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June  2005, 4(2): 367-388. doi: 10.3934/cpaa.2005.4.367

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, Brazil, Brazil

Received  May 2004 Revised  December 2004 Published  March 2005

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.
Keywords: periodic travelling-waves, nonlinear stability., KdV equation, Dispersive equations, cnoidal waves.
Mathematics Subject Classification: 76B25,35Q51,35Q53.
Citation: Jaime Angulo Pava, Borys Alvarez Samaniego. Existence and stability of periodic travelling-wavesolutions of the Benjamin equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 367-388. doi: 10.3934/cpaa.2005.4.367
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