Systematic numerical search of steady-state solutions for the system of coupled Nonlinear Schrodinger Equations with additional double-well potential

Abstract

We study stationary solutions of a system of coupled Nonlinear Schrodinger equations with additional double-well potential. In the theory of Bose-Einstein Condensate (BEC) these equations are called the Gross-Pitaevskii equations and the system describes the dynamics of a sigar-shape BEC cloud that consists of atom of two types. Stationary solutions of the system (called also "nonlinear modes") satisfy a system of two non-autonomous ODEs. The problem is stated as follows: it is necessary (i) to find all localized (i.e., that vanish at plus and minus infinity) solutions of this ODE system that coexist for given parameter values and (ii) to justify the completeness of the search conducted. For this purpose, we employed the method of "filtering out" solutions with singularities developed by our team previously. It consists in scanning of some area in the space of initial data for the Cauchy problem for the system. While this procedure, the solutions that have singularities (i.e., that tend to infinity at a finite value of argument) are detected and excluded from consideration. The diagnostics of the singularities and the criterium for the scanning to stop are based on rigorous mathematical statements. As a result, we identify the parameter regions where nonlinear modes of different types exist ("bright-dark solitons", "dark-dark solitons" etc.)