Hamiltonian Structure and Solitary Wave Dynamics in a Generalized Korteweg–de Vries Equation With Higher Order Dispersion and Dissipation

The Korteweg–de Vries (KdV) equation provides a fundamental framework for describing nonlinear dispersive wave propagation and solitary-wave dynamics. In this study, a generalized KdV model incorporating fifth-order dispersion, weak viscous dissipation, linear damping/gain, and external forcing is investigated. The Hamiltonian structure and conservation properties of the classical KdV equation are first examined, and the generalized model is then analyzed using a weak-perturbation approach. The classical KdV solitary wave is used as the leading-order solution, whereas first-order corrections associated with higher order dispersion and dissipative effects are derived. In particular, an explicit solvability condition is formulated for the fifth-order dispersive correction, and an effective damping rate governing the slow evolution of the solitary-wave amplitude is obtained. The analytical results are complemented by Fourier pseudospectral simulations combined with fourth-order Runge–Kutta time integration. The numerical experiments examine the effects of the perturbation parameters on amplitude, width, propagation speed, energy evolution, and solitary-wave interactions. For the weak perturbation regimes considered, the solitary waves retain a coherent localized structure, whereas higher order dispersion produces weak dispersive radiation and profile modification and dissipation produces gradual amplitude and energy loss. The simulations also indicate a gradual departure from the elastic collision behavior of the classical KdV system, resulting in weakly inelastic interactions. These observations become more pronounced as the perturbation strength increases. The results are therefore interpreted as regime-dependent characteristics of the generalized model within the weakly perturbed parameter ranges investigated. The study provides an analytical and computational framework for examining weakly nonconservative extensions of KdV-type solitary-wave dynamics.