【作者】 黄磊；

【导师】 孙建安；

【作者基本信息】 西北师范大学 ， 理论物理， 2007， 硕士

【摘要】 随着非线性科学的发展,非线性物理学也迅速发展起来.在非线性物理学中,我们常常把复杂的非线性物理系统简化为非线性演化方程来研究,通过对方程的求解来确定物理量之间的定量或定性关系,并可以通过解的图形给出物理量之间关系的直观形象.因此,求解非线性方程并给出解的图形对物理学的发展具有重要意义.本文分别研究了求解非线性演化方程的幂级数展开法和扩展的Riccati方程映射法,并将它们应用于求解动脉血管中血液脉搏波方程、(2+1)维色散长波方程和(3+1)维Burgers方程.最后,由扩展的Riccati方程映射法求得的解,得到了(2+1)维色散长波方程和(3+1)维Burgers方程的局域激发结构.主要工作如下:1.介绍了幂级数展开法的求解步骤,然后将其应用于动脉血管中非线性血液脉搏波方程,得到了方程的周期解、孤波解和激波解.最后得出了结论:在动脉血管中,血液脉搏波在不同条件下会分别以周期波,孤波或激波形式传播.2.对扩展的Riccati方程映射方法作了介绍,然后将此方法分别应用于(2+1)维非线性系统和(3+1)维非线性系统,最终得到了(2+1)维色散长波方程的分离变量解、孤波解、周期解和(3+1)维Burgers方程的分离变量解,并且这些解中含有任意函数.3.在所得到的(2+1)维色散长波方程和(3+1)维Burgers方程解的基础上,通过对解中任意函数的适当选取,得到了它们丰富的局域激发结构和分形结构.并得出结论:(1)分形不仅会出现在不可积系统中,也会出现在可积系统中.(2)(3+1)维非线性系统的局域激发结构比(2+1)维非线性系统的局域激发结构更为丰富.更多还原

【Abstract】 Nonlinear physics developes fastly with the development of nonlinear science. In nonlinear physics, simplified nonlinear evolution equations are often employed to describe the complex nonlinear physics symtem. The quantificational or the qualitative relations between physics quantities can be determined by solving the nonlinear evolution equations. Besides of this, the firsthand impression of the relations between physics quantities can be got by pictures of the solutions of the nonlinear evolution equations. Then, it is very important for the development of physics to solve the nonlinear evolution equations and give the pictures of the solutions. In this dissertation, the power series expansion method and the extended Riccati mapping approach are studied, and employed to solve the nonlinear blood waves in arterial blood vessel, the (2+1)-dimensional dispersive long-water wave equation and the (3+1)-dimensional Burgers equation separately. At last, the localized excitations of (2+1)-dimensional dispersive long-water wave equation and the (3+1)-dimensional Burgers equations are obtained by selecting the arbitrary functions properly in their solutions . There are mainly three sections in this dissertation.1. The power series expansion method is introduced. Applying this method to nonlinear blood waves in arterial blood vessel, the periodic solutions, the solitary solutions and the shock solutions of the equations are obtained. At last, the conclusion is that the blood wave transmits in the arterial blood vessel with the form of periodic wave, solitary wave or shock wave will appear separately under the different conditions.2. The extend Riccati mapping approach is introduced. Then this approach are applied to the (2+1)-dimensional dispersive long-water wave equation and the (3+1)-dimensional Burgers equation. Finally, the variable separation solutions, the solitary solutions and the periodic soltions of the (2+1)-dimensional dispersive long-water wave equation and the variable separation solutions of the (3+1)-dimensional Burgers equation are gained. In addition, there are arbitrary functions in these solutions.3. Based on the solutions obtained above, abundant localized excitations and fractals are received by selecting the arbitrary functions appropriately. And the conclusions are as follows:(1) There are fractals not only in the non-integrable physics systems but also in the integrable physics systems.(2) There are much more abundant localized excitations in the (3+1)-dimensional physics systems than in the (2+1)-dimensional ones.更多还原