periodic, cnoidal-wave solutions u(x, t) = 2δ 2 k 2 m cn 2 [ k(x 4δ 2 k 2 (2m )t, m) ] The solitary waves are solitons: they pass through one another and regain their original shape & velocity. 7

8 Recurrence Recurrence has been explained in terms of the solitons. [Zabusky & Kruskal, 965; Osborne & Bergamasco, 986] Q: How does recurrence depend on the spatial discretization? Hypotheses: H: Higher-accuracy discretization gives better recurrence. H2: Integrable discretization gives better recurrence. 8

9 High-accuracy Discretizations Pseudo-spectral: d dt u = u(d f u) δ 2 D 3 f u Pseudo-spectral (Conservation Form): d dt u = 2 D f u 2 δ 2 D 3 f u Spectral: d dt u = D f F [(Fu) (Fu)] δ 2 D 3 f u D f = F ΩF: Fourier differentiation matrix F: discrete Fourier transform matrix 9

10 Solitons Zabusky Kruskal Pseudosp. Pseudosp. (Cons.) Spectral Initial Value Simulation with N = 28 plotted at t = 9.75

11 Recurrence.9.8 Zabusky Kruskal Pseudosp. Pseudosp. (Cons.) time Simulation with N = 28

12 Recurrence.9.8 Zabusky Kruskal Spectral time Simulation with N = 28 2

13 Recurrence Zabusky-Kruskal Spectral Simulation with N = 28 3

14 Recurrence Surprise(?): Recurrence is worse in pseudospectral and spectral schemes. Higher accuracy does not yield better recurrence. Q: What about a rougher grid (e.g., N = 64)? A: Zabusky-Kruskal and Spectral discretizations don t show recurrence. A2: Pseudospectral discretizations manifest a nonlinear instability. 4

15 No Recurrence.9 Zabusky Kruskal Spectral Simulation with N = 64 5

16 Solitons(?) Zabusky Kruskal Spectral Initial Value Simulation with N = 64 plotted at t =.75 6

17 Nonlinear Instability In pseudospectral discretizations there is rapid uncontrolled growth of the solution for rough grids. The nonlinear terms induce aliasing. Preservation of n u2 n in other discretizations precludes the instability. A similar instability exists in: simple finite-difference discretizations of KdV, discretizations of viscous Burger s equation [Maritz & Schoombie], discretizations inviscid Burger s equation. [Majda & Timofeyev, 22] 7

18 Finer Grid Zabusky-Kruskal N=28 Zabusky-Kruskal N=256 Sp. N=

19 Integrable Discretization of KdV KdV can be associated with the Zakharov-Shabat Scattering problem in the form ψ x = ik u ψ ik Forward difference discrete (Ablowitz-Ladik) scattering problem: ψ n+ = z U n ψ α z n = S n ψ n where U n = hu n, α = h, z = e ikh 9

20 Integrable Discretization of KdV Discrete Compatibility Condition: where d dτ S n = T n+ S n S n T n ψ n+ = S n ψ n Compatibility condition is equivalent to: d dτ ψ n = T n ψ d dτ U n = ( αu n ) [ αu n (U n 2 U n ) αu n+ (U n U n+2 ) α(u n + 2U n + U n+ )(U n U n+ ) +U n 2 2U n + 2U n+ U n+2 ] 2

21 Integrable Discretization of KdV Rescale: U n h 6 u n, α h δ 2 τ 3δ2 h 4 IDKdV: d dt u n = ( ) [ + h2 u n un (u n 2 u n ) 6δ 2 2h + (u n + 2u n + u n+ )(u n u n+ ) 2h Truncated (non-integrable): d dt u n = u n (u n 2 u n ) 2h + u n+(u n u n+2 ) 2h + u n+(u n u n+2 ) 2h + δ 2 u n 2 2u n + 2u n+ u n+2 2h 3 + (u n + 2u n + u n+ )(u n u n+ ) +δ 2 u n 2 2u n + 2u n+ u n 2 2h 2h 3 ] 2

22 Solitons Integrable Truncated Initial Value Simulation with N = 64 plotted at t =

23 Recurrence Zabusky-Kruskal Integrable Truncated Simulation with N = 64 ( h2 6δ 2 =.336) 23

24 Recurrence Zabusky-Kruskal Integrable Truncated Simulation with N = 28 ( h2 6δ 2 =.84) 24

25 Recurrence Zabusky-Kruskal Integrable Truncated Simulation with N = 28 25

26 Recurrence Zabusky-Kruskal Integrable Truncated Simulation with N = 256 ( h2 6δ 2 =.2) 26

27 Observations Presence of solitons in discretization not sufficient to capture recurrence. Recurrence is worse in pseudospectral and spectral discretizations than carefully-chosen (Zabusky-Kruskal) finite-differences. Recurrence not always strengthened by decrease the grid size. Integrable discretization does not capture recurrence better than Zabusky-Kruskal discretization. (Related?) Phenomena: uncontrolled nonlinear instability (modulated) grid-scale oscillations 27