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 = 2 j + 1 , where j 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 σ . 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 G σ in time. However, it belongs to G m σ ( R ) near zero for every x on the circle.