Functions operating on modulation spaces and nonlinear dispersive   equations
  The aim of this paper is two fold. We show that if a complex function $F$ on $\C$ operates in the modulation spaces $M^{p,1}(\R^n)$ by composition, then $F$ is real analytic on $\R^2 \approx \C$. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form $|u|^\alpha u$. We also characterise the functions that operate in the modulation space $M^{1,1}(\R^n)$.   The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonlinearities are also studied in some weighted modulation spaces $M^{p,q}_s(\R^n)$. 
Divyang G. Bhimani, P. K. Ratnakumar
math.AP