Abstract
We consider vector Non-linear Schrödinger Equation (NLSE) with balanced loss-gain (BLG), linear coupling (LC) and a general form of cubic nonlinearity. We use a non-unitary transformation to show that the system can be exactly mapped to the same equation without the BLG and LC, and with a modified time-modulated nonlinear interaction. The nonlinear term remains invariant, while BLG and LC are removed completely, for the special case of a pseudo-unitary transformation. The mapping is generic and may be used to construct exactly solvable autonomous as well as non-autonomous vector NLSE with BLG. We present an exactly solvable two-component vector NLSE with BLG which exhibits power-oscillation. An example of a vector NLSE with BLG and arbitrary even number of components is also presented.
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