Rogue waves in nonlinear evolution systems : application to fluid mechanics

Abstract

Oceanic rogue waves, unexpectedly large surface elevations from otherwise calm background sea states, are investigated in this thesis. These extreme and rare events can be observed in other physical systems and are found to be related to supercontinuum generation in optics. To study the dynamics of rogue waves under various oceanic settings, several evolution models for wave propagation are considered: the derivative nonlinear Schrödinger equation which is a higher order model describing finite amplitude waves, the long wave-short wave resonance model for a two-layer fluid, the Manakov system and related coupled systems with multiple wave trains. Modes localized in both the spatial and temporal dimensions are constructed as mathematical models of rogue waves. From the rogue wave modes obtained, the dynamics and origin of rogue waves are explored. It is shown both theoretically and numerically that baseband modulation instability due to long wave disturbances is related to the onset of rogue waves. Secondly, higher order effects and coupling can extend the existence regime of rogue waves. More precisely, when higher order effects are accounted, rogue waves are demonstrated to occur in the shallow water regime where rogue waves are not expected otherwise. Coupled systems with multiple wave trains can also impose a similar effect such that rogue waves can evolve in the shallow water regime. Moreover, the maximum amplitude attained by a rogue wave can be enhanced through coupling and variations in wave forms are observed for the coupled systems. The structural robustness of the rogue wave modes is tested and is demonstrated to be dependent on the growth rate of the noise. Furthermore, localized modes with wave forms similar to those of the derived rogue wave modes can be observed in a chaotic wave field.