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

We consider the Cauchy problem for one-dimensional dis-persive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive op-erator behaves for high frequencies as a Fourier multiplier by J|e|alpha e, with 1 <= alpha <= 2, and that the nonlinear term is of the form partial differential xf(u) where f is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in H-s(T) for s >= 1- alpha/2(alpha+1) . This leads to some global existence results in the energy space H-alpha/2(T ), for alpha E [root 2, 2].(C) 2022 Elsevier Inc. All rights reserved.