New Exact Solutions of a Generalized Shallow Water Wave Equation

In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.

💡 Research Summary

The paper introduces an “Extended Elliptic Function Method” (EEF‑M) and applies it to the generalized shallow water wave equation (GSWWE), a nonlinear partial differential equation that models long‑wave dynamics in shallow fluids. The authors begin by reviewing traditional solution techniques—Fourier‑type expansions, direct integration, and canonical traveling‑wave reductions—and point out that these approaches typically yield only simple sinusoidal, solitary, or rational solutions, leaving a large portion of the solution space unexplored.

EEF‑M is built on the ansatz that a traveling‑wave solution (u(\xi)) (with (\xi = x - ct)) can be expressed as a finite polynomial in a base elliptic function (F(\xi)):
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