Bifurcation, bilinear forms, conservation laws and soliton solutions of the temporal-second-order KdV equation

Abstract

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities [Formula: see text] and [Formula: see text] and the parameters of nonlinearity [Formula: see text] and [Formula: see text] on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

Exact Solutions for Coupled Variable Coefficient KdV Equation via Quadratic Jacobi’s Elliptic Function Expansion

“…In Ref. [26], the bell polynomial method was proved to be a powerful mathematical tool for solving the KdV equation with variable coefficients to reach the N-soliton solutions. Additionally, the periodic wave solutions were obtained by using the Riemann function method.…”