Dispersive focusing in fractional Korteweg–de Vries-type equations

Abstract

Abstract In this paper we study dispersive enhancement of a wave train in systems described by the fractional Korteweg–de Vries-type equations of the form u t + α n u n u x + β m ( D m { u } ) x = 0 , D m { u } = − | k | m u ( k ) where the operator D m {u} is written in the Fourier space, α n , β m are arbitrary constants and n, m being rational numbers (positive or negative). Using both approximate and exact solutions of these wave equations we describe constructively the process of dispersive focusing. It is based on a time-reversing approach with the expected rogue wave chosen as the initial condition for a solution of these equations. We demonstrate the qualitative difference in the shape of the focused wavetrains for various n and m. Our results can be used for prediction of the rogue wave appearance arising in many types of weakly nonlinear and weakly dispersive wave systems in physical context.