Differential and Integral Equations

Well posedness for Hirota-Satsuma's equation

Rafael Iório and Didier Pilod

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We are interested in the initial-value problem associated to the Hirota-Satsuma equation in the real line $$ u_t+u_x-2uu_t+2u_x\int_x^{\infty}u_tdx'-u_{txx}=0, \quad x \in \mathbb R, $$ where $u$ is a real-valued function. This equation models the unidirectional propagation of shallow water waves as the well-known Korteweg-de Vries and Benjamin-Bona-Mahony equations. Here we show local well posedness for initial data in the space $$ \Omega_s=\{\phi \in H^s(\mathbb R) : -1 \notin \sigma(-\partial_x^2-2\phi)\} \ \text{if}\ s>\tfrac12, $$ and small initial data in $H^s(\mathbb R)$ if $0\le s \le \frac12$. We also prove global well posedness for small energy data in $H^1(\mathbb R)$.

Article information

Differential Integral Equations, Volume 21, Number 11-12 (2008), 1177-1192.

First available in Project Euclid: 14 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: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]


Iório, Rafael; Pilod, Didier. Well posedness for Hirota-Satsuma's equation. Differential Integral Equations 21 (2008), no. 11-12, 1177--1192. https://projecteuclid.org/euclid.die/1355502298

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