PI: Vaninsky DMS-9501002 Three problems of statistical mechanics of one dimensional systems will be investigated. The first is a construction of dynamics for the nonlinear Schrodinger and modified Korteveg-de Vries type equations on the entire line for rough initial data from the support of translation-invariant Gibbsian distribution. The second is construction of the action-angle variables for integrable equations on the entire line for such initial data. The third is the soliton gas conjecture. Nonequilibrium statistical mechanics studies properties of dynamical systems describing big or infinite systems of interacting particles. The Hamiltonian structure of the equations of motion allows one to define in the phase space the so-called Gibbsian measures, invariant under the dynamics. These measures are usually constructed form the basic Hamiltonian and other classical conserved quantities such as the number of particles, integrals of momentum and angular momentum. Statistical mechanics is based on Gibb's postulate which states that this finite parameter family of measures exhausts the class of all translation invariant measures with weak long-range dependence.
