Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Nonlinear partial differential equations

Study on the Decomposition and Interact of Soliton Solutions for Some Nonlinear Evolution Equations

Author YePeng
Tutor GuoYuCui
School Beijing University of Posts and Telecommunications
Course Applied Mathematics
Keywords Nonlinear evolution equations The soliton ( solution ) Hirota bilinear method Boussinesq equation KP equation Soliton interaction
CLC O175.29
Type Master's thesis
Year 2006
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The solitary (Li) sub widespread in many non-linear phenomena, has important applications in the development of modern science and technology. Soliton theory - nonlinear partial differential equations (PDEs) nonlinear theory - more than 40 years ago began to develop this theory more in-depth study not only, but also in the physical and engineering sciences applications driven still in rapid development. The soliton phenomenon exists under two conditions: First, in the case of the absence of any external obstacles, there is a continuous structure (shape, speed, etc.) the stability of the solitary wave propagation. Second, if an isolated wave encounters another similar elevational solitary wave, their interaction, but does not destroy each other characteristic (i.e. the so-called elastic). This solitary wave is defined as solitary (Li) sub. The soliton phenomenon is essentially non-linear phenomena. The main purpose of this paper is to study the Hirota bilinear method and soliton solutions of nonlinear evolution equations as well as the nature of the soliton, decomposition, and explore the interaction, in order to make better use of the soliton. Usually in Applied Mathematics, soliton understood as localized traveling wave solutions of nonlinear evolution equations, after bumping into each other, does not change the waveform and speed (perhaps the phase change). In the field of physics, the soliton is understood solitary wave interaction by waveform and speed only slightly changed, or is understood as limited energy solutions of nonlinear evolution equations, the energy is concentrated in a limited area of ??the space does not change with time increased and spread to an infinite area. Nonlinear development of the soliton solution is widely used, especially in the nonlinear fiber optical soliton communication has been extensively studied in recent years. Therefore, the study of nonlinear evolution method for finding the soliton solution as well as the nature of the soliton, decomposition and interaction is particularly necessary, but also of practical value. Nonlinear evolution equation theory (evolution), combined with the rapidly growing soliton theory, the use of mathematical software to study a class of solutions of the nonlinear evolution equations - soliton solutions. Completed the following work: First, analyzes and summarizes the type of soliton and multi-soliton, soliton in an infinite number of conservation laws, to study soliton and its properties can better analysis and understanding of the basics of these solitons; Secondly, the Hirota bilinear soliton solutions to the Boussinesq equation (31) dimensional KP equation method for nonlinear evolution equations example to discuss the Hirota bilinear method in soliton solutions of nonlinear evolution of the specific application, and gives a simple Hirota method and its application; obtained single-soliton solutions and two-soliton solutions of soliton solutions of a decomposition method and soliton interaction, soliton interaction of the two situations and given through a graphical comparison of the soliton two different role in the process. We hope to learn more about these studies soliton solutions for solving nonlinear evolution equations, properties and applications of the soliton.

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