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Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on \({\mathbb{R}^3}\)

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Abstract

We consider the cubic–quintic nonlinear Schrödinger equation:

$$i\partial_t u = -\Delta u - |u|^2u + |u|^4u.$$

In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region \({{\mathcal{R}}}\) of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095, USA

    Rowan Killip

  2. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Tadahiro Oh

  3. The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Tadahiro Oh

  4. Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ, 08544-1000, USA

    Oana Pocovnicu

  5. School of Mathematics, Institute for Advanced Study, Einstein Dr., Princeton, NJ, 08540, USA

    Oana Pocovnicu

  6. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Oana Pocovnicu

  7. The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, UK

    Oana Pocovnicu

  8. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095, USA

    Monica Vişan

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  1. Rowan Killip
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  2. Tadahiro Oh
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  3. Oana Pocovnicu
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  4. Monica Vişan
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Correspondence to Rowan Killip.

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Killip, R., Oh, T., Pocovnicu, O. et al. Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on \({\mathbb{R}^3}\) . Arch Rational Mech Anal 225, 469–548 (2017). https://doi.org/10.1007/s00205-017-1109-0

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