KIT - Department of Mathematics - Dynamics of a soliton in an external potential

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PDE-Seminar: Talk by Dario Bambusi

PDE-Seminar: Talk by Dario Bambusi

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

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

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.