Soliton dynamics of Boussinesq equation: Lie symmetries, optimal classification, exact solutions, conservation laws, bifurcation structures and sensitivity analysis

Abstract

In shallow water, solitary waves typically arise as a result of the combined effect of nonlinearity and dispersion. The Boussinesq equation plays a pivotal role in study nonlinear solitary wave due to existing dispersive and nonlinear terms. This study aims to conduct a comprehensive investigation on Boussinesq equation through Lie symmetry method. By determining the complete Lie algebra of infinitesimal generators and constructing one dimensional optimal system of subalgebras, the test equation is transformed to ordinary differential equations (ODEs). The ODEs enable to derive a rich variety of exact solutions, which include bright and dark solitons, double soliton configurations, and many others. To examine the qualitative behavior and structural transitions of these solutions, bifurcation theory is applied, offering a robust framework to study the effects of critical parameters on the dynamics and revealing shifts between distinct solution regimes. Stability is analyzed through bifurcation diagrams and phase portraits, capturing influence of perturbations or parameter variations to evolution of wave structures. Additionally, sensitivity and chaotic analysis are introduced to assess the effect of variations in initial conditions and parameters to persistence and robustness of the derived periodic and quasi-periodic solutions, thereby highlighting critical thresholds and transition points. Furthermore, the adjoint equation and conserved vectors are obtained in connection with the identified symmetries within the framework of the Lagrangian formulation. By synthesizing Lie symmetry reductions, bifurcation and stability analysis, sensitivity examination, and conservation law derivation, this work offers a unified framework for probing the intricate dynamics of the Boussinesq equation, advancing the theoretical understanding and practical applicability of nonlinear wave models.