Solitary waves with surface tension. I: Trajectories homoclinic to periodic orbits in four dimensions (Q1188033)

The authors show the existence of trajectories homoclinic to non-trivial orbits of a nonlinear fourth order water wave equation which contains a small parameter \(\varepsilon\) in the highest (fourth order) derivative. They first give a detailed treatment of the existence theory for periodic solutions of this equation; they prove that for sufficiently small parameter there is a circle of even periodic orbits which contains the constant solutions. Then they can show that again for sufficiently small parameters there is an even solution which lies in a vicinity of order \(\varepsilon^ 2\) and of exponential decay in space to the sum of the well-known even sech-squared-solution of the water wave equation for \(\varepsilon=0\) and a translation of a specific periodic solution.