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=-\Delta\psi-\beta(|\psi|^2)\psi+\epsilon V\psi\
,\quad \beta\in C^\omega\ ,\quad V\in{\mathcal S}\cap C^\omega\ ;
$
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 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 $\epsilon^{-r}$ 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 = \epsilon^{1/2}H_{mech}(p, q) + \frac{1}{2} \langle E
L_0 \phi, \phi\rangle + 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 $\phi$ 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.
Primary author
Prof.
Dario Bambusi
(Dipartimento di Matematica, Università degli studi di Milano)
