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) + 1/3 u(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 vertical bar a'(t)vertical bar <= C < t >(-7/6), where < t > = (1 + t(2))(1/2). We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H-1,H-1 = {phi epsilon L-2; vertical bar vertical bar root 1 + x(2) root 1 - partial derivative(2)(x)phi vertical bar vertical bar < infinity}. 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 u(0) is zero. We prove the time decay estimates (3)root t(2) (3)root < t >vertical bar vertical bar u(t)u(x)(t)vertical bar vertical bar infinity <= C epsilon and < t >(1/3 - 1/3 beta)vertical bar vertical bar u(t)vertical bar vertical bar(beta) <= C epsilon for all t epsilon R, where 4 < beta <= infinity. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.