Zero-Dispersion Limit of the Calogero–Moser Derivative NLS Equation

We study the zero-dispersion limit of the Calogero-Moser derivative NLS equation i∂ₜu+∂ₓ² u ±\,2DΠ(|u|²)u=0, x, starting from an initial data u₀∈ L²₊(R)∩ L^∞ (R), where D=-i∂ₓ, and Π is the Szeg{o} projector defined as Π u(ξ)=1[0,+∞)(ξ)u(ξ). 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.