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

Abstract

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation (1/2⩽α⩽1) in any spatial dimension n⩾1 with rough initial data. For 1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces Mp,11-2α(1⩽p⩽∞). For α=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in B˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup e-a+it-Δα to overcome the derivative in the nonlinear term.