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

The time evolution of multi-symplectic PDEs with periodic boundary conditions 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-symplecticity on Fourier space. The spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it — a discrete 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 show that the one-dimensional shallow water equations and the two-dimensional Zakharov–Kuznetsov equation are multi-symplectic and derive spectral discretizations for these systems and present numerical experiments.