Nonlinear Schr$\ddot{o}$dinger equation with timedependent balanced lossgain and spacetime modulated nonlinear interaction
Abstract
We consider a class of one dimensional vector Nonlinear Schr$\ddot{o}$dinger Equation(NLSE) in an external complex potential with Balanced LossGain(BLG) and Linear Coupling(LC) among the components of the Schr$\ddot{o}$dinger field. The solvability of the generic system is investigated for various combinations of time modulated LC and BLG terms, spacetime dependent strength of the nonlinear interaction and complex potential. We use a nonunitary transformation followed by a reformulation of the differential equation in a new coordinate system to map the NLSE to solvable equations. Several physically motivated examples of exactly solvable systems are presented for various combinations of LC and BLG, external complex potential and nonlinear interaction. Exact localized nonlinear modes with spatially constant phase may be obtained for any real potential for which the corresponding linear Schr$\ddot{o}$dinger equation is solvable. A method based on supersymmetric quantum mechanics is devised to construct exact localized nonlinear modes for a class of complex potentials. The real superpotential corresponding to any exactly solved linear Schr$\ddot{o}$dinger equation may be used to find a complexpotential for which exact localized nonlinear modes for the NLSE can be obtained. The solutions with singular phases are obtained for a few complex potentials.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.04802
 Bibcode:
 2021arXiv211204802G
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 Two column, 14 pages,No figure