Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation

Ko-Shin Chen; Cyrill Muratov; Xiaodong Yan; Ko-Shin Chen; Cyrill Muratov; Xiaodong Yan

doi:10.3934/mine.2023090

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-52. doi: 10.3934/mine.2023090

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Research article

Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation

1.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
2.
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
3.
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

Received: 03 March 2022 Revised: 21 April 2023 Accepted: 12 May 2023 Published: 26 May 2023

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 $ \int_0^1 \frac{1}{g(x)} dx < \infty. $ 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.
- layered solutions,
- nonlocal Ginzburg-Landau,
- periodic modulation
Citation: Ko-Shin Chen, Cyrill Muratov, Xiaodong Yan. Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation[J]. Mathematics in Engineering, 2023, 5(5): 1-52. doi: 10.3934/mine.2023090

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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 $ \int_0^1 \frac{1}{g(x)} dx < \infty. $ 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.

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