On Series Expansions Giving Closed-Form Solutions of Korteweg–de Vries-Like Equations

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Abstract
Korteweg–deVries (KdV)-like equations with higher-degree nonlinearity are solved by a direct algebraic technique due to Hereman et al. [J. Phys. A, 19 (1986), pp. 607–628]. For two KdV-like equations, one with fifth-degree nonlinearity, the other a combined KdV and mKdV equation, for particular choices of the coefficients of the nonlinear terms, the kink and antikink solutions found by Dey are recovered. Furthermore, soliton solutions of the combined KdV and mKdV equation are found for all values of the coefficients. Closed-form solutions for the Calogero–Degasperis–Fokas modified mKdV equation are also developed. Applications of the solutions of these equations in quantum field theory, plasma physics, and solid-state physics are mentioned. The Hereman et al. method is illustrated and slightly extended and this direct series method is briefly compared to Hirota’s method.