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Nonrelativistic limit of normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

Zhentao He, Chao Ji

TL;DR

This work analyzes the nonrelativistic limit of normalized solutions to a localized nonlinear Dirac equation on a noncompact metric graph with a compact core. Using a variational framework and a perturbation/reduction approach, the authors obtain existence of normalized solutions for large relativistic parameter $c$ and derive precise asymptotic bounds for the associated minimax levels and Lagrange multipliers. In the nonrelativistic limit, the first component of the spinor converges to a function $g$ solving a nonlinear Schrödinger equation on the graph while the second component vanishes, linking NLDE on graphs to NLSE on graphs under a mass constraint. The results extend the understanding of NLDE on quantum graphs and connect the relativistic problem to a localized NLSE regime across subcritical and certain supercritical parameter ranges.

Abstract

In this paper, we study the nonrelativistic limit of normalized solutions for the following nonlinear Dirac equation (NLDE) on noncompact metric graph $\G$ with finitely many edges and a non-empty compact core $\K$ \begin{equation*} \D u - ωu= χ_\K\abs{u}^{p-2}u, \end{equation*} under the constraint $\int_\G\abs{u}^2\,dx = 1$, where $\D$ is the Dirac operator on $\G$, $u: \G \to \mathbb{C}^2$, the frequency $ω\in \mathbb{R}$ is part of the unknowns which arises as a Lagrange multiplier, $χ_\K$ is the characteristic function of the compact core $\K$, and $2<p<6$. To the best of our knowledge, this is the first study to investigate the nonrelativistic limit of normalized solutions to (NLDE) on metric graphs.

Nonrelativistic limit of normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

TL;DR

This work analyzes the nonrelativistic limit of normalized solutions to a localized nonlinear Dirac equation on a noncompact metric graph with a compact core. Using a variational framework and a perturbation/reduction approach, the authors obtain existence of normalized solutions for large relativistic parameter and derive precise asymptotic bounds for the associated minimax levels and Lagrange multipliers. In the nonrelativistic limit, the first component of the spinor converges to a function solving a nonlinear Schrödinger equation on the graph while the second component vanishes, linking NLDE on graphs to NLSE on graphs under a mass constraint. The results extend the understanding of NLDE on quantum graphs and connect the relativistic problem to a localized NLSE regime across subcritical and certain supercritical parameter ranges.

Abstract

In this paper, we study the nonrelativistic limit of normalized solutions for the following nonlinear Dirac equation (NLDE) on noncompact metric graph with finitely many edges and a non-empty compact core \begin{equation*} \D u - ωu= χ_\K\abs{u}^{p-2}u, \end{equation*} under the constraint , where is the Dirac operator on , , the frequency is part of the unknowns which arises as a Lagrange multiplier, is the characteristic function of the compact core , and . To the best of our knowledge, this is the first study to investigate the nonrelativistic limit of normalized solutions to (NLDE) on metric graphs.
Paper Structure (6 sections, 8 theorems, 103 equations, 1 figure)

This paper contains 6 sections, 8 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gagliardo-Nirenberg-Sobolev inequality
  4. The Perturbation Functional
  5. Proof of Theorem \ref{['th2']}
  6. Further remarks

Key Result

Theorem 1.2

Let $\mathcal{G}$ be any noncompact metric graph with a non-empty compact core $\mathcal{K}$, $2< p < 6$ and $m >m_0(p,\ell_{e_0})$. Then, there exists $c_0>0$ depending only on $m,p$ and $\mathcal{G}$ such that, for any $c>c_0$, there exists a non-trivial $u_c \in \operatorname{dom}(\mathcal{D}_c)$ In addition, $(\omega_c)$ satisfies eqlm.

Figures (1)

  • Figure 1: A noncompact graph $\mathcal{G}$ with finitely many edges and a non-empty compact core.

Theorems & Definitions (15)

  • Definition 1.1: Bo
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 5 more