Study Of The Nonlinear Evolution Equations With Variable Coefficients, Soliton Solution

Nonlinear evolution equations are mathematical models, which are established based on the nonlinear problems in physics, communications and other fields. Exploring the solutions of nonlinear evolution equations and its properties is of great significance to the development of nonlinear science. Because of the variable coefficients can better reflect the interaction mechanism of nonlinear practical problems than constant coefficients, the study of nonlinear science is focused on the variable coefficient nonlinear evolution equations. But the presence of variable coefficients makes the study of nonlinear problems more difficult. In the process of the study of nonlinear evolution equations, solitary wave solution is discovered, which has the elastic scattering properties. Solitary wave have attracted much attention, because of their wide applications in the field of optical fiber communication, fluid mechanics etc. So finding the nonlinear evolution equations’localized excitations of the solitary wave form is a research focus currently. This article’s purpose is to find the soliton solutions of the nonlinear evolution equations with variable coefficients. Firstly, nonlinear evolution equations with variable coefficients’integrability need to be detected; secondly, based on the integrability, the Hirota method and the method of nonlinear separation of variables can be used to explore the linear multi-solitons and localized excitations.In the first part of this article, the nonlinear evolution equations and the process of the soliton theory’s development was introduced; then the research methods of soliton theory was described, and the Painleve analysis, Hirota method, and nonlinear separation of variables method are on the point.In the second part of this article, variable-coefficient (2+1) dimensional breaking soliton equation’s exact multi-soliton solution is exploring. Firstly, Painleve analysis is used to test the equation’s integrability and get the bilinear transformation; next, the exact expression of N-soliton solutions is obtained with the Hirota method; finally, from the image of a single soliton, two solitons, which is obtained with the computer simulation, the equation’s soliton solutions’ development can be felt intuitive.In the third part of this article, coupled variable coefficients (2+1)-dimensional breaking soliton equation’s variable separated solution and localized excitations are exploring. Firstly, Painleve analysis is used to detect the coupled equations’integrability and get the integrable conditions; Secondly, use the variable separated approach to get the equation’s variable separated solution; In the end, with the arbitrary of the low-dimensional functions in the solution, four localized excitation modes are simulated.In the fourth part of this article, the research results of the preceding part are summarized, also the innovation points and the difficulties in the study are pointed out. First, coupled and high-dimensional variable coefficient nonlinear evolution equation’s integrable condition obtained by Painleve analysis; Second, variable coefficients (2+1)-dimensional breaking soliton equation’s N-soliton solution by the promoted the Hirota method; Third, under the integrable condition, the nonlinear separation method’s research is developed, getting coupled variable coefficients (2+1)-dimensional breaking soliton equation’s variable separated solution and its localized excitations. Finally, looks forward to the research prospects of nonlinear evolution equations.