Stationary Solitons of the Fifth Order KdV-type Equations and their Stabilization

TL;DR

This paper derives exact stationary soliton solutions for a fifth order KdV-type equation, analyzes their stability, and discusses their properties, including special solutions for specific parameter conditions and the case p=2.

Contribution

It provides new explicit stationary soliton solutions for the fifth order KdV-type equations under various parameter regimes and examines their stability and properties.

Findings

Solutions are unstable for p ≥ 5.

Solitons are lower and narrower than those with γ=0.

Exact solutions are found for p=2 with all parameters positive.

Abstract

Exact stationary soliton solutions of the fifth order KdV type equation $$ u_t +\alpha u^p u_x +\beta u_{3x}+\gamma u_{5x} = 0$$ are obtained for any p ($>0$) in case $\alpha\beta>0$, $D\beta>0$, $\beta\gamma<0$ (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case $p\geq 5$. Various properties of these solutions are discussed. In particular, it is shown that for any p, these solitons are lower and narrower than the corresponding $\gamma = 0$ solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when $D,\alpha,\beta,\gamma$ are all $>0$ and discuss its various properties.