On Duality, Solitons and Quenches in Generalized Calogero Models in upto-Quartic Polynomial Potentials

The family of Calogero models play a prominent role in Physics and Mathematics.These models are known to be classically integrable for any number of particles even in some external potentials. They can also be remarkably formulated in terms of Matrices evolving in external potentials which makes the underlying structure more transparent thereby addressing questions on integrability. We show the existence of a dual model for the system [1,2].We write the duality relations explicitly between particles and the dual variables. We show that the dual system can contain small number of particles. We argue that such reductions play a role of soliton solutions of the model where the dual variables act like solitonic excitations [1,2].After obtaining such solutions for finite number of Calogero particles, we present the solutions in a collective field theory escription, where the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in the hydrodynamic model are finite dimensional reductions of the integrable field theory and describe the evolution of density bumps and velocity in curved backgrounds. We will also present some results on non-linear dynamics and quenches in these models
[3,4].

[1] M. Kulkarni, A. Polychronakos (2016, in preparation)
[2] A. G. Abanov, A. Gromov, M. Kulkarni, J. Phys. A: Math. Theor. 44, 295203  (2011)
[3] F. Franchini, A. Gromov, M. Kulkarni, A. Trombettoni,  J. Phys. A: Math. Theor. 48, 28FT01  (2015)
[4] F. Franchini, M. Kulkarni, A. Trombettoni (arXiv:1603.03051)