00-28 Ricardo Weder
Center Manifold for Nonintegrable Nonlinear Schroedinger Equations on the Line (40K, Latex) Jan 17, 00
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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 $t \rightarrow \pm \infty$.

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