Minimal Mass-Energy Dynamics for Solitons Arising in Mathematical Physics

It has been over decades for the study of dispersive evolutionary models ranging from water waves to quantum particles and gases, where is applied the theory of Bose-Einstein statistics, Fermi-Dirac statistics or Maxwell-Boltzmann statistics. However the mathematical understanding of large time asymptotic behaviors of those nonlinear waves are rather poor.

We present an overview of recent progress on the rigorous description of such behaviors in term of long time existence and blowup, regularity as well as the solitary waves. This reveals an integral portion of the grand conjecture, the so-called Soliton Resolution Conjecture. In particular, numerical results are presented for the excited states for Gross-Pitaevskii equation (NLSE) with rotation. The initial data is taken to be the ground state, Gaussian or Thomas-Fermi 2 in the observation.