This study derives novel exact traveling wave solutions for the dimensional shallow water wave equation-a pivotal model in coastal hydrodynamics for tsunami prediction and tidal analysis. By employing an enhanced tanh-function method, we obtain a diverse spectrum of solutions, including dark, singular, and periodic solitons, as well as hyperbolic, Jacobi elliptic, rational, and exponential forms, which surpass the variety and generality reported in previous studies. These solutions uncover previously unexplored wave propagation patterns and interaction dynamics. A comprehensive bifurcation analysis elucidates the stability and phase transitions of the wave solutions, providing deeper analytical insight into their behavior. High-resolution graphical visualizations quantitatively demonstrate wave amplification and nonlinear interactions, confirming the superiority of our method in capturing complex physical phenomena. The results not only advance nonlinear wave theory but also enhance predictive models for marine hazard prevention and environmental monitoring strategies.
Partial differential equations (PDEs) are fundamental mathematical tools employed across diverse scientific and engineering disciplines to model complex phenomena involving spatial and temporal variations. These equations are particularly crucial for describing wave propagation, fluid dynamics, and various physical systems where complex behaviors like shock waves, turbulence, and solitons are caused by nonlinear interactions, these equations are especially important. In this regard, the nonlinear Navier-Stokes equations are indispensable for understanding fluid motion, including complex flow patterns, turbulence, and wave breaking. Similarly, the Korteweg-de Vries (KdV) equation has been pivotal in advancing our comprehension of nonlinear wave propagation and characterizing soliton behavior in water waves. Beyond fundamental physics, PDEs find extensive application in environmental science and engineering. They are utilized to simulate wave propagation in atmospheric and marine systems, such as the nonlinear advection-diffusion equations that describe the dispersion of contaminants in air and water. PDEs have a wide range of uses, from engineering applications like shock wave analysis and optical fiber research to environmental modeling, including climate dynamics and extreme weather events. PDEs' wide range of applications underscores their significance in addressing real-world challenges across a multitude of domains.
Nonlinear evolution equations (NLEEs) may be solved exactly using a variety of techniques, such as the extended Sinh-Gordon equation expansion. This technique involves expanding the known solutions of the Sinh-Gordon equation to address a broader class of NLEEs. The modified extended tanh-function approach. This method generalizes the traditional tanh-function method to obtain more comprehensive solutions. The modified extended direct algebraic method. A systematic approach that simplifies finding solutions by transforming NLEEs into more manageable forms. The modified extended mapping method. This method utilizes specific mappings to connect different types of solutions, enhancing the range of solvable equations. The modified Sardar Sub-Equation Method. A methodology designed to extract exact solutions through the use of sub-equations derived from the original NLEEs. The extended F-Expansion method. This method extends the F-expansion approach to formulate solutions for NLEEs more flexibly. Improved Modified Extended Tanh-Function Method, An advancement over the original method that improves computational efficiency and solution variety. Several alternative approaches are employed, including the Homo Balance Method, the Bäcklund Transformation, and Riccati-Bernoulli methods, among others. These methodologies are essential for investigating intricate physical phenomena, such as wave dynamics, fluid dynamics, and thermal interactions, while also connecting theoretical studies to practical implementations, thereby driving progress across numerous disciplines, including various branches of mathematics and physics. The exploration of NLEEs in higher-dimensional spaces continues to be an essential area of research due to its relevance in describing complex physical phenomena such as shallow water waves, plasma dynamics, and nonlinear optics. In particular, the newly proposed -dimensional shallow water wave equation provides a compelling mathematical framework for modeling multidirectional wave propagation in shallow water environments. The (3+1)-dimensional shallow water wave equation, given by Eq. (1), is selected for its ability to model complex multidirectional wave interactions in shallow water environments, where high-order dispersion and nonlinear effects are significant. Unlike lower-dimensional models like the Korteweg-de Vries equation, this equation captures spatial variations in three dimensions, making it ideal for studying phenomena such as tsunamis, tidal waves, and rogue waves in coastal hydrodynamics, justifying its relevance to this study.
where is a real function and the constants govern the nonlinear and dispersive properties of the wave dynamics. A significant advancement in this field was achieved by Liu, who combined symbolic computation with hybrid analytical techniques to derive exact solutions-including lump solitons, breathers, and rogue waves-for the -dimensional shallow water wave equation. While Liu's work established a foundational framework using Hirota bilinear methods and variable separation, the present study extends these results by applying IMETFM to obtain new classes of exact traveling wave solutions. Our approach systematically generates dark soliton and singular solutions, as well as hyperbolic, Jacobi elliptic, periodic, rational, and exponential solutions, thereby expanding the known solution space for this equation. Beyond deriving exact solutions, we conduct a bifurcation analysis to explore stability and dynamic transitions, offering deeper insights into wave propagation mechanisms. The efficacy of IMETFM in handling high-dimensional nonlinear systems is demonstrated, reinforcing its utility for modeling complex wave phenomena in fluid dynamics, plasma physics, and related fields. By bridging gaps in Liu's methodology, this work not only enriches the theoretical understanding of the -dimensional equation but also provides practical tools for analyzing nonlinear wave interactions in physical systems.
This study is structured into six primary sections. The initial section 1 introduces the research, outlining its background and aims. The second section 2 elaborates on the IMETFM approach, while the third section 3 provides an in-depth analysis of exact solutions for the shallow water wave equation. The fourth section 4 provides graphical visualizations of the derived solutions, offering insights into their physical behaviors and wave interactions. The fifth section 5 builds upon these findings by presenting a bifurcation analysis, exploring the stability and qualitative transitions of the wave solutions in the model. The final section, Section 6, provides a conclusion, summarizing the findings and discussing the research's implication