Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

TL;DR

This paper extends refined asymptotic analysis techniques to describe the gradient profile during vortex line reconnection with boundaries in type-II superconductors, incorporating complex boundary conditions and higher-dimensional cases.

Contribution

It introduces a new refined asymptotic description of the gradient profile during vortex reconnection, generalizing previous heat equation blowup results to physical superconducting models.

Findings

Derived the intermediate extinction profile with refined asymptotics.

Established the gradient behavior near extinction time and point.

Confirmed the gradient profile matches previous theoretical predictions.

Abstract

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation $$\frac{\partial h}{\partial t}=\frac{\partial^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta},\quad\beta>0$$ subject to initial boundary value conditions $$h(\cdot,0)=h_0>0,\quad h(\pm1,t)=1.$$ We derive the intermediate extinction profile with refined asymptotics, and with extinction time $T$ and extinction point $0$, the gradient profile behaves as $x\rightarrow0$ like $$\lim_{t\rightarrow T}\,(\nabla h)(x,t)\quad\sim\quad\frac{1}{\sqrt{2\beta}}\frac{x}{|x|}\frac{1}{\sqrt{|\log|x||}}\left[\frac{(\beta+1)^2}{8\beta}\frac{|x|^2}{|\log|x||}\right]^{\frac{1}{\beta+1}-\frac12},$$ agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.