Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation

We propose an algorithm for an asymptotic model of shallow-water wave dynamics in a periodic domain. The algorithm is based on the Hamiltonian structure of the equation and corresponds to a completely integrable particle lattice. In particular, “periodic particles” are introduced in the algorithm for waves travelling through the domain. Each periodic particle in this method travels along a characteristic curve of the shallow-water wave model, determined by solving a system of nonlinear integro-differential equations. We introduce a fast summation algorithm to reduce the computational cost from O(N) to O(N), where N is the number of particles. With the aim of providing a test of the algorithms, we scale the shallow-water wave equation to make it asymptotically equivalent to the KdV equation in the form studied by Zabusky and Kruskal in their seminal 1965 paper, thereby also testing the equivalence of the two models derived under similar asymptotic approximations of shallow water wave dynamics. By using the fast summation algorithm and the asymptotic scaling analysis, we further test this equivalence by investigating the interaction of solitons and recurrence of initial states for the shallow-water wave equation in periodic domains. Finally, to illustrate the hyperbolic nature of the dynamics of the shallow-water wave model, we introduce a particle algorithm and its integral counterpart for the initial-boundary value problem with homogeneous boundary conditions on finite intervals.