| Abstract: |
| In this talk, we will provide an overview of results on extreme events
called rogue waves in nonlinear Schr\odinger (NLS) equations both in
discrete and continuum settings. Motivated by the physics of ultracold
atoms, i.e., atomic Bose-Einstein condensates (BECs), we will attempt
to address the question about what type of experimental initial conditions
should be utilized for producing waveforms which are strongly reminiscent
of the Peregrine soliton. The underlying initial-boundary-value problems
with Gaussian wavepacket initial data will be considered. Then, large
amplitude excitations strongly reminiscent of the Peregrine, Kuznetsov-Ma
breather or regular solitons will be identified when the width of the Gaussian
initial pulse is varied. Then, we will systematically perform a bifurcation
analysis of Kuznetsov-Ma breathers in the Salerno model which itself interpolates
the completely integrable Ablowitz-Ladik (AL) model and discrete NLS equation.
Novel results in the form of nanopteronic solutions will be presented both
at the AL limit but also at the DNLS one where the stability of the identified
solutions will be discussed. Finally, associated open questions and directions
for future study will also be outlined. The findings presented in this talk
might be of particular importance towards realizing experimentally extreme
events in BECs but also in optics. |
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