Breaking Homoclinic Connections for a Singularly Perturbed Differential Equation and the Stokes Phenomenon

Behavior of the separatrix solution y ( t )=−(3/2)/cosh²( t /2) (homoclinic connection) of the second order equation y "= y + y ² that undergoes the singular perturbation ɛ² y ""+ y "= y + y ², where ɛ>0 is a small parameter, is considered. This equation arises in the theory of traveling water waves in the presence of surface tension. It has been demonstrated both rigorously [1, 2] and using formal asymptotic arguments [3, 4] that the above-mentioned solution could not survive the perturbation.The latter papers were based on the Kruskal–Segur method (KS method), originally developed for the equation of crystal growth [5]. In fact, the key point of this method is the reduction of the original problem to the Stokes phenomenon of a certain parameterless "leading-order" equation. The main purpose of this article is further development of the KS method to study the breaking of homoclinic connections. In particular: (1) a rigorous basis for the KS method in the case of the above-mentioned perturbed problem is provided; and (2) it is demonstrated that breaking of a homoclinic connection is reducible to a monodromy problem for coalescing (as ɛ→0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.