Conservation properties of multisymplectic integrators

Recent results on the local and global properties of multisymplectic discretizations of Hamiltonian PDEs are discussed. We consider multisymplectic (MS) schemes based on Fourier spectral approximations and show that, in addition to a MS conservation law, conservation laws related to linear symmetries of the PDE are preserved exactly. We compare spectral integrators (MS versus non-symplectic) for the nonlinear Schrödinger (NLS) equation, focusing on their ability to preserve local conservation laws and global invariants, over long times. Using Lax-type nonlinear spectral diagnostics we find that the MS spectral method provides an improved resolution of complicated phase space structures.