The Research Of Nonlinear Wave And Complex Networks

This thesis includes two parts: one is of solving the analytic solutions of nonlinear partial differential equations;the other is the research and discussion of complex networks.Nonlinear equations mainly describe nonlinear phenomena and complex systems, and their solutions have important help for the understanding of nonlinear interaction and behaviors of complex system. Owing to complexity of solution space of nonlinear equations, there have not normative ways and theories of solving nonlinear equations, and people are compelled to develop many methods for solving the special solutions of nonlinear equations.In the first part of the thesis, two ways of solving nonlinear equations are discussed : homogeneous balance method and Jacobi elliptic function expansion method. First, the process of solving the equations using the homogeneous balance method is introduced simply. And, the variant Boussinesq equation is seen as an example of this method and the soliton solutions of the equation are obtained simply.We mainly study that when the presumed solution function φ = φ(x,t) are otherspecial form solutions, multi-soliton solutions, periodic solutions, rational function solutions, imaginary function solutions, singular solutions and self-similar solutions and so on are obtained. At the same time, the Backlund nonlinear transformation of equations is obtained in the process of deduction. In the second method, firstly, basic Jacobi elliptic function expansion method and applications are introduced simply. Based on this, we have made several new generalizations for the Jacobi elliptic function expansion method. The first generalization is that the expansion of solutions of equations is generalized to negative exponent, and negative exponential elliptic function solutions are obtained. The solutions solved using this generalization include the ones solved using the basic Jacobi elliptic function expansion method. The second is that the expansion of solutions of equations is generalized to include fractionalexponential form (1 + δf~2(ξ))~2 elliptic function solutions, and fractional exponentialform solutions are obtained. The third and fourth is that two different elliptic functions direct coupled forms expand to solve the solutions of equations. In this way, the solutions obtained include the two coupled elliptic function forms. Using these generalized elliptic function expansion methods, lots of new solutions of equations are obtained.In nature and human being society, there are many systems in which there are very complex interactions, for instance, neural network, metabolism, Internet networks, traffic networks and people's relations, and so on. Any complex system, in which there are interactions, can be designed as a complex network under certain abstract.In the second part of this thesis, firstly, the origin of complexity and the basic concepts are introduced, and several familiar complex network models are presented. Secondly, we describe the properties of Chinese railway passenger transport by the way of complex network. Chinese railway passenger transport is seen as complex network: By viewing stations as "nodes", an arbitrary pair of stations is considered to be connected by a "link" when at least one train stops at both stations. In this abstract way, a railway traffic network is generated. It is found that a railway traffic network has the characters of scale free network and small world network. Therefore, this network is believed as small world network with power law distribution. This network is a good practical example for the BA growth network model. Dynamics problems on complex networks are current important research topics. It is important for the understanding of interactions of complex network and complex actions, and the depended relations to the topology characters of networks. Finally, we simply discuss random walk problem on Weighted Lattice embedded Scale-Free network. It is found that when the controlled regional parameter A of the network is less, the impact of regional space on random walk is biggish. When the parameter A is biggish, the characters of the network go to the ones of random network and the impact of regional space on random walk is less.