## Abstract

We study layered solutions in a one-dimensional version of the scalar Ginzburg-Landau equation that involves a mixture of a second spatial derivative and a fractional half-derivative, together with a periodically modulated nonlinearity. This equation appears as the Euler-Lagrange equation of a suitably renormalized fractional Ginzburg-Landau energy with a double-well potential that is multiplied by a 1-periodically varying nonnegative factor g(x) with (Formula presented). A priori this energy is not bounded below due to the presence of a nonlocal term in the energy. Nevertheless, through a careful analysis of a minimizing sequence we prove existence of global energy minimizers that connect the two wells at infinity. These minimizers are shown to be the classical solutions of the associated nonlocal Ginzburg-Landau type equation.

Original language | English (US) |
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Journal | Mathematics In Engineering |

Volume | 5 |

Issue number | 5 |

DOIs | |

State | Published - 2023 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Mathematical Physics
- Applied Mathematics

## Keywords

- layered solutions
- nonlocal Ginzburg-Landau
- periodic modulation