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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 appliedsciences and engineering. We also present exact traveling wave solutions tothis generalized KdV equation using an elliptic function method which can bereadily applied to any scalar evolution or wave equation with polynomial termsinvolving only odd derivatives. We show that the generalized KdV equation stillsupports hump-shaped solitary waves as well as cnoidal wave solutions providedthat the coefficients satisfy specific algebraic constraints.  Analytical solutions in closed form serve as benchmarks for numerical solversor comparison with experimental data. They often correspond to homoclinicorbits in the phase space and serve as separatrices of stable and unstableregions. Some of the solutions presented in this paper correct, complement, andillustrate results previously reported in the literature, while others arenovel.

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