积分分支法和混合函数法在求解非线性发展方程方面的扩展及其应用研究

Many problems in nonlinear science can be summarized as solving nonlinear evolution equations which arise from important models with mathematical and physical significances. Finding exact solutions to these equations has extensive applications in many fields such as hydrodynamics, condensed matter physics, solid-state physics, nonlinear optics, neurodynamics, crystal dislocation and fibre-optic communication. In our research project, based on the integral bifurcation method and mixed function method, which were proposed in our previous research works, we will extend the two methods by catholicity and universality of application. At the same time, using these methods, we will study some classical nonlinear evolution equations with practical background, which includes the continuous nonlinear partial differential equations and discrete difference-differential equations, their different kinds of exact solutions will be obtained. Further more, we will investigate dynamical properties and behaviors of these exact solutions, which in turn help us uncover not only the essentials of things, but the internal essence and laws of matters, eventually we explain dynamical phenomena of our investigated objective. Especially, for some singular nonlinear wave systems, and some complex models of high dimensionality which have both high order terms and more than one nonlinear terms, together with introducing some transformations, we establish individualized solving schemes for them according to their own features. We will obtain the exact solutions whose dynamical behaviors will be demonstrated.

非线性科学中的许多问题都可以归结为求解非线性发展方程的问题。寻找在数学物理模型中具有重要意义的非线性发展方程的精确解，在流体力学、凝聚态物理、固体物理、非线性光学、神经动力学、晶体位错研究以及光纤通信等很多领域有着广泛的应用。本项目在我们前期工作提出积分分支法和混合函数法的基础上，拟扩展这两种方法的普适性和应用方面的广泛性，同时用它们来研究一些经典的和具有实际背景的非线性发展方程，包括连续的非线性偏微分方程（组）和离散的非线性微分差分方程（组），从而获得这些重要的数学物理模型的各种精确解，并进一步研究这些解的动力学性质和动力学行为，用于揭示模型所描述的事物的内在本质和规律，解释所研究问题中存在的各种动力学现象。特别是对于一些奇异类型的非线性波系统和一些既具有高阶项又含有多个非线性项的高维系统的复杂模型，拟结合一些变换，为其量身定做求解方案，最终获得它们的精确解并解释其动力学行为。

本项目基于积分分支法结合齐次平衡原理、Frobenius思想、Lie对称分析以及扩展的F展开法等方法，研究了KdV型高阶波方程、广义的Gardner方程、广义的mKdV类水波模型、具有KdV结构的非线性Cologero-Degasperis-Fokas方程、（4+1）维非线性Fokas方程、高维的浅水波方程、高阶KdV(III)型非线性波方程、非线性Caudrey-Dodd-Gibbon方程、三维Kudryashov-Sinelshchikov方程和耦合的Sine-Gordon方程等一系列非线性发展方程，获得了这些非线性偏微分方程的各种精确解析解，并进一步研究了这些解的动力学性质和动力学行为，从而揭示了这些模型所描述的事物的内在本质和规律性，解释了所研究问题中存在的各种动力学现象。通过精确解的坐标图形，直观地展示了这些模型中存在的各种非线性波动现象。此外，将积分分支法与齐次平衡原理和分离变量法相结合的方法，发展了一套可以求解时间分数阶非线性偏微分方程精确解的新算法，利用这种新设计的求解方法，获得了时间分数阶KdV方程、时间分数阶Burgers方程以及一系列具有扩散与对流项的时间分数阶非线性偏微分方程的精确解，分析了这些精确解的动力学现象和动力学行为，并将分数阶模型的精确解与整数阶模型的精确行波解的动力学行为进行了对比，发现二者决然不同，差别非常大，即便是当分数阶的阶数趋近于整数时，分数阶模型的精确解也无法通过极限的形式得到整数阶模型的精确行波解，尽管当分数阶的阶数趋近于整数时，分数阶模型的方程可以通过极限的形式而得到整数阶模型的方程。