Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations

The time evolution of multi-symplectic PDEs with periodic boundary conditio ns in one and two space dimensions is considered. We introduce the idea of a multi-symplectic Fourier transformfor systems on periodic domains leading to a semi-discretization on Fourier space, and to the concept of multi-sym plecticity on Fourier space. The spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it - a di screte wave model. A by-product of the semi-discretization is that it leads automatically to a system of Hamiltonian ODEs in time when truncated. We s how that the one-dimensional shallow water equations and the two-dimensiona l Zakharov-Kuznetsov equation are multi-symplectic and derive spectral disc retizations for these systems and present numerical experiments. (C) 2001 E lsevier Science B.V. All rights reserved.