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 $(1/2⩽\alpha ⩽1)$ in any spatial dimension $n⩾1$ 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 $M_{p,1}^{1-2\alpha }(1⩽p⩽\infty )$. 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 ${\dot{B}}_{\infty ,1}^0$ $({ℝ}^n)\cap M_{\infty ,1}^0({ℝ}^n)$. 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.