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 in any spatial dimension with rough initial data. For , we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces . For , we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in . The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup to overcome the derivative in the nonlinear term.