Breaking homoclinic connections for a singularly perturbed differential equation and the Stokes phenomenon
Abbreviated Journal Title
Stud. Appl. Math.
ASYMPTOTICS; ORDERS; Mathematics, Applied
Behavior of the separatrix solution y(t) = -(3/2)/cosh(2) (t/2) (homoclinic connection) of the second order equation y" = y + y(2) that undergoes the singular perturbation epsilon (2)y'''' + y" = y + y(2), where epsilon > 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 epsilon --> 0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.
Studies in Applied Mathematics
"Breaking homoclinic connections for a singularly perturbed differential equation and the Stokes phenomenon" (2000). Faculty Bibliography 2000s. 2828.