Final state problem for Korteweg-de Vries type equations

We study the final state problem for the Korteweg-de Vries type equations: ut -1ρ ∫ x ∫ρ -1 ux =λ u2 ux, (t,x) R+ ×R,u(t)- FS (t) L2 →0 as t→∞, where λR, the function FS (t) we call a final state, defined by the final data u+. We show that there does not exist a nontrivial solution of this equation in the case of FS (t)=U(t) u+, where U(t) is the free evolution group of this equation. We construct the modified wave operator for the Korteweg-de Vries type equations under the conditions that the final data u+ arc real-valued functions and the Fourier transform u + () vanishes at the origin.