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Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential

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Abstract

Consider the focusing nonlinear Schrödinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into nine sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Kenji Nakanishi

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  1. Kenji Nakanishi
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Correspondence to Kenji Nakanishi.

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Communicated by W. Schlag

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Nakanishi, K. Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential. Commun. Math. Phys. 354, 161–212 (2017). https://doi.org/10.1007/s00220-017-2902-1

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  • DOI: https://doi.org/10.1007/s00220-017-2902-1

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