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On the Asymptotic Stability of Bound States in 2D Cubic Schrödinger Equation

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Abstract

We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    E. Kirr

  2. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    A. Zarnescu

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  1. E. Kirr
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  2. A. Zarnescu
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Correspondence to E. Kirr.

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Communicated by P. Constantin

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Kirr, E., Zarnescu, A. On the Asymptotic Stability of Bound States in 2D Cubic Schrödinger Equation. Commun. Math. Phys. 272, 443–468 (2007). https://doi.org/10.1007/s00220-007-0233-3

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  • DOI: https://doi.org/10.1007/s00220-007-0233-3

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