Travelling wave solutions in multisymplectic discretisations of wave equations : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
Symplectic integrators for Hamiltonian ODEs have been well studied over the
years and a lot is known about these integrators. They preserve the symplecticity
of the system which automatically ensures the preservation of other
geometric properties of the system, such as a nearby Hamiltonian and periodic
and quasiperiodic orbits.
It is then natural to ask how this situation generalises to Hamiltonian
PDEs, which leads us to the concept of multisymplectic integration. In this
thesis we study the question of how well multisymplectic integrators capture
the long-time dynamics of multi-Hamiltonian PDEs. We approach this question
in two ways|numerically and through backward error analysis (BEA).
As multi-Hamiltonian PDEs possess travelling wave solutions, we wish to see
how well multisymplectic integrators preserve these types of solutions.
We mainly use the leapfrog method applied to the nonlinear wave equation
as our test problem and look for the preservation of periodic travelling waves.
We call the resulting equation the discrete travelling wave equation. It cannot
be solved exactly. Therefore, our analysis begins with numerically solving the
discrete travelling wave equation for simpli ed nonlinearities.
Next, we mov on to analysing periodic solution for a smooth nonlinearity.
This results in the presence of resonances in the solutions for certain combinations
of the parameters. Finally, we use backward error analysis to compare
and back up our results from numerical analysis.