Numerical Solution of Nonlinear Kaup-Kupershmit Equation, KdV-KdV and Hirota-Satsuma Systems

Abstract In this work we investigate the numerical solution of Kaup-Kupershmit (KK) equation, KdV-KdV and generalized Hirota-Satsuma (HS) systems. The proposed numerical schemes in this paper are based on fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in KK and HS equation is diagonal but in KDV-KDV equation is not diagonal. However for KDV-KDV system which is the focus of this paper, we show that the exponential of linear operator and related inverse matrix have definite structure which enable us to implement the methods such as diagonal case. Comparing numerical solutions with exact traveling wave solutions demonstrates that those methods are accurate and readily implemented.