非线性系统的精确求解和可积性是现代数学物理的核心研究课题之一，精确解是探求非线性模型物理本质特性和内在运动规律的有效手段，而可积性能在一定程度上给出解的信息。随着对孤立子碰撞特性认识的趋于成熟和可积系统理论发展的日益完善，从新的角度研究非线性波方程的可积性、寻找孤立波与非纯孤立波的相互作用解及一些具有特殊性质的波越来越受到重视。本项目将：A)发展贝尔多项式可积性理论，将贝尔多项式推广到非线性薛定谔型方程可积性的研究，并构造这类方程的具有特殊性质的解，如周期波解，怪波解等；B)根据从有限变换构造非局域对称的思想，提出与贝尔多项式相关的非局域对称的构造格式，即利用贝尔多项式型贝克隆变换构造非局域对称，这将产生一类具有实际物理背景的孤立波与非纯孤立波的相互作用解，如孤立波与椭圆周期波解等。

One of the central topics in modern mathematical physics is to find exact solution and integrability of nonlinear system, exact solution is an effective tool to study the physical characteristics and motion law of nonlinear model, while integrability can provide some information on the solution. With the development of integrable system theory, the study on the elastic collision of the solitons are common, therefore, understand the integrable system from new ways and find the interaction solutions among solitons and nonsolitons as well as some nonlinear waves with special properties have drawn more and more attention from researchers. In this project we will mainly focus on the following two kinds of problems. A) Extend Bell polynomial to the integrability of nonlinear Schrodinger-type equation, and find their nonlinear waves with special properties, such as rogue waves, periodic waves, et al; B) Based on the idea that the nonlocal symmetry can be obtained from the finite transformation, we will give a scheme for finding nonlocal symmetry related to Bell polynomial, namely, finding nonlocal symmetry from Bell polynomial-type Backlund transformation, which will leads to a kind of interaction solutions among solitons and nonsolitons, such as the interaction solutions with solitons and elliptic periodic waves.