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

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.