Soliton solutions to the fifth-order Korteweg - de Vries equation and their applications to surface and internal water waves

TL;DR

This paper derives exact solitary wave solutions for a complex fifth-order Korteweg-de Vries equation, analyzing how various parameters influence wave properties and stability, with applications to water waves and other media.

Contribution

It presents a new exact solitary wave solution for the fifth-order KdV equation, exploring the effects of additional nonlinear and dispersive terms on wave characteristics and stability.

Findings

Derived exact solitary wave solutions including embedded and regular solitons

Showed nonlinear dispersive terms significantly affect wave existence and stability

Found embedded solitons can be stable against interactions with regular solitons

Abstract

We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived and the dependence of its amplitude, width and speed on the parameters of the governing equation are studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).