Integrable coupled Liénard-type systems with balanced loss and gain

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 Liénard-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 spacedependent 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.

Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction: general results and a case study with Landau Hamiltonian

“…Even for a simpler case of η ik = (−1) i+1 δ ik , Γ i = ∂Γ ∂xi with even N and Γ being the potential of rational Calogero model, no definite answer is known for N > 2 [8]. The Hamiltonian formulation of systems with balanced loss and gain [8,10], which is summarized below, constitute a special case of Eq. (1), albeit encompassing a very large class of such models.…”

“…A Hamiltonian formulation of many-particle systems with space-dependent balanced loss and gain is presented in Refs. [8,10]. The analysis excludes constrained systems, systems with the dissipative term depending nonlinearly on the velocity and any other non-standard Hamiltonian formulations.…”

“…Moreover, all the Hamiltonian systems with balanced loss and gain considered in the literature [1,2,3,4,5,6,7], prior to the general formulation of such systems in Ref. [8,9,10], may be identified as defined in the background of a pseudo-Euclidean metric. The generalized momenta Π is defined by,…”

“…(14). Investigations [8,10,9] on Hamiltonian systems with balanced loss and gain have been so far restricted to a diagonal D which is a symmetric matrix. This is primarily because only the diagonal elements of D are relevant for determining whether a system is dissipative or nondissipative.…”

“…, a particular representation of the matrices is presented in Ref. [10], which is reproduced below for further discussions:…”