Fourier Splitting Method for Kawahara Type Equations

In this work, we integrate numerically the Kawahara and generalized Kawahara equation by using an algorithm based on Strang’s splitting method. The linear part is solved using the Fourier transform and the nonlinear part is solved with the aid of the exponential operator method. To assess the accuracy of the solution, we compare known analytical solutions with the numerical solution. Further, we show that as t increases the conserved quantities remain constant. 1. Introduction The Kawahara equation is a generalized nonlinear dispersive equation which has a form of the Korteweg-de Vries (KdV) equation with an additional fifth-order derivative term [1]. This equation is an evolution partial differential equation (PDE) that describes in one spatial dimension the propagation of shallow water waves with surface tension and magnetoacoustic waves in plasma [1]. Analytical solutions have been investigated [2]; however, no general solutions have been found yet. The inverse scattering transform (IST), a general method used to integrate nonlinear evolution PDEs [3], cannot be applied to this equation because the Painlevé test of integrability fails [4–6]. In this work, we will employ the following generalized version of the Kawahara equation: where the rescaled nondimensional quantities , , , and depend on the physical parameters, is a integer, and the case for corresponds to the magneto-acoustic wave and to the shallow water approximations. When , , , , and , we reduce to a classical representation of the KdV equation. In order to study the equation by means of numerical methods and produce test of such methods, we will make use of the traveling wave ansatz to get a solitary wave (SW) solution to the Kawahara equation. We will adopt the SW solution to be our exact solution to compare with our numerical approximations. Finite differences, finite elements, and radial basis function have been used to get numerical integration of the Kawahara equation [7–9]. The present method has high accuracy; it is fast and easy to implement as seen in other works that deal with nonlinearity—see, for instance, [10]. In this work we employ a split-step (Fourier) method to integrate the equation. The splitting algorithm is a pseudospectral numerical method which consists in approximate iteratively the solution of the nonlinear evolution equation using exponential operators. We advance in time in small steps and treat separately the linear and nonlinear parts. The nonlinear term is handled by using the matrix exponential method, this method was used in [10–12] to solve similar PDEs.