Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in periodic setting. Our main hypotheses are both that operator behaves high frequencies as Fourier multiplier by i|ξ|αξ, 1≤α≤2, and nonlinear term is of form ∂xf(u) where f sum an entire series infinite radius convergence. Under these conditions, we prove unconditional local well-posedness Hs(T) s≥1−α2(α+1). This leads to some global existence results energy space Hα/2(T), α∈[2,2].