The Configuration Solutions Of A Reduced Vortex Dynamical System

Mathematical model of nonlinear wave equation is a kind of important,often used to describe natural phenomena,also is one of nonlinear mathemat-ical physics forefront research subject.Because of its important application background and the resulting nonlinear mathematical difficulties,caused the strong study interest,has wide application and strong vitality.By solving the nonlinear wave equation and the study of qualitative analysis,helps people to understand the nature of the system characteristics,greatly promote the re-lated subjects such as physics,mechanics,and the development of engineering technology.Nonlinear wave equation is a special kind of nonlinear evolution equation-s,which can be regarded as an infinite dimensional dynamical system?that is,using nonlinear evolution equations represented by system?,also has many applications in practical problems,for example,quantum mechanics,the non-linear Schr ddotodinger equation,thermal reaction diffusion phenomenon in the non-linear heat conduction equation,the nonlinear wave equation in elec-tromagnetics,Yang-Mills in gauge field equations and so on.For a long time,because of the nonlinear wave equation with nonlinear and complex physical characteristics,its exact solution problem attracted a large number of mathematicians and physicists,in-depth study directly for ex-act solutions of nonlinear wave equation is difficult,so people try the problem of nonlinear wave equation into ordinary differential equation and solve some special solution of the problem.The main purpose of this paper is to study the nonlinear wave equation is reduced vortex dynamics behavior of power system,using the method of configuration solution Arbody problem,give some collinear reduced vortex dynamical system,configuration and relative equilib-rium solutions.Consider the reduced vorticity dynamical systemand the initial conditionsX_j=?x_j,y_j??R²represents the center of the Jth vortex,mj=±1 is the index of the jtn vortex.First,we try to find the collinear configuration solution of vorticity dy-namical system?1?,Definition 1 The vorticity dynamical system?1?-?2?is shaped likex_j?t?=??t?a_j?j=1,...N?,the solution is called the collinear configuration solu-tion,0?t?iis a quadratic continuous differentiable function,a_j?j=1,2,...,N?is a constant vector.Lemma 1 If there is a scalar function ??t?,constant ?and often vectora_j?j=1,2,...,N?,satisfythen x_j?t?=??t?a_j?j=1,2,...,N?is the vorticity dynamical system?1?-?2?collinear configurational solutions.The vorticity dynamical system?1?-?2?the collinear configuration solutions under some special initial conditions are given in detail in chapter 3.Consider the reduced vorticity dynamical system?1?-?2?the relative equi-librium solution.?? R,order inDefinition 2 Call x_j=e^-?Jtaa_j?j=1,2,…,N?is the reduced vorticity dynamical system?1?-?2?the relative equilibrium solution,in,??R,a_j Is a constant vector.Lemma 2 If a_j satisfyx_j=e^-?Jta_j?j=1,2,…,N?is the reduced vorticity dynamical system?1?-?2?the relative equilibrium solution,Furthermore,the relative equilibrium solutions of the vorticity dynamical sys-tem?1?-?2?under several special initial conditions axe given.