Differential and Integral Equations

The Cauchy problem of the Schrödinger-Korteweg-de Vries system

Yifei Wu

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Abstract

We study the Cauchy problem of the Schrödinger-Korteweg-de Vries system. First, we establish local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show that they are sharp in some well-posedness thresholds. In particular, we obtain local well posedness for the initial data in $H^{-\frac{3}{16}+}({\mathbb{R}})\times H^{-\frac{3}{4}+}({\mathbb{R}})$ in the resonant case; it is almost optimal except for the endpoint. Finally, we establish global well-posedness results in $H^s({\mathbb{R}})\times H^s({\mathbb{R}})$ when $s>\frac{1}{2}$ regardless of whether we are in the resonant case or in the non-resonant case, which improves the results of Pecher (2005).

Article information

Source
Differential Integral Equations, Volume 23, Number 5/6 (2010), 569-600.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019310

Mathematical Reviews number (MathSciNet)
MR2654249

Zentralblatt MATH identifier
1240.35528

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Wu, Yifei. The Cauchy problem of the Schrödinger-Korteweg-de Vries system. Differential Integral Equations 23 (2010), no. 5/6, 569--600. https://projecteuclid.org/euclid.die/1356019310


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