Generalized Painleve equation and superconductivity. Asymptotic behavior of unbounded solutions
Consider the nonlinear scalar differential equation (1)/(p(t))(p(t)y'(t))' = -q(t)f(t, y(t), p(t)y' (t)). where p and q are positive on (0, 1), "singular" at t = 0, 1 and/or y = 0 and f is an element of C(R+ x R+ x R+, R+), associated to the boundary conditions x(0) = a greater than or equal to 0, lim(t-->infinity) p(t)x' (t) = b greater than or equal to 0. We prove the existence of a global, positive and strictly increasing solution x = x(t) of this BVP, such that its "derivative" y = p(t)x'(t) is also a positive and strictly decreasing map, under a natural growth in f of "superlinear" type. Our approach is based on the analysis of the corresponding vector field on the face-plane (x,px') and the well-known Knesser's type (continuum) technique. As an application, we study the generalized Painleve equation in a semi-infinite interval (0, +infinity) x'' = x(2n+1) - (t - c)(2k+1)x(2(n-k)-1), which, in turn, models a superheating field attached to a semi-infinite superconductor. Namely, we prove the existence of a (global) strictly positive solution satisfying x'(0) = 0, lim(t-->+infinity) (x(t))/(tbeta) = M, and lim(t-->+infinity) t(beta)x'(t) = betaM, where 0   0, and n, k is an element of N with k < n are arbitrary constants. (C) 2003 Elsevier Science Ltd. All rights reserved.
painleve equation superconductivity vector field continuum superheating singular boundary value problems positive monotone solution shooting method kneser's property consequent map