Central Limit Theorem for a Soliton Gas

A soliton is a travelling wave with a localised wave profile. When a wave profile has many solitons, the fundamental feature of integrability is the elastic interaction of solitons. 
The concept of soliton gas was introduced by V. Zakharov in 1971 as an infinite collection of weakly interacting solitons in the framework of the  Korteweg-de Vries equation. Later the concept of soliton gas has been extended by El and co-authors to dense gases of solitons in which the interactions are strong and they are collectively described by the Zakharov-El kinetic equations. The Zakharov-El soliton gas equations turn out to coincide with the equations of generalized hydrodynamics derived in the thermodynamic limit of integrable systems out of equilibrium. I will consider a random set of N solitons for the focusing nonlinear Schrodinger equation and I will   derive the law of larger number and the central limit theorem for its solution in the limit N goes to infinity. 
Joint work with M. Girotti, K. McLaughlin and J. Najnudel