Solitary wave interaction and evolution for a higher-order Hirota equation

Abstract

Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and it is found that the tail vanishes when the algebraic relationship from the asymptotic theory is satisfied. Hence a two-parameter family of higher-order asymptotic embedded solitons exists. A comparison between the theoretical predictions and numerical solutions shows strong agreement for both solitary wave interaction, where the higher-order coordinate and phase shifts are compared, and solitary wave evolution, with comparisons made of the solitary wave tail.