Well-Posedness and Regularity for a Higher OrderPeriodic Mkdv Equation

We consider the higher order mKdV equation, so that we are examining those equations with a higher dispersion term of the order m, where m is odd and larger than 3. The corresponding periodic Cauchy problem is in fact well-posed in Sobolev spaces for all s ge 1/2. We then show that the solution to the periodic Cauchy problem for this higher order equation with analytic initial data is analytic in the space variable x at any fixed time t near time zero. However, while this analyticity is not guaranteed in t, the solution does have Gevrey-m regularity with respect to the time variable t.