Travelling wave solutions on a non-zero background for the generalized Korteweg–de Vries equation

Abstract

For the generalized p-power Korteweg–de Vries equation, all non-periodic travelling wave solutions with non-zero boundary conditions are explicitly classified for all integer powers p ⩾ 1. These solutions are shown to consist of: bright solitary waves and static humps on a non-zero background for odd p; dark solitary waves on a non-zero background and kink (shock) waves for even p in the defocusing case; pairs of bright/dark solitary waves on a non-zero background, and also bright and dark heavy-tail waves (with power decay) on a non-zero background, for even p in the focussing case. An explicit physical parameterization is given for each type of solution in terms of the wave speed c, background size b, and wave height/depth h. The allowed kinematic region for existence of the solutions is derived, and other main kinematic features are discussed. Analytical formulas are presented in the higher power cases p = 3, 4, which are compared to the integrable cases p = 1, 2.