高维非线性物理模型局域波特性及其形成机制研究

关于具有新奇性质局部波及其相互作用现象的研究，一直是数学物理、流体力学等领域的重要研究内容。我们将应用孤子理论研究高维非线性物理模型的局域波及其相互作用解，希望通过对具体问题的研究探索和发展关于高维非线性物理模型局域波解的孤子数学理论。拟开展以下三个方面的研究：首先，基于非线性物理模型的双线性形式，发展长波极限和谱参数分析方法研究高维非线性物理模型的局域波特性及其形成机制，并借助符号计算和数值方法，分析波的特殊结构和性质以及振幅与背景之间的关系，讨论局域波的动力学演化；其次，发展KP梯队约化方法得到高维非线性物理模型的Grammian形式的多lump解，并讨论它们的相关性质和动力学演化；最后，发展广义N步达布变换研究高维非线性物理模型的局域波及其相互作用解。本项目将对高维非线性物理模型的局域波有更深入的理解，并归纳和总结出相应的物理机制和数学原理，为实际应用提供理论依据。

The study of local waves with novel properties and their interactions has always been an important topic in the fields of mathematical physics and fluid mechanics. We will apply soliton theory and integrable system theory method to focus on the local wave and its interaction solutions of the high-dimensional nonlinear physical model, and hope to explore and develop the soliton mathematical theory of the local wave solutions of the high-dimensional nonlinear physical model through the study of specific problems. We intend to carry out the research of the following three aspects: firstly, the long wave limit and spectral parameter analysis method are developed to study the local wave characteristics and its formation mechanism of the high-dimensional nonlinear physical model. With the help of symbolic calculation and numerical methods, the special wave structure and properties and the relationship between amplitude and background are analyzed, and the dynamics evolution of the local wave is discussed. Secondly, the KP hierarchy reduction method is developed to get the Grammian form of multi-lump solutions of high-dimensional nonlinear physical models, and their related properties and dynamic evolution are discussed. Finally, the generalized N-fold darboux transform is developed to study the local waves and their interaction solutions of high dimensional nonlinear physical models. This subject will have a deeper understanding of the local wave of the high-dimensional nonlinear physical model, summarize the corresponding physical mechanism and mathematical principle, and provide theoretical basis for practical application.