Multiple singular values of Hankel operators and weak turbulence in the cubic Szeg\H{o} equation

Brown Hall 100

We establish an inverse spectral result on compact Hankel operators on the unit sphere. Namely, we describe the set of symbols of compact Hankel operators having a prescribed sequence of singular values. It is done by constructing a one-to-one correspondence between a symbol of a compact Hankel operator and its sequence of singular values as well as some additional spectral parameters. \\ This one-to-one correspondence plays the role of a non linear Fourier transform for some hamiltonian equation: the cubic Szeg\H{o} equation. It allows to obtain explicit formulae of the solutions and to prove a wave turbulence phenomenon: for a dense $G_\delta$ of initial data, solutions develop large oscillations on small space scales. It is from joint works with Patrick G\'erard.