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 ut+a(t)(u3)x+1/3uxxx=0, (t,x) element of RxR, with initial data u(o,x)=u0(x), x element of R. We assume that the coefficient a(t) element of C1(R) is real, bounded and slowly varying function, such that |a'(t)|C7/6, where =(1+t2)1/2. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H1,1={φ element of L2; parallel√(1+x2)√(1-∂x2)φ||<∞}. 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√t23√||u(t)ux(t)parallel∞Cε and 1/3-1/3β parallel u(t)parallelβCε for all t element of R, where 4<β∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution