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)(u³)_x+1/3u_xxx =0,(t,x)∈R×R, with initial data u(0,x)=u₀(x),x∈R. We assume that the coefficient a(t)∈C¹(R) is a real, bounded and slowly varying function, such that |a´(t)|⩽C(1+|t|)^-7/6. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space. We prove 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