Codimension-one persistence beyond all orders of homoclinic orbits to singular saddle centres in reversible systems

The persistence from a singular limit is studied of homoclinic orbits to a saddle-centre equilibrium in four-dimensional reversible vector fields. The linearisation is assumed to have eigenvalues +i omega, -i omega, + lambda and - lambda, lambda and omega >0, with the singular limit being lambda -> 0. Recent work by E. Lombardi has shown the generic non-persistence of such homoclinic solutions under perturbations that break the integrability of a normal form due to a splitting term (Melnikov function) that is exponentially small in lambda/omega. The arguments of Lombardi are adapted to show that given the correct sign of the coefficient b_2 in the normal form that couples the hyperbolic to the elliptic dynamics, then one should expect persistence in a codimension-one sense. That is, keeping nonlinear terms fixed but treating lambda and omega as independent parameters, all terms in the leading order expression for the Melnikov function has a countable sequence of sign changes as a function of $\omega$. A zero of this function is argued to imply an omega-value at which a curve of homoclinic orbits bifurcates from lambda=0 in the parameter plane. The large-omega asymptotics of this sequence of bifurcation points is determined solely by the value of b2. This approximate analytical result is supported by careful numerical experiments on a class of fourth-order reversible equations with arbitrary quadratic nonlinear terms. The implications of these results are iscussed for the existence of solitary water waves of elevation in the presence of surface tension and for recently discovered embedded solitons in a variety of (non-integrable) nonlinear optics models.