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Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics

We present a first-order formulation of the Calogero model in external potentials in terms of a generating function, which simplifies the derivation of its dual form. Solitons naturally appear in this formulation as particles of negative mass. Using this method, we obtain the dual form of Calogero particles in external quartic, trigonometric and hyperbolic potentials, which were known to be integrable but had no known dual formulation. We derive the corresponding soliton solutions, generalizing earlier results for the harmonic Calogero system, and present numerical results that demonstrate the integrable nature of the soliton motion. We also give the collective fluid mechanical formulation of these models and derive the corresponding fluid soliton solutions in terms of meromorphic fields, commenting on issues of stability and integrability.