Bifurcations and Exact Solitary Wave, Compacton and Pseudo-Peakon Solutions in a Modified Generalized KdV Equation

A modified generalized KdV equation is considered in this paper. Under the given parameter conditions, the corresponding traveling wave system is a singular planar dynamical system with three singular straight lines. The bifurcations and traveling wave solutions of the system are investigated in the parameter space from the perspective of dynamical systems. The existence of solitary wave solutions, periodic peakon solutions, pseudo-peakon solutions, kink and anti-kink wave solutions and compactons is proved. Furthermore, possible exact explicit parametric representations of various solutions are given. Particularly, the model has uncountably infinite many solitary wave and pseudo-peakon solutions.

Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity

This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

Lump-soliton, lump-multisoliton and lump-periodic solutions of a generalized hyperelastic rod equation

In this paper, we will obtain lump-soliton solution for (1[Formula: see text]+[Formula: see text]1)-dimensional generalized hyperelastic rod equation, also known as generalized KdV equation by aid of Hirota bilinear method (HBM). We also obtain lump-multisoliton (which is an interaction of lump with one kink or two kink soliton) and lump-periodic solutions (which is formed by an interaction between lump and periodic waves). The dynamics of these solution are examined graphically by selecting significant parameters.

Instability of the soliton for the focusing, mass-critical generalized KdV equation

<p style='text-indent:20px;'>In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in <inline-formula><tex-math id="M1">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula> norm must eventually move away from the soliton.</p>