Bounds on Sobolev norms for the  defocusing nonlinear
Schrödinger  equation on general flat tori

Catoire, F.; Wang, W. M.

doi:10.3934/cpaa.2010.9.483

Cited by 40 publications

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“…1 Up to an loss of derivative, recovered in some cases in the work of Killip and Visan [40]. 2 For some Strichartz estimates on irrational tori with loss of derivative see also [10,14,37]. where x ∈ T d ω and ∆ is the Laplace-Beltrami operator for the irrational torus 3 .…”

Section: Introductionmentioning

confidence: 99%

Stability of the Cubic Nonlinear Schrodinger Equation on an Irrational Torus

Staffilani¹,

Wilson²

2020

SIAM J. Math. Anal.

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A characteristic of the defocusing cubic nonlinear Schrödinger equation (NLSE), when defined so that the space variable is the multi-dimensional square (hence rational) torus, is that there exist solutions that start with arbitrarily small norms Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE, [18] and [13]. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.

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Section: Introductionmentioning

confidence: 99%

Stability of the Cubic Nonlinear Schrodinger Equation on an Irrational Torus

Staffilani¹,

Wilson²

2020

SIAM J. Math. Anal.

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“…So far, there are also many results about Sobolev norms growth for equations on rectangular tori, see e.g. [3,17,20,21]. Remark 1.3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Analytic solutions of nonlinear elliptic equations on rectangular tori

Shi

2019

Journal of Differential Equations

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In this paper, we consider the nonlinear elliptic equations on rectangular tori. Using methods in the study of KAM theory and Anderson localization, we prove that these equations admit many analytic solutions.Remark 1.2. Regarding the rectangular tori, there have been plenty of results on the study of Schrödinger equations on rectangular tori. Bourgain [4] firstly addressed the question of Strichartz estimates for Schrödinger equations on tori. In [11], Bourgain carried out the study

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“…As mentioned above, the study of the NLS on general rectangular tori was first started in the work of Bourgain [10] where it was shown that certain Strichartz estimates with a loss of derivative hold. Some other partial results for the NLS on irrational tori were obtained in [17,28,41,63]. The combined range of estimates proved for irrational tori in these works are weaker than those proved by Bourgain in [6] due to number-theoretical difficulties.…”

Section: Is a Rational Torus And Formentioning

confidence: 86%

The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

Nahmod

2015

Bull. Amer. Math. Soc.

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Abstract. The field of nonlinear dispersive and wave equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry, analytic number theory and most recently probability, into the existing functional analytic methods. In these lectures we concentrate on the semilinear Schrödinger equation defined on tori and discuss the most important developments in the analysis of these equations. In particular, we discuss in some detail recent work by J. Bourgain and C. Demeter proving the 2 decoupling conjecture and as a consequence the full range of Strichartz estimates on either rational or irrational tori, thus settling an important earlier conjecture by Bourgain.

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Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori

Cited by 40 publications

References 0 publications

Stability of the Cubic Nonlinear Schrodinger Equation on an Irrational Torus

Stability of the Cubic Nonlinear Schrodinger Equation on an Irrational Torus

Analytic solutions of nonlinear elliptic equations on rectangular tori

The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

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