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 + y2 that undergoes the singular perturbation ε2y″″ + y″ = y + y2, where ε > 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 . 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 ε → 0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.
Studies in Applied Mathematics
Number of Pages
Source API URL
Tovbis, Alexander, "Breaking Homoclinic Connections For A Singularly Perturbed Differential Equation And The Stokes Phenomenon" (2000). Scopus Export 2000s. 1040.