Zero dispersion limit of the Calogero-Moser derivative NLS equation

Abstract

We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation $$i{\partial }_{t}u+{\partial }_x^{2}u\pm 2D\Pi (|u{|}^2)u=0,x\in \mathbb{R},$$ 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 Szegő projector defined as $\widehat{\Pi u}(\xi )=1_{[0,+\infty )}(\xi )\hat{u}(\xi ).$ 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.

24 pages

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