A legendre dual-petrov-galerkin spectral element method for the kawahara-type equations

An efficient and accurate Legendre dual-Petrov-Galerkin spectral element method for solving the Kawahara-type equations is proposed. The Sobolev bi-orthogonal basis functions are constructed, which reduce the num-ber of nonzero elements in the matrix and improve the computational effi-ciency. The generalized stability of the fully discrete spectral element scheme is analyzed. For applications, we consider the Kawahara, KdV-Kawahara and modified Kawahara equations, and discuss numerically the motion of single solitary wave, conservation laws, the phenomena of wave generation and inter-action of two and three solitons. Numerical results illustrate the effectiveness of the suggested approach.