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Article

Dispersive focusing in fractional Korteweg-de Vries-type equations

Journal of Physics A: Mathematical and Theoretical. 2020. Vol. 53. No. 34. P. 345703.
Tobisch E., Pelinovsky E.

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 ut + αn un ux + βm(Dm{u})x = 0,Dm{u} = −|k|m u(k) where the operator Dm{u} is written in the Fourier space, αn, βm 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.