Rogue waves in discrete and continuous nonlinear evolution systems : theoretical studies and applications to fluid flows with shear current and stratification

Abstract

Rogue waves are unexpectedly large displacements from equilibrium positions or tranquil background and are intensively studied as extreme and rare events in Nature and engineering systems recently. There are many examples in different context of physics, e.g. fiber optics and oceanic waves. Here both theoretical modeling of dynamical systems and issues of practical applications are considered. The first part focuses on the dynamics of rogue waves in discrete systems, using the discrete Hirota equation as the basic model. The second part of thesis concentrates on fluid mechanics. The effects of shear currents on the existence of rogue waves are analyzed. Another aspect of fluid mechanics studied is the dynamics of density stratification. Properties of coupled triad resonance which can occur for a stratified fluid with a constant buoyancy frequency are studied through (i) an analytical approach using a Hamiltonian and (ii) a computational perspective via numerical simulations. The discrete coupled Hirota system including the discrete third order dispersion has a wide application in optical field. It is found that rogue waves can exhibit some unexpected features. Low order conservation laws are established and breather (pulsating) solutions are derived analytically. New regimes of modulation instability of discrete evolution equation can be produced by the coupling. The robustness test is elucidated numerically to check the sensitivity to initial conditions. In the context of water waves, the modulation instability will be influenced in the presence of nonlinear shear current. For the no shear case, the modulation instability can only occur in sufficient deep water, which is kh>1.363. (k is wave number, h is water depth) For the concave to the right shear, the modulation instability will be enhanced (or suppressed) when wave packet is moving along (or against) the shear current. While for the convex shear, the result is opposite. Besides, it is found the growth rate of disturbance for concave shear is much larger than the linear and convex case. Streamline patterns for rogue waves are also illustrated as the preliminary study. Two examples of coupled triads are investigated. The first one is the stratified fluid with constant buoyancy frequency, the second one is two-layer fluid in the presence of linear shear. The derivation of evolution equation of the first example is provided. The variations of interaction coefficients with water depth and buoyancy frequency are illustrated. A simple mathematical model of coupled triads is set up to investigate the evolution of amplitudes with different combination of interaction coefficients. Finally, the spontaneous generation of modes due to coupling and the dependence on the interaction coefficients are also investigated numerically. (426 words)