Title: Spectral theory for Soliton gases of focusing and defocusing NLS equation

Abstract: In this talk, I will briefly introduce the concept of soliton gases and their spectral theory.  In particular, we will discuss how to derive the Nonlinear-Dispersion relation (NDR) equations for a dense soliton gas. By generalizing a theory developed by Forest-Lee, we use the solutions of the NDR equations to compute the kurtosis of the gas. Some further questions in this field will be touched at the end.