Large time asymptotics of solutions to the generalized Korteweg-de Vries equation

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg-de Vries (gKdV) equation u,+ (\u\(rho-1) u)(x) + 1/3u(xxx) = 0, where x, t is an element of R when the initial data are small enough. If the power rho, of the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior For large time, More precisely, we show that the solution u(t) satisfies the decay estimate parallel to u(t)parallel to(L)beta less than or equal to C(1 + t)(-(1/3)(1 - 1/beta)) for beta is an element of (4, infinity], parallel to uu(x)(t)parallel to(L)infinity less than or equal to Ct(-2/3)(1 + t)(-1/3) and using these estimates we prove the existence of the scattering state u(+) is an element of L-2 such that parallel to u(t) - U(t)u(+) parallel to (L2) less than or equal to Ct(-(rho-3)/3) for any small initial data belonging to the weighted Sobolev space H-1,H-1 = {f is an element of L-2; parallel to(1 + \x\(2))(1/2) (1 - partial derivative(x)(2))(1/2) f parallel to (L2) < infinity}, where U(t) is the Airy Free evolution group. (C) 1998 Academic Press.