### 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

## Abstract

Consider the nonlinear Schrödinger equation

it is well known that, when , under suitable conditions on , the NLS admitts traveling wave solutions (soliton for short). When , 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 . Numerical investigation, done in the case of a 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 for any . 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.