Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-04-15 DOI:10.1007/s00041-024-10084-0
X. Carvajal, M. Panthee
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Abstract

We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion

$$\begin{aligned} \partial _{t}u+i\alpha \partial ^{2}_{x}u- \partial ^{3}_{x}u+i\beta |u|^{2}u = 0, \quad x,t \in \mathbb R, \end{aligned}$$

for given data in the Sobolev space \(H^s(\mathbb R)\). This IVP is known to be locally well-posed for given data with Sobolev regularity \(s>-\frac{1}{4}\) and globally well-posed for \(s\ge 0\) (Carvajal in Electron J Differ Equ 2004:1–10, 2004). For given data in \(H^s(\mathbb R)\), \(0>s> -\frac{1}{4}\) no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in (Carvajal in Electron J Differ Equ 2004:1–10, 2004).

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具有三阶分散性的立方非线性薛定谔方程的尖锐全局解析性
我们考虑与具有三阶分散性的立方非线性薛定谔方程相关的初值问题(IVP) $$\begin{aligned}\partial _{t}u+i\alpha \partial ^{2}_{x}u- \partial ^{3}_{x}u+i\beta |u|^{2}u = 0, \quad x,t \in \mathbb R, \end{aligned}$$对于给定数据在 Sobolev 空间 \(H^s(\mathbb R)\)中。众所周知,对于具有 Sobolev 正则性的(s>-\frac{1}{4})给定数据,这个 IVP 是局部好求的;对于(s\ge 0\)给定数据,这个 IVP 是全局好求的(Carvajal 在 Electron J Differ Equ 2004:1-10, 2004)。对于给定数据在 \(H^s(\mathbb R)\), \(0>s> -\frac{1}{4}\) 中的全局好摆性结果尚不清楚。在这项工作中,我们为这类数据推导出了一个几乎守恒的量,并得到了一个尖锐的全局可好求解结果。我们的结果回答了 (Carvajal in Electron J Differ Equ 2004:1-10, 2004) 中留下的问题。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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