Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

The solutions of the Cauchy problem of the KdV equation on a periodic domain , [see formula in PDF] possess neither the sharp Kato smoothing property, [see formula in PDF] nor the Kato smoothing property, [see formula in PDF] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain , (See Eq. (1) below) [see formula in PDF] where g ∈ C ∞ ( ) is a real value function with the support [see formula in PDF] It is shown that (1) if ω ≠ ∅, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; (2) if g is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property.