A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota’s perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.

💡 Research Summary

The paper introduces a systematic “multiple exp‑function method” for constructing exact multi‑wave solutions of nonlinear partial differential equations (PDEs). The method can be regarded as a direct generalization of Hirota’s perturbation scheme, but it is designed to be readily implemented in computer algebra systems (CAS) such as Maple. The core idea is to assume that the unknown field can be expressed as a rational function of several exponential terms:

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