Asymptotic expansion of solutions in a rolling problem

Asymptotic methods in the theory of differential equations and in nonlinear mechanics are commonly used to improve perturbation theory in the small oscillation regime. However, in some problems of nonlinear dynamics, in particular for the Higgs equation in field theory, it is important to consider not only small oscillations but also the rolling regime. In this article we consider the Higgs equation and develop a hyperbolic analogue of the averaging method. We represent the solution in terms of elliptic functions and, using an expansion in hyperbolic functions, construct an approximate solution in the rolling regime. An estimate of accuracy of the asymptotic expansion in an arbitrary order is presented.

Funding Agency	Grant Number
Russian Foundation for Basic Research	11-01-00894-a 11-01-00828-a 11-01-12114-ofi-m
Ministry of Education and Science of the Russian Federation	NSh-4612.2012.1 NSh-2928.2012.1
The work was partially supported by the Russian Foundation for Basic Research (project nos. 11-01-00894-a (I.A.) and 11-01-00828-a, 11-01-12114-ofi-m-2011 (I.V.)) and by grants of the President of the Russian Federation (project nos. NSh-4612.2012.1 (I.A.) and NSh-2928.2012.1 (I.V.)).