| There are many methods to solve nonlinear integrable systems,such as inverse scattering transformation,Darboux transformation,B(?)cklund transformation,Hirota’s direct method,mapping and deformation method,standard and extended truncated Painlev(?) analysis and so on.In addition to this,the symmetry analysis is also one of many effective methods for solving nonlinear systems.Not only Lie point symmetry group can be obtained,the symmetry reduction solutions of the original system can be gained by applying the standard Lie symmetry method to integrable differential systems.Applying potential symmetry,inverse recursion operators,Darboux transformations,B(?)cklund transformations and Lax pair methods,we can obtain many nonlocal symmetries for many differential systems but the nonlocal symmetries cannot be used to construct finite transformations directly.In order to overcome this obstacle,professor Lou proposed that by introducing new dependent variables,we can localize nonlocal symmetry to a local one in a new prolonged system and apply this method to many integrable systems,abundant new symmetry reduction solutions and their properties could be gained in detail.In this paper,the residual symmetry and interaction solutions of two kinds of nonlinear evolution equations are studied mainly by using residual symmetry and consistent Riccati expansion method.The contents are as follows.Firstly,the nonlocal residual symmetry of the(3+1)-dimensional KadomtsevPetviashvili(KP)equation and modified Jaulent-Miodek(JK)equation are obtained by using truncated Painlev(?) expansion.Since nonlocal symmetry cannot be directly reduced symmetrically,a new dependent variable is introduced to transform the nonlocal residual symmetry into Lie point symmetry of the prolonged symmetry.Then the theory of symmetric group transformation is obtained by solving the initial value problem according to Lie’s first fundamental theorem.Secondly,we use the consistent Riccati expansion method to find the exact solutions of the(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation and modified JaulentMiodek(JK)equation.The forms of the solutions are obtained through the analysis of the lead termFinally,we substitute the set solutions and Riccati equation into the equations and use maple to get the compatibility equation about W,so as to get the solutions of the equations.The Jacobian elliptic function is used to obtain the interaction solutions of different forms of soliton and elliptic periodic wave according to different forms of elliptic equations. |
