本文選題：代數(shù)測(cè)地線 + Hamilton-Jacobi方法�。� 參考：《東北石油大學(xué)》2017年碩士論文

【摘要】：本文主要研究三個(gè)數(shù)學(xué)物理中的非線性問題,即n維二次曲面上的代數(shù)測(cè)地線的精確構(gòu)造,微擾KdV方程和微擾Burgers方程的大范圍漸近解和變形Bossinesq方程所有單行波解的分類。在第一章和第二章利用Hamilton-Jacobi方法研究n維二次曲面上的測(cè)地線問題。首先,得到了三維二次曲面上測(cè)地線的代數(shù)表達(dá)式,該測(cè)地線是兩個(gè)二維曲面的交線,然后利用隱函數(shù)定理和數(shù)值的方法,證明了這些測(cè)地線是真實(shí)存在的。最后,將上述結(jié)果推廣到n維,得到n維二次曲面上的測(cè)地線的精確代數(shù)表達(dá)式,并利用隱函數(shù)定理證明了這些測(cè)地線的存在性。第三章中,將重整化群方法應(yīng)用于兩個(gè)流體力學(xué)中的著名方程,即微擾KdV方程和微擾Burgers方程,得到了大范圍一致有效漸進(jìn)解。我們的方法是利用Kunihiro的基于微分幾何中包絡(luò)理論的重整化群方法,去消除近似解中的久期項(xiàng),使其在無窮遠(yuǎn)處收斂,進(jìn)而得到大范圍漸近解。最后一章,利用多項(xiàng)式完全判別系統(tǒng)來研究變形Bossinesq方程精確解問題。首先利用行波變換,將方程化成積分形式,通過討論根與系數(shù)的關(guān)系,將方程的解進(jìn)行分類,得到了Bossinesq方程的所有單行波解。
[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.
【學(xué)位授予單位】：東北石油大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175
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本文編號(hào)：1946051