On the well-posedness of the intermediate nonlinear Schr\"{o}dinger equation on the line

Abstract

We consider a family of intermediate nonlinear Schr\"{o}dinger equations (INLS) on the real line, which includes the continuum Calogero-Moser models (CCM). We prove that INLS is locally well-posed in $H^{s}(\mathbb{R})$ for any $s>\frac 14$, which improves upon the previous best result of $s>\frac 12$ by de Moura-Pilod (2008). This result is also new in the special case of CCM, as the initial condition is not required to lie in any Hardy space. Our approach is based on a gauge transformation, exploiting the remarkable structure of the nonlinearity together with bilinear Strichartz estimates, which allows to recover some of the derivative loss. This turns out to be sufficient to establish our main results for CCM in the Hardy space. For INLS and CCM outside of the Hardy space, the main difficulty comes from the lack of the Hardy space assumption, which we overcome by implementing a refined decomposition of the solutions, which observes a nonlinear smoothing effect in part of the solution. We also discover a new Lax pair for INLS and use it to establish global well-posedness in $H^{s}(\mathbb{R})$ for any $s>\frac 14$ under the additional assumption of small $L^2$-norm.