Traveling Wave Solutions of Degenerate Coupled Multi-KdV Equations

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.