Traveling Wave Solutions of Degenerate Coupled Multi-KdV Equations

TL;DR

This paper investigates traveling wave solutions of degenerate coupled multi-KdV equations, reducing them to polynomial ODEs and introducing novel methods, including Chebyshev's Theorem and polynomial factorization, to find explicit solutions.

Contribution

It develops two new methods for solving high-degree polynomial ODEs from coupled KdV equations, including Chebyshev's Theorem application and polynomial factorization techniques.

Findings

Derived explicit solutions including solitary waves.

Applied Chebyshev's Theorem to find solutions.

Factorization method yields multiple solution forms.

Abstract

Traveling wave solutions of degenerate coupled $\ell$-KdV equations are studied. Due to symmetry reduction these equations reduce to one ODE, $(f')^2=P_n(f)$ where $P_n(f)$ is a polynomial function of $f$ of degree $n=\ell+2$, where $\ell \geq 3$ in this work. Here $\ell$ is the number of coupled fields. There is no known method to solve such ordinary differential equations when $\ell \geq 3$. For this purpose, we introduce two different type of methods to solve the reduced equation and apply these methods to degenerate three-coupled KdV equation. One of the methods uses the Chebyshev's Theorem. In this case we find several solutions some of which may correspond to solitary waves. The second method is a kind of factorizing the polynomial $P_n(f)$ as a product of lower degree polynomials. Each part of this product is assumed to satisfy different ODEs.