Mathematics

We study the defocusing Calogero-Moser derivative non-linear Schrodinger equation (CMNLS), which was proved by Pelinovsky and Grimshow to be integrable. First, we prove global well-posedness of the solutions, for any intial data in $H^s$, with $s \geq 1$, and then we study their smoothing properties. Precisely, following a recent work by Tao and awork by Gerard, Kappeler and Topalov on the Benjamin-Ono equation, we construct a variant ofTao’s Gauge trasformation in order to prove that the flow is essentially a smoothing perturbationof the linear flow. To this end, we consider the Lax-Pair representation of the equation. Thenwe prove that the adaptation of the Tao’s Gauge transform is a high frequency approximationof Birkhoff-map for the CMNLS.This is a joint work with P. Gerard.