Unveiling Novel Exact Soliton Structures and Bifurcation Dynamics in the Higher-Order Boussinesq–Burgers Equation via the Modified Extended Mapping Method

Résumé

This study addresses the challenging problem of deriving exact analytical solutions for the higher-order Boussinesq-Burgers (HOBB) equation, a fundamental nonlinear model that captures complex wave propagation phenomena arising from the intricate interplay between nonlinearity, dispersion, and higher-order effects in various physical contexts, including fluid dynamics and plasma physics. Despite its significance, extracting explicit solutions for such models remains highly nontrivial. To tackle this difficulty, we implement the Modified Extended Mapping Method (MEMM) as a robust and systematic analytical framework. By applying an appropriate traveling wave transformation, the original nonlinear partial differential equation is converted into an ordinary differential equation, which facilitates the derivation of a wide class of exact analytical solutions in explicit form. In parallel, a rigorous bifurcation analysis is conducted to investigate the qualitative behavior of the corresponding dynamical system and to classify the stability properties of its equilibrium states.

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