Geometry of Integrable systems

Dynamics of a soliton in an external potential

Jun 8, 2017, 9:30 AM

40m

005 (SISSA)

Speaker

Prof. Dario Bambusi (Dipartimento di Matematica, Università degli studi di Milano)

Description

Joint work with A. Fusé, A. Maspero, S. Sansottera Consider the nonlinear Schrödinger equation $-i{\psi }_t=-\mathrm{\Delta }\psi -\beta (|\psi {|}^2)\psi +ϵV\psi ,\beta \in C^{\omega },V\in \mathcal{S}\cap C^{\omega };$ it is well known that, when $ϵ=0$, under suitable conditions on $\beta$, the NLS admitts traveling wave solutions (soliton for short). When $ϵ\ne 0$, heuristic considerations suggest that the soliton should move as a particle subject to a mechanical force due to an effective potential computed from $V$. The problem is to understand if this is true or not. In this talk I will show that the soliton does not exchange neither energy nor angular momentum with the rest of the field for times of the order ${ϵ}^{-r}$ for any $r$. This allows to deduce some informations on the trajectory of the soliton. The proof is composed by two steps: first one shows that the Hamiltonian of the NLS can be rewritten in suitable coordinates as follows $H={ϵ}^{1/2}{H}_{mech}(p,q)+\frac{1}{2}⟨E{L}_0\varphi ,\varphi ⟩+h.o.t.$ where $H_{mech}(p,q)$ is the hamiltonian of a mechanical particle in a central potential and describes the motion of the soliton's barycenter, while $\varphi$ is a function representing the "free'' field. The second step consistes in applying a Nekhoroshev type theorem; in turn this requires to verify a nondegeneracy hypothesis on $H_{mech}(p,q)$ (quasiconvexity). This is obtained by applying a result that we recently got ensuring that such an assumption is always satisfied in the central motion problem except for Harmonic end Keplerian potentials. In the talk I will also try to give the ideas of the proof of this second result.