Discrete Integrable Systems and Geometric Numerical Integration

This thesis deals with discrete integrable systems theory and modified Hamiltonian equations in the field of geometric numerical integration. Modified Hamiltonians are used to show that symplectic schemes for Hamiltonian systems are accurate over long times. However, for nonlinear systems the series defining the modified Hamiltonian equation usually diverges. The first part of the thesis demonstrates that there are nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. Specifically, they arise as reductions of a lattice version of the Korteweg-de Vries (KdV) partial differential equation. We present cases of one and two degrees of freedom symplectic mappings, for which the modified Hamiltonian equations can be computed as a closed form expression using techniques of action-angle variables, separation of variables and finite-gap integration. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example is a system of an implicit dependence on the time step, which is obtained by dimensional reduction of a lattice version of the modified KdV equation. The second part of the thesis contains a different class of discrete-time system, namely the Boussinesq type, which can be considered as a higher-order counterpart of the KdV type. The development and analysis of this class by means of the B{\"a}cklund transformation, staircase reductions and Dubrovin equations forms one of the major parts of the thesis. First, we present a new derivation of the main equation, which is a nine-point lattice Boussinesq equation, from the B{\"a}cklund transformation for the continuous Boussinesq equation. Second, we focus on periodic reductions of the lattice equation and derive all necessary ingredients of the corresponding finite-dimensional models. Using the corresponding monodromy matrix and applying techniques from Lax pair and $r$-matrix structure analysis to the Boussinesq mappings, we study the dynamics in terms of the so-called Dubrovin equations for the separated variables.