  Models that remain integrable even in confining potentials are extremely rare
and almost non-existent. Here, we consider a one-dimensional hyperbolic
interaction model, which we call as the Hyperbolic Calogero (HC) model. This is
classically integrable even in confining potentials (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 studied the
physics of soliton collisions and quenches using numerical simulations. We have
examined the motion of dual complex variables and found 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. Analytical expressions of soliton solutions have been
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.
