高维可积模型及特殊解的机械化算法研究

The study on higher dimensional, supersymmetric and generalized nonlinear equations is important and hot topic in nonlinear mathematical physics. In this project, based on the soliton theory and integrable system theory, we shall study some mechanical algorithms of higher integrable models and their special solutions. This project concentrates on the following four aspects: 1) develop a constructive algorithm for the Painlevé test of supersymmetric nonlinear equations and test the Painlevé integrability automatically. 2) Perform the Painlevé classification for higher dimensional nonlinear equations with multiple parameters, and derive some new integrable models. 3) For the Painlevé integrable models in higher dimensions, we will propose systematic algorithms for constructing Bäcklund transformation and Lax pairs, which are based on the improved singular manifold method. 4) For the bilinear integrable models, the binary Bell polynomials method will be extended to study higher dimensional nonlinear equations, nonlinear coupled systems and some special types of equations. 5) Combining the bosonization method with direct algebraic methods, we will construct several types of travelling wave solutions, solitary wave solutions and localized solutions. By means of bilinear method and truncated Painlevé expansion method, we will derive quasi-periodic wave solutions, explode-decay solutions and lump solutions for higher dimensional nonlinear equations. The results of this project will not only promote the applications of mathematical mechanization in nonlinear partial differential equations, but also provide an effective analytic methods and research tools.

高维、超对称、广义非线性方程是非线性数学物理的研究热点。本项目以孤立子理论和可积系统理论为基础，以符号计算为工具，建立和发展高维可积模型及特殊解的若干机械化算法。内容包括：1) 建立超对称方程Painlevé检验的机械化算法，实现可积性判定的自动推导；2)发展高维广义非线性方程Painlevé可积归类的机械化算法，获得新可积模型；3) 针对Painlevé可积的高维模型，改进奇异流形方法，提出Bäcklund变换和Lax对的构造性算法；4) 针对双线性可积模型，拓展Bell多项式方法，建立高维、耦合及特殊类型方程的多种可积性质的系统化算法；5) 结合玻色化方法和直接代数方法，构造超对称方程的行波解、孤子解、局域相干解；发展双线性方法和截断展开法，寻求高维可积模型的拟周期解、爆破熄灭解及有理解等。研究成果有助于推进数学机械化在微分领域的应用，也为实际问题提供有效的分析方法和研究工具。

本项目在高维、广义非线性演化方程可积性质及特殊解的机械化算法研究领域，提出并发展了一些行之有效的新算法，获得了若干创新性的研究成果。主要包括：（1）发展了高维广义非线性方程Painlevé可积归类的机械化算法，获得了若干3+1维、4+1维及n+1维可积模型。（2）结合奇异流形方法，拓展Bell多项式方法，建立高维非线性演化方程的多种可积性质的系统化算法，系统推导得到了3+1维、4+1维及n+1维可积模型的N-孤子解、双线性Backlund变换、Lax对、无穷守恒律等可积性质。（3）基于Darboux变换理论，结合双线性方法，发展了继承求解策略与齐次平衡方法，建立了高维非线性演化方程特殊解的系统化构造算法，获得了双向孤子、暗-暗孤子、呼吸子、有理块解、多瞬子解等多种类型的特殊解。依托项目的资助，课题组顺利完成了预期的研究任务和预定目标。通过研究，在Nonlinear Dynamics、Applied Mathematics Letters、Computers and Mathematics with Applications、International Journal of Computer Mathematics等权威期刊上发表课题相关学术论文20篇，其中SCI论文16篇，中文核心期刊论文3篇，ESI论文1篇，Web of Science核心集引用180余次。