 |
Center for Nonlinear Studies |
In this talk, we will focus on the formation of rogue waves in the nonlinear Schr¨odinger (NLS)equation as well as the Salerno model which contains the integrable Ablowitz-Ladik (AL) and(non-integrable) discrete NLS (DNLS) models. We will first consider generic Gaussian initialdata that can be supported by experiments in attractive Bose-Einstein Condensates for boththe NLS and Salerno models, where novel spatio-temporal dynamics will be presented as theGaussian’s width changes. Indeed, we will show that large amplitude excitations stronglyreminiscent of the Peregrine soliton, (time-periodic) Kuznetsov-Ma (KM) breather or regularsolitons appear to form. Then, we will focus on the existence, stability and dynamics ofdiscrete Kuznetsov-Ma breathers in the Salerno model. Through the use of pseudo-arclengthcontinuation techniques, we will explore the configuration space of KM breathers by varyingthe period of the solution and the homotopy parameter associated with the Salerno model(connecting the AL and DNLS models). We will show that on the one hand, the KM breatherin the AL model is not the only one solution since more KM solutions bearing oscillatorytails are shown to be present therein. On the other hand, and as per the DNLS model, novelKM breathers will be shown through pseudo-arclength continuation by starting from the anticontinuum limit. The results will be complemented by discussing the stability of the solutions using Floquet theory and numerical simulations. More recent results on the DNLS equation will be presented too and open problems and questions will be discussed.
Host: Avadh Saxena