Analytical And Numerical Studies Of Several Nonlinear Mathematical Physics Problems

Nonlinear problems in mathematics and physics are arousing great interests from scientists as the theories on linear problems have been well developed. Most of the problems in both basic science and engineering-oriented fields are nonlin-ear, and how to handle these complicated problems is a significant challenge for mathematicians and physicists. In this dissertation, two topics are discussed. On one hand, we study a few important nonlinear problems and obtain some meaningful results by using some known mathematical tools. On the other hand, we also propose a new approximate approach for nonlinear problems by abstract-ing and summarizing the basic idea which underlies some well-known exact and approximate mathematical methods.Chapter 1 is an introduction which is devoted to reviewing the mathematical and physical backgrounds of some important nonlinear equations discussed in this dissertation. The significance and development of the nonlinear science are reviewed, too. The mathematical tools involved are introduced in this chapter, and wre also briefly report the main wrorks of this dissertation.In chapter 2, we derive a variable coefficient KdV equation and a variable coefficient modified KdV(mKdV) equation from a nonlinear inviscid nondissipa-tive and equivalent barotropic vorticity equation in a beta plane. We also derive a coupled variable coefficient mKdV system from a two-layer model of stratified fluid. By constructing exact solutions of these derived equations, we explain a crucial nonlinear phenomena in atmospheric system, that is. the evolution of atmospheric blocking life cycles thoroughly. Based on the multiple scale expan-sion method, we propose a systemic approach for deriving variable coefficient equations from the original models. We also develop a direct method to con-struct Backlund transformations which connect the derived variable coefficient equations with their corresponding constant coefficient ones. Then taking ad-vantages of the Backlund transformations and the known results of the constant coefficient nonlinear equations, the solutions of the variable coefficient nonlinear systems can be generated. This direct method is also used in other chapters in this dissertation as a fundamental way to solve variable coefficient nonlinear equations.In chapter 3, we derive a multiple vortex interaction model from the (2+1)-dimensional nonlinear inviscid nondissipative and equivalent barotropic vorticity equation in a beta plane by supposing four reasonable constraints (vortex local-ity, short range interaction, one-one interaction and total system). The classical Lie symmetries and conservation laws of the model are discussed. Exact so-lutions such as the vortex sources and Bessel vortex solutions are given, too. Furthermore, we numerically simulate the vortex interactions in the atmospheric systems without rotation based on the vortex interaction model. It is found that this model can generate several interaction patterns such as merging, separation, mutual orbiting and absorption. The results of these numerical simulations are well consistent with some known experiments and meteorologic observations. The model is also used to explain the interaction between a typhoon and a subtropical high.In chapter 4, the direct method developed in chapter 2 is used to solve a (3+1)-dimensional variable coefficient nonlinear Schrodinger(NLS) equation and a coupled (1+1)-dimensional variable coefficient NLS system via constructing the Backlund transformations between the variable coefficient equations and the constant variable ones. Solitary wave solution and vector solitary wave solution are obtained for the two variable coefficient equations respectively, and the rela-tions between those variable coefficients under which the exact solutions exist are also revealed. These two variable coefficient nonlinear equations are theoretically meaningful in Bose-Einstein condensation and nonlinear optics. We also study a novel nonlinear equation, the resonant Davey-Stewartson(DS) equation, which is related to the NLS equation, and three types of exact solutions of the resonant DS equation are constructed via the classical Lie group theory.Different with chapter 2,3 and 4, chapter 5 is devoted to developing a novel approximate method, nonsensitive homotopy approach, which is based on the homotopy analysis method. The nonsensitive homotopy approach builds linear or nonlinear homotopy relations between the hard-to-solve original model and a simplified model which has exact solutions, while some auxiliary parameters are also introduced to generate highly accurate approximate solutions. This ap-proach does not rely on small perturbation parameters. Compared with the known liomotopy analysis method, the nonsensitive homotopy approach shows that it is possible to introduce nonlinear homotopy relations, and it gives a non-sensitive principle to chose the reasonable auxiliary parameters. Furthermore, by constructing the nonsensitive quantities, an explicit process for choosing aux-iliary parameters is given. The nonsensitive homotopy approach is used to solve two nonlinear differential equations and to calculate the energy levels of several quantum anharmonic oscillators. The results of the calculations and the error analysis show the validity of the nonsensitive homotopy approach. This approach can be used to solve nonlinear differential equations approximately, and can also be used as an effective nonperturbative technique in physics.The last chapter concerns the summary and discussion for the whole disser-tation, and the prospect for the further work is also discussed in this chapter.