## Differential and Integral Equations

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

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

#### 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