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 CC operates in the modulation spaces Mp,1(Rn)Mp,1(Rn) by composition, then F   is real analytic on R2≈CR2≈C. This answers negatively, the open question posed in Ruzhansky et al. (2012) [21], regarding the general power type nonlinearity of the form |u|αu|u|αu. We also characterise the functions that operate in the modulation space M1,1(Rn)M1,1(Rn).The local well-posedness of the NLS, NLW and NLKG equations for the ‘real entire’ nonlinearities are also studied in some weighted modulation spaces Msp,q(Rn).