非均匀介质中随机可积系统的研究

本项目拟研究若干在数学和物理上重要的非均匀谱随机可积系统的各种精确解及解的动力学特征分析。主要涉及用双线性方法和Darboux变换方法构造非均匀谱随机可积系统（如非均匀谱随机KdV方程，非均匀谱随机Schr?dinge方程，一些2+1维非均匀谱随机可积方程等）各种有物理意义的精确解，并深入分析这些精确解在统计意义下的期望均值函数的动力学特征。本项目研究的内容将试图解决：1. 非均匀谱随机可积系统精确解的构造；2. 解的动力学特征分析；3. 其它可积性质的揭示。从而使人们能够更加深刻的理解非均匀谱随机可积系统所描述的物理模型的本质。

The research project seeks to find new explicit solutions for some nonisospectral and variable coefficient stochastic integrable system. The dynamical properties of these new solutions are discussed in detail. Our aim is to construct explicit solutions by using the bilinear method for a nonisospectral and variable coefficient stochastic KdV equation. By using the Darboux transformation, we obtain some exact solutions of a nonisospectral and variable coefficient stochastic Schr?dinger equation. Analysis of properties of the statistical average of the solutions will show that the obtained explicit solutions for these nonisospectral and variable coefficient stochastic integrable system possess new dynamical characters which are different from the ones of classical integrable systems.