Existence and stability of periodic travelling-wavesolutions of the Benjamin equation

Jaime Angulo Pava; Borys Alvarez Samaniego

doi:10.3934/cpaa.2005.4.367

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2005, Volume 4, Issue 2: 367-388. Doi: 10.3934/cpaa.2005.4.367

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Existence and stability of periodic travelling-wavesolutions of the Benjamin equation

Jaime Angulo Pava^1, and
Borys Alvarez Samaniego^1,

1.
Department of Mathematics, IMECC-UNICAMP, C.P. 6065, CEP 13083-970, Campinas, SP

Received: May 2004

Revised: December 2004

Published online: March 2005

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Abstract

A family of steady periodic water waves in very deep fluids when the surface tension is present and satisfying the following nonlinear pseudo-differential equation $ u_t + u u_x + u_{x x x} +l \mathcal{H} u_{x x}=0$, known as the Benjamin equation, is shown to exist. Here $\mathcal{H}$ denotes the periodic Hilbert transform and $l \in\mathbb{R}$. By fixing a minimal period we obtain, via the implicit function theorem, an analytic curve of periodic travelling-wave solutions depending on the parameter $l$. Moreover, by making some changes in the abstract stability theory developed by Grillakis, Shatah, and Strauss, we prove that these travelling waves are nonlinearly stable to perturbations with the same wavelength.

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Mathematics Subject Classification: 76B25,35Q51,35Q53.

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