On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation (1/2 <=alpha <= 1) in any spatial dimension n >= 1 with rough initial data. For 1/2 <alpha <= 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 <= p <=infinity). 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 B-infinity,1(0) (R-n) boolean AND M-infinity,1(0) (R-n). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg- Landau semigroup e(-(a+i)t(-Delta)alpha) to overcome the derivative in the nonlinear term.