BIFURCATION OF A PLETHORA OF MULTIMODAL HOMOCLINIC ORBITS FOR AUTONOMOUS HAMILTONIAN-SYSTEMS

We consider a class of fourth-order, reversible, autonomous Hamiltonian systems for which an orbit homoclinic to a node on the stable manifold is known to exist. If parameters are varied so that on the stable manifold the node becomes a focus (i.e. when 4 real eigenvalues become complex) it is shown that generic homoclinic orbits of systems in this class bifurcate into a countably infinite set of distinct symmetric homoclinic orbits. As an application we show that the problem u(t) + Pu(t) + u(t) - u(t)2 = 0 u(0) = u(0) = 0, u not-equal 0 u(+/- infinity) = u(+/- infinity) = u(+/- infinity) = u(+/- infinity) = 0 which has a unique (up to translations) solution, of large amplitude, for P less-than-or-equal-to -2 has a countably infinite set of distinct large amplitude solutions for each P is-an-element-of (-2, -2 + epsilon) for some epsilon > 0. For P is-an-element-of (2 - epsilon, 2) it has at least two small amplitude solutions and for all P < 2 it has at least one solution. The contrast between uniqueness for P < -2 and large multiplicity for P > -2 is a remarkable bifurcation which has particular implications for the solitary water-wave problem with surface tension. In particular, it leads to a conjecture that for fixed Froude and Bond numbers in a particular region of parameter space there exist very large numbers of multi-modal, distinct solitary water-wave solutions of the Euler equations.