Spontaneous soliton generation in the higher order Korteweg–de Vries equations on the half-line

Abstract

Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.