On the analyticity for the generalized quadratic derivative complex Ginzburg-Landau equation (Q1724512)

Summary: We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation \((1 / 2 \leqslant \alpha \leqslant 1)\) in any spatial dimension \(n \geqslant 1\) with rough initial data. For \(1 / 2 < \alpha \leqslant 1\), we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces \(M_{p, 1}^{1 - 2 \alpha}(1 \leqslant p \leqslant \infty)\). For \(\alpha = 1 / 2\), we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in \(\dot{B}_{\infty, 1}^0\)\((\mathbb{R}^n) \cap M_{\infty, 1}^0(\mathbb{R}^n)\). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup \(e^{- \left(a + i\right) t \left(- \Delta\right)^\alpha}\) to overcome the derivative in the nonlinear term.