Special Session 53: Qualitative and Quantitative Techniques for Differential Equations arising in Applied and Natural Sciences

Extreme nonlinear excitations in lattice and continuum models

Efstathios Charalampidis
California Polytechnic State University
USA
Co-Author(s):    
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.