Quantization effects for multi-component Ginzburg-Landau vortices

In this paper, we are concerned with n-component Ginzburg-Landau equations on ℝ². 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 L² 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. Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of n-component equations.