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