Solitons, Shock Waves, Conservation Laws and Bifurcation Analysis of Boussinesq Equation with Power Law Nonlinearity and Dual Dispersion

Abstract

This paper obtains the soliton solutions to the Boussinesq equation with the effect of surface tension being taken into account. The power law nonlinearity is considered. Three integration tools are adopted in order to extract the soliton solutions. They are the traveling wave hypothesis, ansatz method and the semi-inverse variational principle. Finally, the Lie symmetry approach is adopted to extract the conservation laws of this equation.

Analytical Solutions for the Elastic Circular Rod Nonlinear Wave, Boussinesq, and Dispersive Long Wave Equations

“…where 1 , 1 are integral constants. Substituting the second formula of (7) into the first one and assuming 1 = 1 − 2 , 2 = (3/2) , 3 = −(1/2), = 1 + 1 , we can obtain (3). So the solutions of (7) are equivalent to that of (3) and the second formula of (7).…”

“…The dynamics of shallow water waves, which are seen in various places like sea beaches, lakes, and rivers, are governed by the Boussinesq equation. In recent years there has been much interest in some variants of the Boussinesq systems [2][3][4][5][6]. These coupled Boussinesq equations [7] arise in shallow water waves for two-layered fluid flow.…”

Singular traveling wave solutions for Boussinesq equation with power law nonlinearity and dual dispersion

“…For example, the velocity variation or the higher derivative term is introduced to adjust the linear dispersion performance of the equation. This paper is devoted to the study of Boussinesq equation with power law nonlinearity and dual dispersion that is investigated in fluid dynamics [11][12][13] as follows:…”

Soliton Solution of Good Boussinesq Equation

“…In addition, the phenomenon of shock waves associated with this Boussinesq-type equation was examined in [2,20]. Two-dimension versions of the Boussinesq equation have been treated in the recent work [21] by the Ansatz method, which turns out equivalent to the tanh method.…”