The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces

This paper studies the periodic Cauchy problem for a KdV equation whose dispersion is of order m=2j+1m=2j+1, where jj is a positive integer, (KdVm). Using Bourgain–Gevrey type analytic spaces and appropriate bilinear estimates, it is shown that local in time well-posedness holds when the initial data belong to an analytic Gevrey spaces of order σσ. This implies that in the space variable the regularity of the solution remains the same with that of the initial data. It also implies that the size of the uniform radius of analyticity is preserved. Moreover, the solution is not necessarily GσGσ in time. However, it belongs to Gmσ(R)Gmσ(R) near zero for every xx on the circle.