## Abstract

We study the asymptotic behavior for large time of small solutions to the Cauchy problem for the modified Benjamin-Ono equation: u_{t} + (u^{3})_{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 + x^{2})^{s/2}(1 - δ_{x}^{2})^{m/2}φ_{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.

Original language | English |
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Pages (from-to) | 237-255 |

Number of pages | 19 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Jan |

## Keywords

- Asymptotic behavior of solutions
- Modified B-O equation
- Modified scattering states