On the dynamical behaviours of solitons and rogue waves in mechanical and hydrodynamical systems

Abstract

Solitons are spatially localized waves that can propagate without changing their properties, shapes and velocities, even though after their mutual collisions. They are found in nonlinear dynamical systems particularly when the dispersive and the nonlinear effects are balanced. In the context of optics, pulses generated in the form of solitons can help to achieve long-distance communication. On the other hand, rogue waves are spatially and temporally localized waves of large displacement that appear unexpectedly and disappear without a trace. The appearance of rogue waves, in the context of hydrodynamics, may cause tremendous damage to ships and casualties. Both solitons and rogue waves are of fundamental importance in nonlinear media. To study the dynamical behaviours of the soliton or rogue wave solutions in nonlinear dispersive systems, several evolutionary models are considered in this thesis: the derivative nonlinear Schrödinger system, the coupled discrete Hirota system, the coupled nonlinear Schrödinger-type system, and the coupled complex short pulse system. Hirota bilinear method is used to find the multi-soliton solutions. Rogue wave solutions are obtained by taking the long-wave limit of the breather solutions (waves of localized pulsating modes). The instabilities due to plane wave modulations are also discussed. In the derivative nonlinear Schrödinger system, conditions of dark solitons, anti-dark solitons and the travelling direction of solitons are found. Interactions of breather and soliton are presented as well. The rogue wave solutions are obtained for the coupled discrete Hirota system, the coupled nonlinear Schrödinger-type system and the coupled complex short pulse system. It is shown that because of the cross-phase modulations, rogue waves in the coupled systems may exist while they are not found in the single-component systems. The existence of rogue waves is suggested to be related to the growth rates due to modulational instabilities. Also, the amplitudes of rogue waves can be enhanced due to the coupling effects compared with that in the single-component systems. As the dispersive parameter or nonlinear parameter varies, the coupled system may display the polarity reversal property where the dark rogue waves change to bright rogue waves and vice versa.