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) equationut+(|u|ρ−1 u)x+13uxxx=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, ∞], ‖uux(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 spaceH1, 1={f∈L2; ‖(1+|x|2)1/2(1−∂2x)1/2 f‖L2<∞}, whereU(t) is the Airy free evolution group.