An algebro-geometric solution for a Hamiltonian system with application to dispersive long wave equation

Abstract

By using an iterative algebraic method, we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation. It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure. Two commutators, with zero curvature and Lax representations, for the hierarchy are constructed, respectively, by using two different systematic methods. Under a Bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional Hamiltonian system. By introducing the Abel–Jacobi coordinates, an algebro-geometric solution for the dispersive long wave equation is derived by resorting to the Riemann theta function.