ZERO-DISPERSION LIMIT OF THE CALOGERO--MOSER DERIVATIVE NLS EQUATION

We study the zero-dispersion limit of the Calogero--Moser derivative nonlinear Schro"\dinger equation iDtu + D x 2 u \pm 2D\Pi(|u|2) \cdot u = 0, x E \BbbR, starting from initial data u 0 E L 2 + (\BbbR) f-L\infty(\BbbR), where D =-iDxand \Pi is the SzegoH\ projector defined as \Piehat\widu(\xi) = 1 [0,+ \infty ) ( \xi ) u widehat\( \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.