Center Manifold for Nonintegrable Nonlinear Schrödinger Equations on the Line

Abstract:

In this paper we study the following nonlinear Schrödinger equation on the line,

where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x→±∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0, ∞). 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→±∞. In general, the periodic orbits are different for t→±∞. 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→±∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→±∞.