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) ε R × R, with initial data u(0, x) = u₀(x), x ε R. We assume that the coefficient a(t) ε C¹(R) is real, bounded and slowly varying function, such that |a′(t)| ≤ C〈t〉^-7/6, where 〈t〉 = (1 + t²)^1/2. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H^1,1 = {φ ε L²; || √1+x² √1-∂_x²φ|| < ∞}. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395-418), here we exclude the condition that the integral of the initial data u₀ is zero. We prove the time decay estimates ³√t^{2 3}√〈t〉||u(t)u_x(t)||_∞ ≤ Cε and 〈t〉^1/3-1/3β||u(t)||β ≤ Cε for all t ε R, where 4 < β ≤ ∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.