Coexistence of non trivial solutions of the one-dimensional Ginzburg-Landau equation: a computer-assisted proof

In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of exact solutions nearby numerical approximations. Coexistence of as many as seven non trivial solutions is proved.

The following links provide the programs sufficient to carry out the main result of the above mentioned work.