Hamiltonian Conservation Laws and Finite-Difference Simulations for a Nonlinearly Coupled Generalized Schrödinger–KdV System

Source abstract

This paper studies a Hamiltonian generalized Schrödinger–KdV system generated by a nonlinear long-wave coupling potential. Starting from an energy functional with a smooth coupling function , we derive a matched coupled system involving the terms and , with the quadratic case used for numerical simulations. Under periodic boundary conditions, we prove conservation of the Schrödinger mass, the long-wave mean, a generalized momentum, and the Hamiltonian energy. A fully discrete finite-difference method based on central differences and the implicit midpoint rule is then constructed. The scheme exactly preserves the discrete mass and discrete long-wave mean, while the generalized momentum and energy are monitored as diagnostic quantities. Numerical experiments confirm the expected second-order convergence behavior and illustrate the long-time boundedness of the monitored invariant errors.