Numerical integration of a coupled Korteweg-de Vries system

Abstract

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to the non-integrable case,

Numerical solution of a coupled Korteweg–de Vries equations by collocation method

“…And as far as I know, very little numerical works have been done for CKdV equations . Halim et al [13,14] have introduced a numerical scheme for general CKdV systems. The scheme is valid for solving an arbitrary number of equations with arbitrary constant coefficients, the method is applied to the Hirota system and compared with its known analytical solution investigating the influence of initial conditions and grid sizes on accuracy.…”

A numerical treatment based on Haar wavelets for coupled KdV equation

“…These equations describe interaction of two long waves with different dispersion relations, it is introduced by Hirota and Satsuma [1] in 1981. A lot of long waves with weak dispersion such as internal, acoustic, and planetary waves in geophysical hydrodynamics are related with (cKdV) equation [2,3].…”

“…Ismail solved cKdV system by using finite difference and finite element methods [12][13][14]. Halim et al [2,3] introduced a numerical scheme for general cKdV systems. For the periodic initial boundary value problem of the cKdV system a finite difference scheme produced by Wazwaz [15].…”

Finite difference method with different high order approximations for solving complex equation

“…Such as the Korteweg-de Vries (KdV) equation [3,4,5,6] and the nonlinear Schrodinger equation has been solved by [7,8]. Numerical solution of coupled partial differential equations, as an example, the coupled nonlinear Schrodinger equation admits soliton solution and it has many applications in communication, this system has been solved numerically by Ismail [9,10,11,12] and the coupled Korteweg-de Vries equation has been solved numerically [13,14,15,16]. The complex nonlinear partial differential equations have been solved in [17,18,19,20,21].…”