Quantization effects for multi-component Ginzburg-Landau vortices

Abstract

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.

Keywords

• n-component ginzburg-landau equations
• quantization effect; pohozaev identity
• 35b40
• 35j60
• 35q60