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The elliptic equation Au = F(u) possesses non-trivial solutions in R n which are exponentially small at infinity, for a large class of functions F. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.

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We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions.

The peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods. We give a very simple proof of the orbital stability of the peakons in the H1 norm. © 2000 John Wiley & Sons, Inc.

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The peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods. We give a very simple proof of the orbital stability of the peakons in the H 1 norm.