Two Results For The Derivative Schr(o|¨)dinger Type Equations

This paper contains two parts. In the first part, we consider the periodic modified Beniamin-Ono (mBO) equation Using the gauge transform correspond to modified Beniamin-Ono equation to weak-en the low-high frequencies interaction of the derivative nonlinear term and Bourgain space frame to absorb partial derivative of the nonlinear term, we prove the periodic modified Benjamin-Ono equation is globally well-posed in the energy space H1/2. The gauge transform correspond to modified Benjamin-Ono equation is not invertible which is a technical difficulty we will overcome in this paper.In the second part, we consider the derivative nonlinear Schrodinger equation which is similar to modified Beniamin-Ono equation. We prove that the modified scat-tering operator is well-defined as a map from the neighborhood of the origin in H1,α+to the neighborhood of the origin in H1,α+, where α>1/2and γ>0is small. The weighted Sobolev space is defined by This result firstly prove that the modified scattering operator for the derivative nonlinear Schrodinger equation is well-defined.