Barbara Prinari, Department of Mathematics, State University of New York (SUNY) at Buffalo

Novel soliton and breather solutions of scalar and coupled Ablowitz-Ladik systems

In this talk we will present novel soliton and breather solutions of scalar and coupled Ablowitz-Ladik systems. In the first part, we will discuss solutions of the focusing and defocusing Ablowitz–Ladik lattice on a non-zero background which are discrete analogs of the Kuznetsov–Ma (KM) breathers of the focusing nonlinear Schrödinger equation. In the focusing case, we present novel, purely discrete KM breathers which are regular for all choices of the breather parameters. In the defocusing case with large background, we will discuss the conditions on the background and on the spectral parameters that guarantee the KM solution to be non-singular on the lattice for all times. Furthermore, we will present a novel KM-type breather solution which is also regular on the lattice under the same conditions.  In the second part of the talk, we will present novel breather solutions of an integrable discretization of the Manakov system (a coupled Ablowitz-Ladik system) and discuss their interesting interaction properties.