Traveling Wave Solutions to Fifth- and Seventh-order Korteweg-de Vries Equations: Sech and Cn Solutions

TL;DR

This paper reviews higher-order KdV equations used in science and engineering, presenting exact elliptic function solutions that include solitary and cnoidal waves, serving as benchmarks and revealing new solutions.

Contribution

It introduces novel exact traveling wave solutions for fifth- and seventh-order KdV equations using an elliptic function method, expanding the set of known solutions.

Findings

Supports hump-shaped solitary waves under specific conditions

Provides closed-form solutions as benchmarks for numerical methods

Identifies new solutions and corrects previous results

Abstract

In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higher-order dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this generalized KdV equation using an elliptic function method which can be readily applied to any scalar evolution or wave equation with polynomial terms involving only odd derivatives. We show that the generalized KdV equation still supports hump-shaped solitary waves as well as cnoidal wave solutions provided that the coefficients satisfy specific algebraic constraints. Analytical solutions in closed form serve as benchmarks for numerical solvers or comparison with experimental data. They often correspond to homoclinic orbits in the phase space and serve as separatrices of stable and unstable regions. Some of the solutions presented in this paper correct, complement, and illustrate results previously reported in the literature, while others are novel.