若干高维可积和不可积模型的严格解

可积和不可积模型可以描述自然科学中的诸多现象, 寻找高维非线性模型的严格解已成为可积系统的一个重要研究内容. 结合达布变换法和多线性分离变量法, 可以得到多个(2+1)维非线性模型包含任意函数的严格解, 通过选取不同的任意函数, 构造这些非线性模型新的相互激发模式. 进一步推广了形变映射理论, 建立了变系数 场和sine-Gordon以及双sine-Gordon场的形变映射关系, 从而得到高维不可积模型包含任意函数的新严格解. 对任意函数的不同选择, 构造了sine-Gordon和双sine-Gordon可积模型丰富的局域解和周期解, 如多solitoff解及其周期波推广、周期形变的蛇形孤波解以及变模的拟周期解等.

Exact solutions for some high dimensional integrable and nonintegrable systems

How to find more exact solutions of the integrable and nonintegrable systems is one of the most important issues in the soliton theory. With the help of Darboux transformation and variable separation solutions of initial problem and Lax pair of the PDEs, we obtain some new types of soliton solutions of the dispersive long wave equation, the asymmetric Nizhnik-Novikov-Veselov equation, the Nizhnik-Novikov-Veselov equation. Considering the entrance of the arbitrary variable separated functions, we can construct many new types of exact solutions and the multiple localized excitations through proper selections of the arbitrary functions. In this study, we have developed the mapping relations between one or more constrained fields and high dimensional sine-Gordon or double sine-Gordon equation. From the mapping relations, we can obtain many new kinds of the periodic wave solutions, the quasi-periodic wave solutions and periodic-kink solutions of high dimensional sine- Gordon and double sine-Gordon systems.