| Abstract: |
| This talk will focus on the formation of rogue waves, in the form of time-periodic and spatially localized solutions known as Kuznetsov-Ma (KM) breathers in the discrete, focusing nonlinear Schr\odinger (DNLS) equation and Ablowitz-Ladik (AL) systems. In doing so, we will investigate the existence, stability and dynamics of KM breathers in the Salerno model which itself interpolates between the DNLS and AL systems. We will explore the configuration space of KM breathers by varying the homotopy parameter associated with the Salerno model (connecting the AL and DNLS models) as well as the period of the solution. We will show that on one hand, the KM breather in the AL model is not the only one solution since more KM solutions bearing oscillatory tails are shown to be present therein. On the other hand, and as per the DNLS model, novel KM breathers will be presented in this case. Then, upon using a proximity argument, KM breathers on a flat background will be shown to exist for the DNLS case. The results will be complemented by discussing the stability of the solutions using Floquet theory and direct dynamical simulations. More recent results on the defocusing AL system will be presented too (if time permits) and open problems and questions will be discussed. |
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