A NOTE ON DECAY PROPERTY OF NONLINEAR SCHRÖDINGER EQUATIONS

Chenjie Fan; Zehua Zhao

doi:10.1090/proc/16296

A NOTE ON DECAY PROPERTY OF NONLINEAR SCHRÖDINGER EQUATIONS

Chenjie Fan
, Zehua Zhao

数学学院

CAS - Academy of Mathematics and System Sciences
Beijing Institute of Technology

科研成果: 期刊稿件 › 文章 › 同行评审

7 引用（Scopus）

摘要

In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in H^s for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all > 0, one has sup_t>₀ tu(t) − e^it^Δu⁺_H 1/2 = ∞. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L¹. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L¹ condition of initial data will get rid of this phenomena.

源语言	英语
页（从-至）	2527-2542
页数	16
期刊	Proceedings of the American Mathematical Society
卷	151
期	6
DOI	https://doi.org/10.1090/proc/16296
出版状态	已出版 - 6月 2023

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title = "A NOTE ON DECAY PROPERTY OF NONLINEAR SCHR{\"O}DINGER EQUATIONS",

abstract = "In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in Hs for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all > 0, one has supt>0 tu(t) − eitΔu+H 1/2 = ∞. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L1. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L1 condition of initial data will get rid of this phenomena.",

keywords = "Nonlinear Schr{\"o}dinger equation, convergence rate, decay estimate, scattering, scattering rate",

author = "Chenjie Fan and Zehua Zhao",

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year = "2023",

month = jun,

doi = "10.1090/proc/16296",

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AU - Fan, Chenjie

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N2 - In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in Hs for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all > 0, one has supt>0 tu(t) − eitΔu+H 1/2 = ∞. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L1. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L1 condition of initial data will get rid of this phenomena.

AB - In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in Hs for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all > 0, one has supt>0 tu(t) − eitΔu+H 1/2 = ∞. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L1. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L1 condition of initial data will get rid of this phenomena.

KW - Nonlinear Schrödinger equation

KW - convergence rate

KW - decay estimate

KW - scattering

KW - scattering rate

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 6

ER -

A NOTE ON DECAY PROPERTY OF NONLINEAR SCHRÖDINGER EQUATIONS

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