Large time asymptotics of solutions to the generalized Benjamin-Ono equation

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: $u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal{H} u_{xx} = 0$, where $\mathcal{H}$ is the Hilbert transform, $x, t \in {\mathbf{R}}$, 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. In the case $\rho=3$ critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same $L^{\infty }$ time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.