Studies On Power Series Expansion Method And Extend Riccati Mapping Approach In Nonlinear Physics Equations

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.