Dynamics of Solitons, Lie Symmetry, Bifurcation, and Stability Analysis in the Time-regularized Long-wave Equation

The time-regularized long-wave equation is pivotal in understanding diverse wave dynamics, such as shallow water waves, pressure waves in liquids and gas bubbles, ion-acoustic waves in plasma, and nonlinear transverse waves in magnetohydrodynamics. The analysis is initiated by deriving the infinitesimal generators of the Lie group symmetries, followed by constructing a commutator table and adjoint table to study the algebraic structure of the equation’s symmetries. Using this symmetries, the time-regularized long-wave equation is systematically reduced and solved for invariant solutions associated with each symmetry. These solutions give insight into the inherent properties of the system and wave behavior. The extended hyperbolic function method is employed to derive exact solutions to the simplified versions of the time-regularized long-wave equation. This method enhances the solution process by generating soliton solutions including dark, singular, bright and periodic-singular solutions. To illustrate their behavior of obtained solutions, graphical representations are given by using different values of parameters. A comprehensive bifurcation analysis is also conducted to explore the qualitative changes in the dynamics of system, while the Hamiltonian structure of the equation is constructed to investigate its conservation properties. The study also investigates the phase portraits of the system to offer a visual interpretation of the solution trajectories in phase space. Finally, the modulation instability of the time-regularized long-wave equation through linear stability analysis is analyzed to provide a clearer understanding of the conditions that lead to the onset of instability in wave propagation.