Non-degenerate localised waves beyond Manakov system and their new perspectives

Abstract

We study the non-degenerate dynamics of localised waves beyond Manakov system and offer their new perspectives based on the wave component analysis. Our investigation is in the framework of the coupled Hirota (CH) equations. An exact multi-parameter family of solutions for the localised waves is derived within a new Lax pair which is necessary for producing the new types of solutions describing the non-degenerate localised waves, such as the non-degenerate general breathers, non-degenerate Akhmediev breathers, non-degenerate Kuznetsov-Ma solitons and non-degenerate rogue waves. Especially, the degenerate and non-degenerate solutions for rogue waves are different from previous ones, even within the context of the Manakov system. A new technique of wave mode analysis (or the characteristic line analysis) is provided to classify degenerate and non-degenerate solutions beyond the eigenvalue perspectives, namely the critical relative wave number. Such technique is suitable for both the CH equations as well as Manakov system. Hereby, we redefine the non-degenerate localised waves from a fully different view. We further prove that a transition between the non-degenerate localised waves to various types of solitons appears in the CH equations due to the higher-order effects and there is no analogue in Manakov system. In order to further understand such transition dynamics and physical properties of the non-degenerate solutions, the physical spectra are presented analytically. The higher-order terms take impacts on the spectra, for which the state transition solutions as well as a new type of breathers are found. Furthermore, we investigate the relation between non-degenerate modulation instability and higher-order effects. We also offer an exact initial condition to excite the degenerate and non-degenerate localised waves using the numerical simulation and test the stability for the excitation of such solutions by adding a weak perturbation. Since the CH equations can model a large number of physical phenomena in the deep ocean, in the birefringent fibre as well as in the nonlinear channel, our results may provide insights for the related experimental studies.