Integrable coupled Li$\acute{e}$nard-type systems with balanced loss and gain

TL;DR

This paper develops a Hamiltonian framework for many-particle systems with space-dependent balanced loss and gain, revealing integrability properties, exact solutions, and quantization aspects for coupled Lie9nard-type differential equations.

Contribution

It introduces a Hamiltonian formulation for such systems, identifies conditions for partial and complete integrability, and provides explicit solutions and quantization methods.

Findings

Partially integrable for systems with translational or rotational symmetry.

Complete integrability for two-particle systems with involutive integrals.

Exact and quasi-exact solutions including stable periodic states.

Abstract

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient.The resulting equations of motion from the Hamiltonian are a system of coupled Li$\acute{e}$nard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of $m+1$ integrals of motion are constructed for a system of $2m$ particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and space-dependent gain/loss co-efficients, which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain-loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.