Zero dispersion limit of the Calogero-Moser derivative NLS equation

Abstract

We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation $$i\partial_tu+\partial_x^2 u \pm\,2D\Pi(|u|^2)u=0, \qquad x\in\mathbb{R},$$ starting from an initial data $u_0\in L^2_+(\mathbb{R})\cap L^\infty (\mathbb{R}),$ where $D=-i\partial_x,$ and $\Pi$ is the Szeg\H{o} projector defined as $\widehat{\Pi u}(\xi)=1_{[0,+\infty)}(\xi)\widehat{u}(\xi).$ We characterize the zero-dispersion limit solution by an explicit formula. Moreover, we identify it, in terms of the branches of the multivalued solution of the inviscid Burgers-Hopf equation. Finally, we infer that it satisfies a maximum principle.