Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we introduce the Hamiltonian boundary value method(HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectralmethod in spatial direction, which leads to the finite-dimensional Hamiltoniansystem. The HBVM, which can preserve the Hamiltonian effectively, is applied in timedirection. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries(KdV) equation are taken as examples to show the validity of the proposed method.Numerical results confirm that the proposed method can simulate the propagation andcollision of different solitons well. Meanwhile, the corresponding errors in Hamiltonianand other intrinsic invariants are presented to show the good preservation property ofthe proposed method during long-time numerical calculation.