Abstract
In this paper, the non-linear for the small long amplitude waves in two dimensional (2D) shallow water waves propagation with free surface are considered. The shallow water wave problem leads to the non-linear Hamiltonian model equation. Based on the binary Bell-polynomials approach, the bilinear form, bilinear Bäcklund transformation and multiple wave solutions are obtained. The conservation laws are constructed using two different techniques, namely, the Ibragimov's theorem and the multiplier method. The Noether's approach was applied to the non-linear Hamiltonian model equation to obtain the conservation laws. Also, we show that the non-linear Hamiltonian model equation is nonlinearly self-adjoint. Conserved quantities of Hamiltonian model equation are illustrated. Finally, with the help of the extended homogeneous balance method, and an exponential method, a set of new exact solutions for the non-linear Hamiltonian model equation are obtained.
•The Hamiltonian mechanics and Bäcklund transformation are obtained.•Weakly multiple wave solution and conservation laws are deduced.•Mathematical methods applied for NPDE.