Quantization effects for multi-component Ginzburg-Landau vortices
In this paper, we are concerned with n-component Ginzburg-Landau equations on R2 ${\mathbb{R}}^{2}$ . 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-Mer...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-02-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0165 |
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| Summary: | In this paper, we are concerned with n-component Ginzburg-Landau equations on R2
${\mathbb{R}}^{2}$
. 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 Nn
${\mathbb{N}}^{n}$
. Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of n-component equations. |
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| ISSN: | 2169-0375 |