Study On Some Problems In Nonlinear Physics

Nonlinear physics is one of the important branches in nonlinear science. In this paper we have investigated the soliton theory and chaotic theory. We have also investigated the hyperbola function method, which is based on the soliton theory, to solve the nonlinear evolution equations. We have introduced the hyperbola function method as well, and generalized the method to solve some nonlinear evolution equations. The main results are as follows:(l)We got the damping KdV equation u, + 6uux + u + su = 0 by using the reductive perturbation method from the incompressible fluid filled thin elastic tube. The theoretical and numerical investigation for this equation was done. The results indicate that the amplitude of the initial solitary wave will decrease exponentially with respect to time t when s > 0 and will increase exponentially when s < 0 . Some new soliton(s) will be generated in the latter case.(2) We studied the hyperbola function method and generalized it. At first, we took the KdV equation and the KdV-Burgers equation as examples to point out the fact that it is not necessary to obtain the value of the order of solitary wave solutions by equating the highest nonlinear term and the highest-order partial derivative term in this method. Meanwhile, we obtained some exact solitary wave solutions of the KdV-Burgers equation. That two conditions of the expansion functions should be satisfied was presented by solving the nonlinear vibrating string equation and the combined KdV equation. Many exact solutions were obtained by selecting the other expansion functions based on these two conditions.(3) We successfully derived some exact solitary wave solutions of the nonlinear approximate equations with long waves in shallow water by using the hyperbola function method. Some results are different from the previous work. The fact shows that the method can be applied to solve not only the single nonlinear equation but also a set of coupled nonlinear equations.(4) The study on solving higher-dimensional nonlinear equations was done. We took the (3+l)-dimensional NNV equation as an example, reduced it by using the travelling wave method. The translation relation between the NNV equation and one-dimensional KdV equation and that between the NNV equation and the one-dimensional mKdV equation were found. This approach can be generalized to find the relationship between a higher dimensional equation and a lower dimensional nonlinear equation. Some exact solutions of the higher dimensional nonlinear equation can be obtained by solving the lower one.(5) We studied the forced and damping system of the spring pendulum, and both theoretically and numerically investigated this nonlinear wave equation and obtained some interesting results. They indicate that the spring pendulum system shows variety of bifurcation and chaotic phenomena as the parameters of the system vary in an appropriate range.