# Automatic exploration techniques for the numerical bifurcation study of the Ginzburg-Landau equation

## TL;DR

This paper introduces a robust numerical continuation method for exploring the solution landscape of the Ginzburg-Landau equations, using bifurcation analysis to understand superconducting states.

## Contribution

It develops an automatic exploration algorithm combining Lyapunov-Schmidt reduction and equivariant branching, implemented in Python for two-dimensional superconductivity models.

## Key findings

- Successfully constructed complete solution landscapes for multiple examples.
- Demonstrated robustness and effectiveness of the algorithm.
- Showed applicability to a class of grids using symmetry-based methods.

## Abstract

This paper considers the extreme type-II Ginzburg-Landau equations, a nonlinear PDE model that describes the states of a wide range of superconductors. For two-dimensional grids, a robust method is developed that performs a numerical continuation of the equations, automatically exploring the whole solution landscape. The strength of the applied magnetic field is used as the bifurcation parameter. Our branch switching algorithm is based on Lyapunov-Schmidt reduction, but we will show that for an important class of grids an alternative method based on the equivariant branching lemma can be applied as well. The complete algorithm has been implemented in Python and tested for multiple examples. For each example a complete solution landscape was constructed, showing the robustness of the algorithm.