Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory

Models that remain integrable even in confining potentials are extremely rare and almost non-existent. Introducing an external potential in integrable models, breaks the conservation of even simple quantities such as total linear momentum, and hence, the system no longer remains integrable in general. The Calegero family provides a unique and rare opportunity to study models, integrable even in external potentials. Here, we consider a one-dimensional hyperbolic interaction model, which we call as the hyperbolic Calogero (HC) model. This is classically integrable even in a confining potential (which have box-like shapes). We present a first-order formulation of the HC model in an external confining potential. Using the rich property of duality, we find multi-soliton solutions of this confined integrable model. Absence of solitons correspond to the equilibrium solution of the model. We demonstrate the dynamics of multi-soliton solutions via brute-force numerical simulations. We study the physics of soliton collisions and quenches using numerical simulations. We examine the motion of dual complex variables and find an analytic expression for the time period in a certain limit. We give the field theory description of this model and find the background solution (absence of solitons) analytically in the large-N limit (where N is the number of particles). Analytical expressions of soliton solutions are obtained in the absence of external confining potential. Our work is of importance to understand the general features of trapped interacting particles that remain classically integrable and can be of relevance to the collective behaviour of trapped cold atomic gases as well.