Non-linear Schrödinger equation with time-dependent balanced loss-gain and space–time modulated non-linear interaction

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