A local discontinuous Galerkin method for solving Korteweg-de Vries (KdV) type equations with non-homogeneous boundary effect is developed. We provide a criterion for imposing appropriate boundary conditions for general KdV type equations. The discussion is then focused on the KdV equation posed on the negative half plane, which arises in the modelling of transition dynamics in the plasma sheat...

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In...

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context Schamel equation. Our study reveals different regimes characterized by behavior wave interactions. In certain regimes, maintain well-separated crests consistently over time, while other number local maxima undergoes variations following patterns 2→1→2→1→2 or 2→1→2. These findings d...