Topological solitons, cnoidal waves and conservation laws of coupled wave equations

In this paper a few coupled wave equations that arise in the dynamics of two-layered shallow water waves in ocean shores and beaches have been studied. The mapping method is applied to extract cnoidal waves and solitary wave solutions to the coupled Korteweg-de Vries (KdV) equation, coupled Boussinesq equation and the coupled Whitham-Broer-Kaup equation. The ansatz method is also applied to obtain topological 1-solution to the coupled KdV equation with power law nonlinearity. The multiplier method then gives a few conserved quantities of the coupled KdV equation.