Ricardo Weder
Center Manifold for Nonintegrable Nonlinear Schroedinger Equations on the Line
(40K, Latex)
ABSTRACT. In this paper we study the following nonlinear Schr\"{o}dinger equation on the line,
$$
i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(t,x) +
V(x) u(t,x) + f(x, |u|)\frac{u(t,x)}{|u(t,x)|}, u(0,x)=\phi (x),
$$
where $f$ is real-valued, and it satisfies suitable conditions on regularity, on grow as a function of
$u $ and on
decay as $ x \rightarrow \pm \infty$.
The {\it generic} potential, $ V$, is real-valued and it is chosen so that the spectrum of $H:= -\frac{d^2}{ d x^2} +V$
consists of one simple negative eigenvalue and absolutely-continuous spectrum filling $[0, \infty)$. The
solutions to this equation have, in general,
a localized and a dispersive component. The nonlinear bound states, that
bifurcate from the zero solution at the energy of the eigenvalue of $H$, define an invariant center manifold that
consists of the orbits of time-periodic localized solutions . We
prove that all small solutions approach a particular periodic orbit in the center manifold as
$t \rightarrow \pm \infty$. In general,
the periodic orbits are different for $ t \rightarrow \pm \infty$. Our result implies also that the
nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear
bound state is asymptotic as $ t \rightarrow \pm \infty $ to the periodic orbits of nearby nonlinear bound states that
are, in general, different for $ t \rightarrow \pm \infty$.