Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation

TL;DR

This paper introduces a non-unitary transformation method to map vector nonlinear Schrödinger equations with balanced loss and gain to solvable forms, enabling the construction of exactly solvable models with power-oscillation.

Contribution

The authors develop a generic non-unitary transformation approach to construct exactly solvable vector NLSE models with balanced loss and gain, including autonomous and non-autonomous cases.

Findings

Mapped vector NLSE with BLG to simpler equations

Constructed an exactly solvable two-component vector NLSE with power-oscillation

Presented an example with arbitrary even number of components

Abstract

We consider vector Non-linear Schrodinger 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.