On the modified Korteweg de Vries equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg - de Vries equation u(t) + a(t) (u(3))(x) + 1/3u(xxx) = 0. (t, x) epsilon R x R, with initial data u(0, x)= u(0)(x), x epsilon R. We assume that the coefficient a(t) epsilon C-1 (R) is real, bounded and slowly varying function, such that \a'(t)\ less than or equal to C(1 + \t\)(-7/6). We suppose that the initial data are real - valued and small enough, belonging to the weighted Sobolev space. We Drove the time decay estimates of the solutions. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.