On the integrability and quasi-periodic wave solutions of the Boussinesq equation in shallow water

In this paper, the complete integrability of the Boussinesq equation in shallow water is systematically investigated. By using generalized Bell's polynomials, its bilinear formalism, bilinear Backlund transformations, Lax pairs of the Boussinesq equation are constructed, respectively. By virtue of its Lax equations, we find its infinite conservation laws. All conserved densities and fluxes are obtained by lucid recursion formulas. Furthermore, based on multidimensional Riemann theta functions, we construct periodic wave solutions of the Boussinesq equation. Finally, the relations between the periodic wave solutions and soliton solutions are strictly constructed. The asymptotic behaviors of the periodic waves are also analyzed by a limiting procedure.