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2008 Well posedness for Hirota-Satsuma's equation
Rafael Iório, Didier Pilod
Differential Integral Equations 21(11-12): 1177-1192 (2008).

Abstract

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 $H^1(\mathbb R)$.

Citation

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Rafael Iório. Didier Pilod. "Well posedness for Hirota-Satsuma's equation." Differential Integral Equations 21 (11-12) 1177 - 1192, 2008.

Information

Published: 2008
First available in Project Euclid: 14 December 2012

zbMATH: 1224.35364
MathSciNet: MR2482501

Subjects:
Primary: 35Q53
Secondary: 35B30

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.21 • No. 11-12 • 2008
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