Traveling Wave Solutions of Degenerate Coupled KdV Equation

TL;DR

This paper thoroughly analyzes traveling wave solutions of a specific coupled KdV equation, classifying solutions based on polynomial roots and providing explicit elliptic function solutions with graphical illustrations.

Contribution

It offers a detailed classification and explicit forms of solutions for the Kaup-Boussinesq type coupled KdV equations, including solitary and periodic waves, and discusses extensions to higher orders.

Findings

Existence of no nontrivial solutions in certain cases

Explicit elliptic function solutions for solitary and periodic waves

Graphical representations of exact solutions

Abstract

We give a detailed study of the traveling wave solutions of $(\ell=2)$ Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. Extension of our study to the cases $\ell=3$ and $\ell=4$ are also mentioned.