The Application Of General Symmetry Group Method Of Solving High-order And High-dimensional Nonlinear Equations With Variable Coefcients

This paper mainly studies the exact solution of the high-dimensional nonlinearKadomtsev-Petviashvili equation and the high-order nonlinear Schrodinger equation withvariable coefcients.Firstly, this paper briefly introduced some method used widely for solving nonlinearevolution equations. For example:(1) Hirota bilinear direct method.(2) Improved hyper-bolic function method.(3) Multi-linear variable separation method.(4) Multiple scales(5) Classical Lie group method.(6) Extending symmetry group method.Secondly, it is the application of high-dimensional and high-order variable coefcien-t nonlinear evolution equations by the extending of symmetric group methods mentionedabove, the extending symmetry group method applied to the Kadomtsev-Petviashvili (KP)equation with high-dimensional variable coefcient. I obtained the extend symmetry groupof this model and the corresponding finite transformation and solved any variable coef-cients KP equation using the transform. This result has the generalizability. And thenobtained some exact solution by choosing some special function for variable coefcients.when seeking the extending symmetry group of the cubic-quintic nonlinear variablecoefcient nonlinear Schrodinger equation (CQNLS), we believed that W have equal statuswith u as independent variables§It is not only a function of z, t, but also|u|,|u|2,|u|4.Selecting exponential function as variable coefcients§we obtained some new CQNLSequation with variable coefcients as well as its exact solution. These solutions can bewell explained dispersion characteristics of the soliton propagation in fibers with slowlydecreasing. These solutions can be well explained characteristics of the soliton propagation in fibers with slowly decreasing and can also provide a theoretical for the design of fiberamplifiers.Finally, some problems are presented which are to be solved in this paper.