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Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation

Emanuela Gussetti, Martina Hofmanová

TL;DR

This work proves the existence of non-trivial statistically stationary solutions to the 1D Schrödinger map equation (SME) with null Neumann boundaries by a Kuksin-type limiting procedure based on the stochastic Landau–Lifshitz–Gilbert equation (LLG). It constructs invariant measures for the stochastic approximations, passes to a limit to obtain a stationary SME solution that preserves its law, and shows the solution is genuinely random with spatial and temporal dynamics when the noise is space-dependent. The authors also establish stationary solutions to the stochastic SME itself, including a stationary spherical Brownian motion as a special case, and demonstrate how SME trajectories can be transformed into stationary solutions for the binormal curvature flow (BCF) via a Hashimoto-type map; they also discuss the link to the cubic nonlinear Schrödinger equation (CNSE) and provide a lower bound on the prevalence of nontrivial space-time dynamics. Overall, the paper extends the fluctuation-dissipation framework to dispersive geometric PDEs, yielding new stationary stochastic solutions and clarifying their geometric relations and nontriviality properties.

Abstract

We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow.

Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation

TL;DR

This work proves the existence of non-trivial statistically stationary solutions to the 1D Schrödinger map equation (SME) with null Neumann boundaries by a Kuksin-type limiting procedure based on the stochastic Landau–Lifshitz–Gilbert equation (LLG). It constructs invariant measures for the stochastic approximations, passes to a limit to obtain a stationary SME solution that preserves its law, and shows the solution is genuinely random with spatial and temporal dynamics when the noise is space-dependent. The authors also establish stationary solutions to the stochastic SME itself, including a stationary spherical Brownian motion as a special case, and demonstrate how SME trajectories can be transformed into stationary solutions for the binormal curvature flow (BCF) via a Hashimoto-type map; they also discuss the link to the cubic nonlinear Schrödinger equation (CNSE) and provide a lower bound on the prevalence of nontrivial space-time dynamics. Overall, the paper extends the fluctuation-dissipation framework to dispersive geometric PDEs, yielding new stationary stochastic solutions and clarifying their geometric relations and nontriviality properties.

Abstract

We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow.

Paper Structure

This paper contains 31 sections, 27 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. The first main result: existence of non-trivial statistically stationary solutions to the SME
  3. The second main result: existence of stationary solutions to the stochastic SME
  4. The third main result: existence of non-trivial statistically stationary solutions to the binormal curvature flow.
  5. Relations with the cubic focusing non-linear Schrödinger equation.
  6. Literature on statistically stationary solutions.
  7. Organization of the paper.
  8. Acknowledgements:
  9. Notations and setting.
  10. Some notations
  11. Some results on the Landau-Lifschitz-Gilbert equation in 1D
  12. A priori estimates and equalities on the perturbed equation.
  13. Some Lemmata on the space average of the solution.
  14. Proof of Theorem \ref{['th:existence_statistical_solutions']}
  15. Proof of Theorem \ref{['th:existence_statistical_solutions']} a., b.
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $h\in W^{1,\infty}(D;\mathbb{R})$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 33 more