非局部Bell多项式及其在非局部系统中的应用

Recently, the non-local integrable systems has become a hot topic in the field of nonlinear science due to its origin in the hot research field of modern physics: parity-time symmetry. This project mainly studies the following problems : (1) Extend the Bell polynomial to non-local ones, and then studies the integrability of non-local nonlinear equations by establishing the relationship between non-local Bell polynomial and Hirota bilinear operator; (2) Extend the Riemann-theta function to the study of quasi-periodic solutions of non-local integrable equations, and discuss the asymptotic relation between quasi-periodic solutions and soliton solutions as well as the relation with the quasi-periodic solutions of local integrable equations; (3) Transform the non-local integrable equation into a multi-linear system by using the bilinear transformation or the Painleve truncation expansion, and then find their interaction solutions and analyse the physical phenomena described by these interactions.

非局部可积系统的发现来源于现代物理的热门研究领域：宇称-时间对称，从而成为近年来非线性科学领域的一个研究热点。本项目主要研究如下问题：（1）将Bell多项式推广到非局部的情形，通过建立非局部Bell多项式与Hirota双线性算子之间的关系，进而研究非局部非线性方程的可积性；（2）将Riemann-theta函数推广到非局部可积方程拟周期解的研究，并讨论在此意义下的拟周期解与孤子解的渐进关系以及与局部可积方程的拟周期解之间的联系；（3）基于双线性变换或Painleve截断展开，将非局部可积方程转化为多线性形式，进而构造其相互作用解，并分析这些相互作用所描述的物理现象。

局部与非局部可积系统的研究因有广泛的实际应用而具有非常重要的理论价值。本项目研究的主要内容如下：（1）提出非局部Bell多项式理论；（2）基于双线性方法构造可积方程的lump解和相互作用解，讨论时间变量对于解的动力学行为的影响；（3）利用黎曼－希尔伯特方法构造三分量四阶薛定谔系统的多孤子解，讨论了呼吸子解与孤子解的空间结构和碰撞等动力学行为。项目的研究丰富了非局部可积系统的研究，并进一步推广了Bell多项式在微分方程领域的应用。