On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear schrödinger equation

We study the asymptotic behavior for large time of small solutions to the Cauchy problem for the modified Benjamin-Ono equation: u_t + (u³)_x + Hu_xx = 0, where H is the Hilbert transformation, x,t ∈ R. We investigate the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Shrödinger equation and then apply techniques developed in [11] - [14] to the resulting cubic nonlocal nonlinear Schrödinger equation. Our method is simpler than that used in [10] because we can use the factorization of the free Schrödinger group. Our purpose in this paper is to show that solutions have the same L^∞ time decay rate as in the corresponding linear Benjamin-Ono equation and to prove the existence of modified scattering states, when the initial data are sufficiently small in the weighted Sobolev spaces H^2,0 ∩ H^1,1, where H^m,s = {φ ∈ S': ||φ||_m,s = ||(1 + x²)^s/2(1 - δ_x²)^m/2φ_L2 < ∞}, m,s ∈ R. This is an improvement of the previous result, where we considered small initial data from the space H^3,0 ∩ H^1,2. Our method is based on a certain gauge transformation and an appropriate phase function.