Invariant measures for the dnls equation
Abstract
We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ data. The measures are absolutely continuous with respect to suitable weighted Gaussian measures supported on Sobolev spaces of increasing regularity. These results have been obtained in collaboration with Giuseppe Genovese (University of Z\"urich) and Daniele Valeri (University of Glasgow).