On the Smoothing Properties of Solutions to the Modified Korteweg-de Vries Equation

We study the smoothness of solutions of the initial value problem for the generalized Korteweg-de Vries equation The solution u can be written in its integral form u(x, t) = u1(t) + u2(t), where u1(t) and u2(t) are the linear and the integral parts of u, respectively. It is established that for data in Hs with s ≥ 1, u2 possesses better regularity in the persistence properties and the smoothing effects than the solution u. Our main tools are recent results obtained by C. Kenig, G. Ponce, and L. Vega (Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math.46 (1993), 527-620) related to the well-posedness of solutions of the generalized Korteweg-de Vries equation.