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Quantization effects for multi-component Ginzburg-Landau vortices

Rejeb Hadiji, Jongmin Han, Juhee Sohn

Abstract

In this paper, we are concerned with $n$-component Ginzburg-Landau equations on $\rtwo$.By introducing a diffusion constant for each component, we discuss that the $n$-component equations are different from $n$-copies of the single Ginzburg-Landau equations.Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case.First, we show that if the solutions have their gradients in $L^2$ space, they are trivial solutions.Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and $\nat^n$.Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of $n$-component equations.

Quantization effects for multi-component Ginzburg-Landau vortices

Abstract

In this paper, we are concerned with -component Ginzburg-Landau equations on .By introducing a diffusion constant for each component, we discuss that the -component equations are different from -copies of the single Ginzburg-Landau equations.Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case.First, we show that if the solutions have their gradients in space, they are trivial solutions.Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and .Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of -component equations.
Paper Structure (3 sections, 7 theorems, 98 equations)

This paper contains 3 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main1']} and Theorem \ref{['thm:main-lambda']}
  3. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

Let $( u_1,\cdots,u_n)$ be a solution of eq:SGL-rtwo satisfying Then, either $( u_1,\cdots,u_n)\equiv(0,\cdots,0)$ or $( u_1,\cdots,u_n)=(c_1,\cdots,c_n)$ for some constants $c_1,\cdots, c_n \in \mathbb C$ with $|c_1|^2+\cdots+|c_n|^2=n$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 2 more