Stability and Dynamics of Traveling and Shock Wave Solutions to the Generalized KdV–Burgers Equation on Curved Manifolds

Abstract

This paper investigates a generalized Korteweg–de Vries–Burgers (KdV–Burgers) equation posed on compact Riemannian manifolds, with dissipation and dispersion provided via the Laplace–Beltrami operator and its higher powers. We first establish local well-posedness of weak and strong solutions in high-order Sobolev spaces under precise structural assumptions on the coefficients, and derive energy estimates that make explicit how curvature and the spectrum of Δ𝑔 control dissipation. On this analytic basis, we study the stability of traveling wave and shock-type (viscous) profiles propagating along the flow of a divergence-free vector field, combining linearization with 𝐿2 -energy methods. The analysis extends the Rankine–Hugoniot condition for the leading-order shock speed, while showing that geometric dissipation and dispersion govern the relaxation of perturbations toward these profiles. In particular, we identify conditions on the coefficients and on the geometric setting under which traveling waves and shock-type solutions remain stable in the presence of small disturbances, and we highlight how curvature influences soliton and shock dynamics through the spectral properties of the underlying manifold.