## Differential and Integral Equations

- Differential Integral Equations
- Volume 10, Number 4 (1997), 777-796.

### On the generalized Korteweg-de Vries-type equations

#### Abstract

We show well-posedness results for the generalized Korteweg-de Vries equation with nonlinear term $F(u)\partial_xu$. We assume $F(u)$ is a $C^4$ function and $F(0)=0$. Using a version of the chain rule for fractional derivatives and some estimates on the evolution group, we prove existence, uniqueness and regularity properties of the solution of the equation when the space of the initial data is $H^s(\mathbb{R}), \, s>1/2$. The theorem we prove is sharp. We obtain all the above results also for a mixed KdV and Schrödinger type equation proposed as a model for the propagation of a signal in an optic fiber.

#### Article information

**Source**

Differential Integral Equations, Volume 10, Number 4 (1997), 777-796.

**Dates**

First available in Project Euclid: 1 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367438641

**Mathematical Reviews number (MathSciNet)**

MR1741772

**Zentralblatt MATH identifier**

0891.35135

**Subjects**

Primary: 35A07

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Staffilani, Gigliola. On the generalized Korteweg-de Vries-type equations. Differential Integral Equations 10 (1997), no. 4, 777--796. https://projecteuclid.org/euclid.die/1367438641