Speaker: Barbara Prinari (Department of Mathematics, University of Colorado, Colorado Springs)
Title: Scalar, vector and matrix nonlinear Schrödinger equations: a bright and dark soliton zoo
Abstract: In this talk we will discuss solitons and rogue wave solutions for various nonlinear Schrödinger (NLS) systems, which model a plethora of physically interesting phenomena in areas ranging from fluid dynamics and nonlinear optics, to low temperature physics and Bose-Einstein condensates. Particular emphasis will be given to the soliton interaction properties. While interaction of scalar solitons is trivial, the interaction of vector and matrix solitons is much richer: the collision is still elastic (the total energy of each soliton is conserved), but there can be redistribution of energy among the components, i.e., a polarization shift. This phenomenon will be discussed both for bright solitons (rapidly decaying to zero as x→±∞ all components), and for dark and dark-bright solitons (with at least one component that decays asymptotically to a constant (non-zero) background). Focusing NLS systems on a non-zero background also admit rogue wave solutions and, in the matrix case, non-zero background can give rise to domain-wall type solitons. These solutions and their properties will also be illustrated.