Abstract
Recent results on numerical integration methods that exactly preserve the symplectic structure in both space and time for Hamiltonian PDEs are discussed. The Preissman box scheme is considered as an example, and it is shown that the method exactly preserves a multi-symplectic conservation law and any conservation law related to linear symmetries of the PDE. Local energy and momentum are not, in general, conserved exactly, but semi-discrete versions of these conservation laws are. Then, using Taylor series expansions, one obtains a modified multi-symplectic PDE and modified conservation laws that are preserved to higher order. These results are applied to the nonlinear Schrödinger (NLS) equation and the sine-Gordon equation in relation to the numerical approximation of solitary wave solutions.
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