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Nonlinear Sciences > Exactly Solvable and Integrable Systems
Title: Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schrödinger equation with balanced loss-gain
(Submitted on 6 Dec 2022 (v1), last revised 18 Sep 2023 (this version, v2))
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.
Submission history
From: Supriyo Ghosh [view email][v1] Tue, 6 Dec 2022 06:53:15 GMT (386kb,D)
[v2] Mon, 18 Sep 2023 10:46:57 GMT (1541kb,D)
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