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<p>This study explores analytical soliton solutions for the Benney-Luke equation using three newly advanced techniques: the modified Sardar-sub equation method, the extended Sinh-Gordon equation expansion method, and the bifurcation theory. The Benney-Luke equation describes water wave propagation, essential for analyzing wave tension in physical systems. Through the application of these techniques, a wide variety of solutions has been formulated, comprising bright solitons, singular solitons, hyperbolic solitons, dark solitons, combinations of dark and bright solitons, combinations of kink and soliton solutions, periodic solitons, exponential solitons, lumps, and rational solitons. To demonstrate the behavior of these solutions, 3D and 2D plots are generated for selected parameter values. The analysis highlights diverse stable and efficient solitary waveforms, such as lumps, kink-shaped solitons, periodic waves, and bell-shaped waves. The validity of each solution is examined by inserting it back into the original model. Furthermore, the problem is converted into an ordinary differential equation via wave transformations, and it is then examined in more detail as a planar dynamical system connected to a one-dimensional Hamiltonian function. Phase portraits are plotted, and parameter analysis is carried out to find any potential chaotic behavior. The findings demonstrate that the solutions obtained can be applied to study complex phenomena within the framework of this model. Moreover, the techniques demonstrate efficacy, dependability, and the ability to produce precise analytical solutions for further nonlinear systems.</p>

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Dynamics of Different Soliton Solutions, Phase Portraits, Bifurcation Analysis, and Chaotic Behavior in the Benney-Luke Equation Represent Surface Waves in a Shallow Channel with Surface Tension Effects

  • 期刊论文

  • 发表日期：

    2025年8月12日

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  • Nikita Mann,
  • Sachin Kumar

摘要

This study explores analytical soliton solutions for the Benney-Luke equation using three newly advanced techniques: the modified Sardar-sub equation method, the extended Sinh-Gordon equation expansion method, and the bifurcation theory. The Benney-Luke equation describes water wave propagation, essential for analyzing wave tension in physical systems. Through the application of these techniques, a wide variety of solutions has been formulated, comprising bright solitons, singular solitons, hyperbolic solitons, dark solitons, combinations of dark and bright solitons, combinations of kink and soliton solutions, periodic solitons, exponential solitons, lumps, and rational solitons. To demonstrate the behavior of these solutions, 3D and 2D plots are generated for selected parameter values. The analysis highlights diverse stable and efficient solitary waveforms, such as lumps, kink-shaped solitons, periodic waves, and bell-shaped waves. The validity of each solution is examined by inserting it back into the original model. Furthermore, the problem is converted into an ordinary differential equation via wave transformations, and it is then examined in more detail as a planar dynamical system connected to a one-dimensional Hamiltonian function. Phase portraits are plotted, and parameter analysis is carried out to find any potential chaotic behavior. The findings demonstrate that the solutions obtained can be applied to study complex phenomena within the framework of this model. Moreover, the techniques demonstrate efficacy, dependability, and the ability to produce precise analytical solutions for further nonlinear systems.