Modified wave operators for the derivative nonlinear Schrödinger equation (Q1322092)

We consider the scattering problem for the derivative nonlinear Schrödinger equation: \[ i\partial_ t\psi+ \partial_ x^ 2\psi= 2i\delta\partial_ x (|\psi|^ 2 \psi), \qquad (t,x)\in\mathbb{R}\times\mathbb{R},\tag{1} \] where \(\delta\neq 0\). It is shown that modified wave operators for (1) exist on a neighborhood of zero in the weighted Sobolev space \(H^{4,0}\cap H^{0,4}\), where \[ H^{m,s}= \{f\in S';\;\| f\|_{m,s}= \|(1+ | x|^ 2)^{s/2} (1- \Delta)^{m/2} f\|_ 2<\infty\}, \qquad m,s\in\mathbb{R}. \] {}.