Zero dispersion limit of the Calogero-Moser derivative NLS equation

We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation $$i{\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x^{2}u\pm 2D\mathrm{\Pi }(\mathrm{\mid }u{\mathrm{\mid }}^2)u=0,x\in \mathbb{R},$$ starting from an initial data $u_0\in L_{+}^2(\mathbb{R})\cap L^{\mathrm{\infty }}(\mathbb{R}),$ where $D=-i{\mathrm{\partial }}_x,$ and $\mathrm{\Pi }$ is the Szegő projector defined as $\widehat{\mathrm{\Pi }u}(\xi )=1_{[0,+\mathrm{\infty })}(\xi )\widehat{u}(\xi )\mathrm{.}$ 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.