Differential and Integral Equations

The Cauchy problem for a nonlocal perturbation of the KdV equation

Borys Alvarez Samaniego

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Let $\mathcal H$ denote the Hilbert transform and $\eta \ge 0$. We show that the initial-value problems $ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ and $ u_t + \frac{1}{2} (u_x)^2 + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ are globally well-posed in $H^s(\mathbb{R})$, $s \ge 1$, $\eta>0$. We study the limiting behavior of the solutions of the first equation as $\eta$ tends to zero in $H^s(\mathbb{R})$ and $s \ge 2$. Moreover, we prove a unique continuation theorem for the first equation in $\mathcal F_{3,3}(\mathbb{R})=H^3(\mathbb{R}) \cap L^2_3(\mathbb{R})$, $\eta>0$.

Article information

Differential Integral Equations, Volume 16, Number 10 (2003), 1249-1280.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35G25: Initial value problems for nonlinear higher-order equations


Samaniego, Borys Alvarez. The Cauchy problem for a nonlocal perturbation of the KdV equation. Differential Integral Equations 16 (2003), no. 10, 1249--1280. https://projecteuclid.org/euclid.die/1356060547

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