## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 2 (2006), 163-188.

### Large data wave operator for the generalized Korteweg-de Vries equations

#### Abstract

We consider the generalized Korteweg-de Vries equations : \[ u_t + (u_{xx} + u^p)_x =0, \quad t,x \in \mathbb R, \] for $p \in (3,\infty)$. Let $U(t)$ be the associated linear group. Given $V$ in the weighted Sobolev space $H^{2,2} = \{ f \in L^2 : (1+|x|)^2(1-\partial_x^2)f \|_{L^2} < \infty\}$, possibly large, we construct a solution $u(t)$ of the generalized Korteweg-de Vries equation such that : \[ \lim_{t \to \infty} \| u(t) - U(t) V \|_{H^1} =0. \] We also prove uniqueness of such a solution in an adequate space. In the $L^2$-critical case ($p=5$), this result can be improved to any possibly large function $V$ in $L^2$ (with convergence in $L^2$).

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 2 (2006), 163-188.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050523

**Mathematical Reviews number (MathSciNet)**

MR2194502

**Zentralblatt MATH identifier**

1212.35408

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Côte, Raphaël. Large data wave operator for the generalized Korteweg-de Vries equations. Differential Integral Equations 19 (2006), no. 2, 163--188. https://projecteuclid.org/euclid.die/1356050523