Computation of Homoclinic Solutions to Periodic Orbits in a Reduced Water-wave Problem

This paper concerns homoclinic solutions to periodic orbits in a fourth-order Hamiltonian system arising from a reduction of the classical water-wave problem in the presence of surface tension. These solutions correspond to travelling solitary waves which converge to non-decaying ripples at innnity. An analytical result of Amick and Toland, showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit, is extended numerically. Also, a related result by Amick and McLeod, showing the non-existence of homoclinic solutions to zero, is motivated geometrically. A general boundary-value method is constructed for continuation of homoclinic orbits to periodic orbits in Hamiltonian and reversible systems. Numerical results are presented using the path-following software auto, showing that the Amick{Toland solutions persist well away from the singular limit and for large-amplitude periodic orbits. Special account is taken of the phase shift between the two periodic solutions in the asymptotic limits. Furtermore, new multi-modal homoclinic solutions to periodic orbits are shown to exist under a transversal-ity hypothesis, which is veriied a posteriori by explicit computation. Continuation of these new solutions reveals limit points with respect to the singular parameter.