Geometric numerical schemes for the KdV equation

Abstract

Abstract. Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.

Macroscopic dynamics of incoherent soliton ensembles: Soliton gas kinetics and direct numerical modelling

“…It follows from (13), (15) and (16) We note that, in the degenerate case η 1 = η 2 ≡ η 0 the ansatz (12) reduces to the one-component distribution (8) with f 0 = f 1 + f 2 as expected. Expressing s 1,2 in terms of f 1,2 from (15) we obtain…”

“…The effect p-1 of two-soliton collisions on the properties of the statistical moments of nonlinear wave field associated with soliton gas -the soliton turbulence -was studied in [14]. An effective method for the numerical computation of soliton gas was developed in [15] and was applied to the modelling of soliton gases in the KdV and the KdV-BBM (BenjaminBona-Mahoni) equations in [16].…”

Numerical Simulation of Dispersive Shallow Water Waves with an Efficient Method

“…It looms large in the study of non-linear dispersive waves. KdV equation was derived to model shallow water waves with weak nonlinearities by Korteweg and de Vries as the following: [9][10][11][12][13] U t + UU x + U xxx = 0 ( 1 ) * Author to whom correspondence should be addressed.…”

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

“…The KdV equation is the pioneer model in soliton wave theory, which gives rise to solitons, given by with recursion operator R is given by where D denotes the total derivative with respect to x , and D − 1 is its integration operator.…”

“…The work on nonlinear evolution equations, which model these phenomena, has long been a major concern for solutions with concrete physical meaning such as the multiple soliton solutions and the interaction between the resulting solitons. The integrable equations, such as the Korteweg-de Vries (KdV), modified Korteweg-de Vries, and nonlinear Schrodinger equations, possess sufficiently large number of conservation laws and give rise to multiple soliton solutions [1][2][3][4][5][6][7][8][9][10]. For any nonlinear evolution equation, the presence of three soliton solutions is believed to be an important indication for the integrability of that equation [5], but not sufficient to prove the integrability of this equation.…”