Bifurcation scenarios and quasi-periodic dynamics in a dispersive-dissipative medium

Abstract

This article presents a comprehensive analytical and dynamical investigation of a nonlinear long-wave model in which wave propagation is governed by the interplay of nonlinear steepening, regularizing dispersion, linear drift and a second-order term controlled by the parameter ϱ. The sign of ϱ determines the physical regime of the medium: (ϱ > 0) corresponds to weak dissipation, $(\varrho =0)$ recovers a nondissipative or conservative propagation environment, while (ϱ < 0) models an anti-dissipative or energy-feeding regime in which small disturbances may grow. Such a structure captures a wide range of physical situations including unidirectional shallow-water waves, weakly viscous channels and long-wave transport in media where energy may be damped, conserved, or injected. Using the Khater II technique, we construct new explicit traveling-wave solutions such as solitary, periodic and kink-type profiles. The traveling-wave reduction is rewritten as a Hamiltonian system, enabling a detailed stability analysis and a qualitative description of the phase-space geometry. A full bifurcation classification is provided, distinguishing periodic, homoclinic and heteroclinic orbits across different parameter regimes. To investigate sensitivity to external fluctuations, a weak time-periodic perturbation is introduced, generating quasi-periodic and chaotic dynamics demonstrated through numerical simulations and Poincaré sections. The results reveal how dispersion, nonlinearity and the sign of ϱ jointly shape the onset of complex long-wave behavior and provide exact analytical benchmarks for validating numerical solvers in dissipative and non-dissipative dispersive systems.