Nonlinear WKB Method, Asymptotic Soliton-Like Solutions of Variable Coefficients Korteweg–de Vries Equations with Singular Perturbation and Rankine–Hugoniot-Type Conditions

The main focus of the paper consists of getting and studying asymptotic soliton-like solutions of a singularly perturbed (a small parameter at the highest derivative) Korteweg–de Vries equation with variable coefficients. The solutions appear similar to the well-known soliton-solutions of the corresponding Korteweg–de Vries equation with constant coefficients. Derivation of these solution is based on the nonlinear Wentzel-Kramers-Brillouin (WKB) method. The corresponding algorithm is described and justified. The existence of a global asymptotic soliton-like solution is considered as well. The theoretical results are illustrated by examples. The constructed solutions are used to derive the Rankine-Hugoniot–type conditions.