## Advances in Differential Equations

- Adv. Differential Equations
- Volume 9, Number 5-6 (2004), 509-538.

### Schrödinger group on Zhidkov spaces

#### Abstract

We consider the Cauchy problem for nonlinear Schrödinger equations on $\mathbb{R}^{n}$ with nonzero boundary condition at infinity, a situation which occurs in stability studies of dark solitons. We prove that the Schrödinger operator generates a group on Zhidkov spaces $X^{k}(\mathbb{R}^{n})$ for $k>n/2$, and that the Cauchy problem for NLS is locally well-posed on the same Zhidkov spaces. We justify the conservation of classical invariants which implies in some cases the global well-posedness of the Cauchy problem.

#### Article information

**Source**

Adv. Differential Equations Volume 9, Number 5-6 (2004), 509-538.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867934

**Mathematical Reviews number (MathSciNet)**

MR2099970

**Zentralblatt MATH identifier**

0476.03047

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72] 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]

#### Citation

Gallo, Clément. Schrödinger group on Zhidkov spaces. Adv. Differential Equations 9 (2004), no. 5-6, 509--538. https://projecteuclid.org/euclid.ade/1355867934.