Mathematical Sciences: On Some Problems Which Arise in the Study of Invariant Measures of Nonlinear Evolutionary Equations.

将研究一维系统的三个统计力学问题。首先，在平移不变吉本分布的支持下，对粗糙初始数据在整条线上的非线性薛定谔方程和修正Korteveg-de Vries型方程进行了动力学构造。二是构造整条线上初始数据的可积方程的作用角变量。第三个是孤子气体猜想。非平衡统计力学研究描述大的或无限的相互作用粒子系统的动力系统的性质。运动方程的哈密顿结构允许人们在相空间中定义所谓的吉布斯测度，它在动力学下是不变的。这些测量通常是由基本的哈密顿量和其他经典的守恒量构成的，比如粒子的数量、动量和角动量的积分。统计力学基于吉布的假设，该假设表明，这种有限参数测度族耗尽了所有具有弱远程依赖性的平移不变测度类。

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.