Small data well-posedness for derivative nonlinear Schrödinger equations

Abstract

We study the generalized derivative nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=P(u,\overline{u},{\partial }_{x}u,{\partial }_x\overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg }P\ge 3$, the equation is locally well-posed in $H^{\frac{1}{2}}$ when each term in $P$ contains only one derivative, otherwise we have a local well-posedness in $H^{\frac{3}{2}}$. If $\text{deg }P\ge 5$, the solution can be extended globally. By restricting to equations of the form $i{\partial }_{t}u+\Delta u={\partial }_{x}P(u,\overline{u})$ with $\text{deg }P\ge 5$, we were able to obtain the global well-posedness in the critical Sobolev space.

41 pages

References in corpus (1)

• Well-posedness for a generalized derivative nonlinear Schrödinger equation

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