Finite energy solutions of nonlinear Schrdinger equations of derivative type

This paper is concerned with the initial value problem for nonlinear Schrodinger equations of the form \[ (\dag)\qquad \left\{ \begin{gathered} i\partial _t \psi + \partial \psi = i\lambda \partial \left( {| \psi |^2 \psi } \right) + \lambda _1 | \psi |^{p_1 - 1} \psi + \lambda _2 | \psi |^{p_2 - 1} \psi ,\quad (t,x) \in \mathbb{R} \times \mathbb{R}, \hfill \\ \psi (0,x) = \phi (x),\quad x \in \mathbb{R}, \hfill \\ \end{gathered} \right.\] where $\partial = \partial _x = {\partial /{\partial x}},\lambda ,\lambda _1 ,\lambda _2 \in \mathbb{R}$ and $2 \leq p_1 < p_2 < 5$. It is shown that if $\phi \in H^1 (\mathbb{R})$ and $\| \phi \|_2^2 < {{2\pi } /{| \lambda |}}$, then there exists a unique global solution $\psi $ of (†) such that $\psi \in C(\mathbb{R};H^1 (\mathbb{R}))$. This paper introduces a new method to obtain the result.