Local and global well-posedness for the extended Schrödinger-Benjamin-Ono system

Abstract

We study the well-posedness problem for the extended Schrödinger-Benjamin-Ono system (eSBO) on the real line. This system couples a Schrödinger field $u$ with a Benjamin-Ono type field $v$, including a term of the form ${\partial }_x(v^2)$. This latter term, just as in the case of the Benjamin-Ono equation, causes the system to become quasilinear and unsolvable via Picard iteration. We prove that eSBO is locally well-posed in $H^{s+\frac{1}{2}}(\mathbb{R})\times H^s(\mathbb{R})$ for any $s\ge 0$. In particular, this result covers the energy space at $s=\frac{1}{2}$, yielding global well-posedness in $H^1(\mathbb{R})\times H^{\frac{1}{2}}(\mathbb{R})$ with a small $L^2$-assumption on the Schrödinger part of the initial data.

41 pages