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

PI：Vaninsky DMS-9501002 本文将研究一维系统统计力学的三个问题。 第一种是在粗糙初始数据下，在非线性不变的Gibbsian分布的支持下，构造了非线性Schrodinger和修正的Korteveg-de弗里斯型方程在整条直线上的动力学。 第二个是建设的作用角变量的整条线上的可积方程的初始数据。 第三个是孤立子气体猜想。 非平衡统计力学研究描述大的或无限大的相互作用粒子系统的动力学系统的性质。 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，在动力学下不变。 这些测度通常由基本哈密顿量和其他经典守恒量如粒子数、动量积分和角动量构成。 统计力学是基于吉布斯的假设，其中指出，这一有限参数家庭的措施用尽类的所有平移不变措施与弱长程依赖。

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.