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October 22, 2025Open Access

Elliptic Functions and Advanced Analysis of Soliton Solutions for the Dullin–Gottwald–Holm Dynamical Equation with Applications of Mathematical Methods

Key Points

Soliton solutions were derived through explicit algebraic constraints, leading to various profiles including hyperbolic and Jacobi-elliptic functions.
The analysis specified parameter regimes that produced distinct symmetry properties among the solutions, enhancing understanding of the DGH equation.
Employing a sub-ODE construction, the study exemplified closed-form families of solutions such as periodic and rational types, visualized using 2D and 3D plots.
Findings highlight the potential applications of advanced mathematical methods for generating solitary waves in complex systems.

Abstract

We studied traveling-wave solutions of the Dullin–Gottwald–Holm (DGH) equation via a sub-ODE construction. Under explicit algebraic constraints, the approach yielded closed-form families—bell-shaped, hyperbolic (sech/tanh), Jacobi-elliptic function (JEF), Weierstrass-elliptic function (WEF), periodic, and rational—and classified their symmetry properties. Optical solitons (bright and dark) arose as limiting cases of the elliptic solutions. We specified the parameter regimes that produced each profile and illustrated representative solutions with 2D/3D plots to highlight symmetry. The results provide a unified, reproducible procedure for generating solitary and periodic DGH waves and expand the catalog of exact solutions for this model.

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Cite This Study

Rizvi et al. (Tue,) studied this question.

synapsesocial.com/papers/68f8a381c0c01e5ef8abdb58 https://doi.org/https://doi.org/10.3390/sym17101773

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