On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system

In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Kortewegde vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation.