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arXiv · 2212.02786

Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schr\"odinger equation with balanced loss-gain

Supriyo Ghosh·Pijush K. Ghosh
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Abstract

We consider a class of one dimensional Vector Nonlocal Non-linear Schr\"odinger Equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schr\"odinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power, $\cal{PT}$-symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc. are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The cordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.

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BibTeXRIS

Supriyo Ghosh, Pijush K. Ghosh. 2022-12-06. Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schr\"odinger equation with balanced loss-gain. https://doi.org/10.1088/1402-4896%2Facfc88

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