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Dynamics of a soliton in an external potential

  • Speaker: Prof. Dr. Dario Bambusi
  • Place: Seminar room 1.067, Building 20.30
  • Time: 30.6.2016, 14:30 - 15:30
  • Invited by: CRC 1173


Consider the nonlinear Schrödinger equation

-i\psi_t=-\Delta\psi-\beta(|\psi|^2)\psi+\epsilon V\psi\
 ,\quad \beta\in C^\infty(\mathbb{R})\ ,\quad V\in{\cal S}\ ;

it is well known that, when \epsilon=0, under suitable conditions on \beta, the NLS admitts traveling wave solutions (soliton for short). When \epsilon\not=0, heuristic considerations suggest that the soliton should move as a particle subject to a mechanical force due to the potential. The problem of understanding if this is true or not has attracked a remarkable amount of work and it has been show that in the most favorable cases, the dynamics of the soliton is close to the dynamics of a mechanical particle at least for times of order \epsilon^{-3/2}. Numerical investigation, done in the case of a V=\delta have shown that this is not true for longer times.

In will show that the orbit of the soliton remains close to the mechanical orbit of a particle for much longer times, namely for times of the order \epsilon^{-r} for any r. The main point is that one has to renounce to control the position of the soliton on the orbit.

The proof is composed by three steps: introduction of Darboux coordinates, development of Hamiltonian perturbation theory and use of Strichartz estimates.