Smoothing Properties and Soliton Waves for Hamiltonian Systems, I: Schrödinger Equations with Magnetic Fields

As is known, nonlinear Schrödinger equations (NLS) are generated by the Hamiltonian, certain conservation law in physics. We consider the magnetic NLS that models rotating BEC. We show some working examples including harmonic potentials and more general magnetic potentials. In the focusing case there may exhibit soliton wave and finite time blowup. However the conditions on the blowup are subtle where the electric and magnetic potentials are involved. The quantized vortex is another phenomena that arises in a magnetic field. This is a remarkable signature for the superfluidity of the BEC. If time permitting I will also indicate further study on the use of invariant measure associated to the magnetic Hamiltonian in the presence of random initial data. This has been an efficient method in dealing with low regularity data for Hamiltonian systems, a subject with growing interest in the field in recent years.