## Differential and Integral Equations

- Differential Integral Equations
- Volume 16, Number 7 (2003), 841-864.

### Well-posedness for the fourth-order nonlinear Schrödinger-type equation related to the vortex filament

#### Abstract

We consider the time-local well-posedness for the initial-value problem of the fourth-order nonlinear Schrödinger-type equation in one space dimension which describes the motion of the vortex filament. By using the method of Fourier restriction norm introduced by Bourgain [3] and Kenig-Ponce-Vega [17]--[19], we show the time-local well-posedness in the Sobolev space $H^s(\mathbb R)$ with $s\ge1/2$ under certain coefficient conditions.

#### Article information

**Source**

Differential Integral Equations, Volume 16, Number 7 (2003), 841-864.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060600

**Mathematical Reviews number (MathSciNet)**

MR1988728

**Zentralblatt MATH identifier**

1042.35077

**Subjects**

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

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 76D17: Viscous vortex flows

#### Citation

Segata, J. Well-posedness for the fourth-order nonlinear Schrödinger-type equation related to the vortex filament. Differential Integral Equations 16 (2003), no. 7, 841--864. https://projecteuclid.org/euclid.die/1356060600