Local Well-posedness of the Derivative Schrödinger Equation in Higher Dimension for Any Large Data

Abstract

In this paper, the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ${u_t} - {\rm{i}}\Delta u + {\left| u \right|^2}\left( {\overrightarrow \gamma \cdot \nabla u} \right) + {u^2}\left( {\overrightarrow \lambda \cdot \nabla \overline u } \right) = 0,\,\,\,\,\left( {x,t} \right) \in {\mathbb{R}^n} \times \mathbb{R},\,\,\overrightarrow \gamma ,\overrightarrow \lambda \in {\mathbb{R}^n};\,\,n \ge 2.$

It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in ${H^s}\left( {{\mathbb{R}^n}} \right)\,\,\left( {s > {n \over 2}} \right)$ for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space ${H^{{n \over 2}}}$.

Keywords

Well-posedness / Derivative Schrödinger equation in higher dimension / Short-time X _s,b / Large initial data

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