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) equationu_t+(u^ρ-1u)_x+13u_xxx=0, wherex,t∈Rwhen the initial data are small enough. If the powerρof 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 solutionu(t) satisfies the decay estimate u(t)_Lβ≤C(1+t) ^{-(1/3)(1-1/β)}forβ∈(4,∞], uu_x(t)_L∞≤Ct^-2/3(1+t)^-1/3and using these estimates we prove the existence of the scattering stateu₊∈L²such that u(t)-U(t)u_+L2≤Ct^-(ρ-3)/3for any small initial data belonging to the weighted Sobolev spaceH^1,1={f∈L²; (1+x²)^1/2(1-∂²_x) ^1/2f_L2<∞}, whereU(t) is the Airy free evolution group.