Breather Wave Solutions for the (3+1)-D Generalized Shallow Water Wave Equation with Variable Coefficients

• Published in: Qualitative theory of dynamical systems 2023-12, Vol.22 (4), Article 127
• Main Authors: Dawod, Lafta Abed, Lakestani, Mehrdad, Manafian, Jalil
• Format: Article
• Language: English
• Subjects: Breathers; Difference and Functional Equations; Dynamical Systems and Ergodic Theory; Mathematical analysis; Mathematics; Mathematics and Statistics; Oceanography; Shallow water; Solitary waves; Traveling waves; Water waves; Wave equations
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-023-00826-8

The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a known equation. To achieve this, an illustrative example of the VC generalized shallow water wave equation is provided to demonstrate the feasibility and reliability of the used procedure in this study. It is shown that Hirota bilinear method is an important scheme. So, it has plenty of classes of rational solutions by selecting the interaction breather and dark soliton solutions and homoclinic breather wave solutions. Here many types of rough and breather solutions are obtained. The mentioned equation is transformed into the Hirota bilinear form with help of the Hirota direct method. In this process, the Hirota bilinear operator plays a significant role. Based on the Hirota bilinear form, the breather wave forms solutions of the equation are obtained. Meanwhile, the figures of the breather wave forms solutions and periodic wave solutions are plotted. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of the free parameters on the behavior of the acquired figures to a few of the obtained solutions for two nonlinear rational exact cases was also discussed. In addition to addressing a scientific explanation of the analytical work, the results are graphically presented to make it simple to recognize the dynamical aspects. Many new types of traveling-wave solutions are revealed, including the breather wave, the dark kink singular, the periodic solitary singular and the singular soliton solutions. By comparing the proposed method with the other existing methods, the results show that the execution of this method is concise, simple, and straightforward.