Finite Dimensional Hamiltonians and Almost-Periodic Solutions for 2+1 Dimensional Three-Wave Equations

Abstract

For the 2+1 dimensional three-wave equation, by using the known nonlinear constraints from 2+1 dimensions to 1+1 dimensions, we reduce it further to 0+1 dimensional (finite dimensional) Hamiltonian systems with constraints of Neumann type. These Hamiltonian systems are proved to be Liouville integrable by finding a full set of involutive conserved integrals and proving their functional independence. Moreover, almost-periodic solutions of the 2+1 dimensional three-wave equation are obtained by solving these Hamiltonian systems explicitly.