Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring

We study a Ginzburg-Landau energy in a one-dimensional ring with nonuniform thickness, where the nonuniformity is expressed by a piecewise-constant function. That is a simplified model describing a supercurrent in the superconducting ring. Then the Ginzburg-Landau equation with a discontinuous coefficient subject to periodic boundary conditions is derived as the Euler-Lagrange equation of the energy functional. Since the unknown variable of the equation is complex-valued, we can define the phase of a solution if the solution has no zero. The purpose of this article is to establish the existence of nontrivial solutions with no zero and to reveal the configuration of the phase of the solutions as the coefficient converges to zero in a set of subintervals. More precisely we control the convergence of the coefficient with a small positive parameter $\varepsilon$ having various orders in the subintervals and prove the convergence of the solutions to those of a limiting equation as $\varepsilon\to0$ together with the convergence rate. As a consequence, for small $\varepsilon$ most of the phase variation takes place on the subintervals where the coefficient converges to zero with the highest order. Finally we show the stability of those solutions.