【作者】 王辉；

【导师】 耿献国；

【作者基本信息】 郑州大学 ， 基础数学， 2014， 博士

【摘要】 本文主要分为如下两个部分：其一，借助于Lenard递推序列，推导出分别与一个4×4、两个3×3矩阵谱问题相联系的孤子方程族，对于某些方程族或者方程，我们给出了它们的广义Hamilton结构和无穷守恒律；其二，我们给出了相应孤子方程的精确解。其中第二章，我们给出了相应CH型方程的尖孤子解；第四、五章基于三角曲线理论及代数几何知识，我们构造出了相应孤子方程的代数几何解。第二章中，我们通过引入负幂流，得到三类CH型方程。其中两个具有N peakon形式解。我们借助广义函数δ，给出了N peakon解所满足的动力系统。孤子方程的代数几何解揭示解的内部结构，描述了非线性现象的拟周期行为。本文第三章主要介绍黎曼面以及Theta函数的相关知识，其中的概念，引理以及定理可以更好地帮助我们理解三角曲线。第四章和第五章，我们采取一套很系统的方法去构造三角曲线，再通过引入适当的Baker-Akhiezer函数，亚纯函数及椭圆变量，从而将孤子方程分解为可解的Dubrovin-type常微分方程组。进一步，根据亚纯函数及Baker-Akhiezer函数零点和极点的性质，我们定义第二类和第三类Abel微分，结合Riemann定理及Riemann-Roch定理，得到了亚纯函数以及Baker-Akhiezer函数的黎曼theta函数表示。最后，我们再结合亚纯函数以及Baker-Akhiezer函数的渐近性质，给出了孤子方程族的代数几何解。更多还原

【Abstract】 The thesis can be mainly divided into two parts. First, with the help of Lenardrecursion equations and the zero-curvature equation, we derive three different soliton hi-erarchies, which are associated with one4×4and two3×3matrix spectral problemsrespectively. Moreover, generalized Hamiltonian structure and infinite conservation lawsof the hierarchy and soliton equation are established; On the other hand, We present theexplicit solutions of soliton equations above. In chapter two, we derive peaked solutionof some CH type equations. And based on the theory of trigonal curve and the knowledgeof algebraic geometry, we construct the algebro-geometric solutions of two hierarchies ofsoliton equations associated with two different3×3matrix spectral problems in chapterfour and five, respectively.In chapter two, by introducing the negative flow, we derive three CH type equations,two of which admit N-peakons.Algebro-geometric solutions of soliton equations reveal inherent structure mecha-nism of solutions, and describe the quasi-periodic behavior of nonlinear phenomenon.Chapter three mainly concentrates on Riemann surface and Theta function, and the con-cepts, lemmas and theorems do a good favor to understand the trigonal curve. In chapterfour and five, we propose a systemic method to construct the trigonal curve, and thenintroduce the appropriate Baker-Akhiezer function, meromorphic function and ellipticvariables on the three-sheeted Riemann surface, from which soliton equations are decom-posed into the system of solvable Dubrovin-type ordinary differential equations. Further-more, in accordance with the properties of the zeros and singularities of the meromor-phic function and Baker-Akhiezer function, we get their Riemann theta function repre-sentations by means of the second and third Abel differentials, Riemann theorem andRiemann-Roch theorem. Combining the Riemann theta function representations of the meromorphic function and the Baker-Akhiezer function with their asymptotic properties,we finally obtain the algebro-geometric solutions of soliton equations.更多还原