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On the Well-Posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on \(\mathbb {T}^3\)

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Abstract

In this paper, we study cubic and quintic nonlinear Schrödinger systems on three-dimensional tori, with initial data in an adapted Hilbert space \(H^s_{{\underline{\lambda }}},\) and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusing systems. We show that local solutions of the defocusing cubic system with initial data in \(H^1_{{\underline{\lambda }}}\) can be extended for all time. Additionally, we prove that global well-posedness holds in the quintic system, focusing or defocusing, for initial data with sufficiently small \(H^1_{{\underline{\lambda }}}\) norm. Finally, we use the energy-Casimir method to prove the existence and uniqueness, and nonlinear stability of a class of stationary states of the defocusing cubic and quintic nonlinear Schrödinger systems.

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Acknowledgements

We thank the anonymous referees for very helpful comments. T.C. gratefully acknowledges support by the NSF through grants DMS-1151414 (CAREER), DMS-1716198, DMS-2009800, and the RTG Grant DMS-1840314 Analysis of PDE. A.U. was supported by NSF grants DMS-1716198 and DMS-2009800 through T.C., and by the RTG Grant DMS-1840314 Analysis of PDE.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

    Thomas Chen & Amie Bowles Urban

  2. Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA

    Amie Bowles Urban

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  1. Thomas Chen
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  2. Amie Bowles Urban
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Correspondence to Thomas Chen.

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Communicated by Nader Masmoudi.

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Chen, T., Bowles Urban, A. On the Well-Posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on \(\mathbb {T}^3\). Ann. Henri Poincaré 25, 1657–1692 (2024). https://doi.org/10.1007/s00023-023-01371-5

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  • DOI: https://doi.org/10.1007/s00023-023-01371-5

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