Structure and Gevrey asymptotic of solutions representing topological defects to some partial differential equations

Abstract

We consider the positive real-valued solutions of a particular type of ordinary differential equations that arise when considering defect solutions to semilinear partial differential equations. We provide sufficient conditions on the nonlinear term to ensure the existence, uniqueness and monotonicity of solutions to the ordinary differential equation with the prescribed boundary conditions. We then focus on the behaviour of such solutions at infinity and we prove that there is a unique formal expansion at infinity of the Gevrey type, i.e. the coefficients of the expansion grow as a power of a factorial. Moreover, we show that the actual solution is indeed asymptotic 1-Gevrey to this formal expansion. We also present a numerical algorithm to compute the solution for arbitrary values of the degree n in the particular case of the Ginzburg–Landau equation. In particular, we address the difficulty in the numerical computations when n is relatively large due to the fact that the shooting parameter becomes exponentially small for the whole class of nonlinearities considered in this work.