Long time stability of plane wave solutions to Schrödinger equation on Torus

Mi, Lufang; Sun, Yingte; Wang, Peizhen

doi:10.1080/00036811.2020.1823375

Cited by 2 publications

(2 citation statements)

References 22 publications

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“…Regarding higher Sobolev norms, most results are in the periodic case. See [FGL13] (polynomial bounds for Sobolev initial data) and the preprint [MSW18] (subexponential bounds for Gevrey initial data). A dual point of view is to construct special orbits for which the Sobolev norms grow as fast as possible (thus giving an upper bound on the stability times).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Biasco

Massetti

Procesi

2019

Commun. Math. Phys.

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We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Contents 1. Introduction and main results 1.1. Stability results 1.2. The abstract Birkhoff Normal Form Acknowledgements Part 1. An abstract framework for Birkhoff normal form on sequences spaces 2. Symplectic structure and Hamiltonian flows 3. Immersions for spaces of Hamiltonians. 4. Small divisors and homological equation 5. Abstract Birkhoff Normal Form Part 2. Applications to Gevrey and Sobolev cases 6. Immersions 7. Homological equation 8. Birkhoff normal form 9. Gevrey stability. Proof of Theorem 1.1 10. Sobolev stability Part 3. Appendices Appendix A. Constants. Appendix B. Proofs of the main properties of the norms Appendix C. Small divisor estimates References 1 Namely g is a holomorphic function on the domain Ta := {x ∈ C/2πZ : |Im x| < a} with L 2 -trace on the boundary. 2 A vector ω ∈ R n is called diophantine when it is badly approximated by rationals, i.e. it satisfies, for some γ, τ > 0, |k • ω| ≥ γ|k| −τ , ∀k ∈ Z n \ {0} .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Biasco

Massetti

Procesi

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Regarding higher Sobolev norms, most results are in the periodic case. See [FGL13] (polynomial bounds for Sobolev initial data) and the preprint [MSW18] (subexponential bounds for subanalytic initial data). A dual point of view is to construct special orbits for which the Sobolev norms grow as fast as possible (thus giving an upper bound on the stability times).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Exponential and sub-exponential stability times for the NLS on the circle

Biasco

Massetti

Procesi

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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In this note we study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we state a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Complete proofs are given elsewhere (see [BMP18]).

show abstract

Long time stability of plane wave solutions to Schrödinger equation on Torus

Cited by 2 publications

References 22 publications

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Exponential and sub-exponential stability times for the NLS on the circle

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