Soliton dynamics and stability in the Boussinesq equation for shallow water applications

This manuscript deals with the Fourth-order Boussinesq water wave equation, which is integrable and possesses soliton solutions. Boussinesq water wave equation is a vital tool for investigating nonlinear phenomena in various waves and shallow water phenomena in fluid dynamics, such as diffraction, refraction, weak nonlinearity, and shoaling. Along with fluid dynamics, it is essential in many disciplines of physics, including the transmission of long waves in shallow waters, vibrations in a nonlinear string, acoustics, laser optics, and one-dimensional nonlinear lattice waves. The Generalized Arnous approach, the new Kudryashov method, and the Modified Sub-equation method are applied to this objective. The resultant diverse solutions consist of trigonometric and hyperbolic functions. These approaches generate accurate analytical curves for soliton waves, which comprise kink, bright, and dark waves. The graphical aspects of the produced solutions are investigated using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mn>3</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:math> -surface graphs, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:math> -line graphs, and contour and polar plots, in addition to theoretical derivations. This work is novel in its integrated use of three symbolic methods to derive a broad spectrum of exact soliton solutions for the fourth-order Integrated Boussinesq water wave equation, including compound and hybrid waveforms. The inclusion of the graphical visualization, stability analysis, and open source code resources further strengthens its contribution to nonlinear wave modeling.