某些非线性问题的精确解与渐近解

发布时间：2018-05-28 08:42

  本文选题：代数测地线 + Hamilton-Jacobi方法　； 参考：《东北石油大学》2017年硕士论文

【摘要】：本文主要研究三个数学物理中的非线性问题,即n维二次曲面上的代数测地线的精确构造,微扰KdV方程和微扰Burgers方程的大范围渐近解和变形Bossinesq方程所有单行波解的分类。在第一章和第二章利用Hamilton-Jacobi方法研究n维二次曲面上的测地线问题。首先,得到了三维二次曲面上测地线的代数表达式,该测地线是两个二维曲面的交线,然后利用隐函数定理和数值的方法,证明了这些测地线是真实存在的。最后,将上述结果推广到n维,得到n维二次曲面上的测地线的精确代数表达式,并利用隐函数定理证明了这些测地线的存在性。第三章中,将重整化群方法应用于两个流体力学中的著名方程,即微扰KdV方程和微扰Burgers方程,得到了大范围一致有效渐进解。我们的方法是利用Kunihiro的基于微分几何中包络理论的重整化群方法,去消除近似解中的久期项,使其在无穷远处收敛,进而得到大范围渐近解。最后一章,利用多项式完全判别系统来研究变形Bossinesq方程精确解问题。首先利用行波变换,将方程化成积分形式,通过讨论根与系数的关系,将方程的解进行分类,得到了Bossinesq方程的所有单行波解。
[Abstract]:In this paper, we study three nonlinear problems in mathematics and physics, that is, the exact construction of algebraic geodesic on n-dimensional Quadric surface, the large-scale asymptotic solutions of perturbation KdV equation and perturbation Burgers equation and the classification of all one-way wave solutions of deformed Bossinesq equation. In the first and second chapters, the geodesic problem on n-dimensional Quadric surfaces is studied by using Hamilton-Jacobi method. First, the algebraic expression of geodesic on 3D Quadric surface is obtained. The geodesic line is the intersection of two two-dimensional surfaces. Then, by using implicit function theorem and numerical method, it is proved that these geodesic lines are real. Finally, the above results are extended to n-dimensional, and the exact algebraic expressions of geodesic on n-dimensional Quadric surfaces are obtained, and the existence of these geodesic lines is proved by using implicit function theorem. In chapter 3, the renormalization group method is applied to two famous equations in hydrodynamics, that is, perturbation KdV equation and perturbation Burgers equation. Our method is to use Kunihiro's renormalization group method based on envelope theory in differential geometry to eliminate the duration term in the approximate solution, to make it converge at infinity, and then to obtain the asymptotic solution in a large range. In the last chapter, the exact solution of deformed Bossinesq equation is studied by using polynomial complete discriminant system. First, the equation is transformed into integral form by using traveling wave transformation. By discussing the relation between root and coefficient, the solutions of the equation are classified and all the one-way wave solutions of Bossinesq equation are obtained.
【学位授予单位】：东北石油大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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本文编号：1946051