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 $H^{s+\frac 12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\geq 0$. In particular, this result covers the energy space at $s=\frac 12$, yielding global well-posedness in $H^{1}(\mathbb{R})\times H^{\frac 12}(\mathbb{R})$ with a small $L^2$-assumption on the Schrödinger part of the initial data.
From: Justin Forlano [view email]
[v1]
Wed, 17 Jun 2026 16:27:25 UTC (43 KB)
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