The Application Of Lie Symmetry Method In Solving The Nonlinear Evolution Equations

In this paper, by applying the classical Lie group method, the compatibility methodand the modified CK direct reduction method, we consider the Lie symmetries of four setsof nonlinear evolution equations include the (2+1)-dimensional Kadomtsov-Petviashvili-Joseph-Egri equation, the mKdV-KP equation, the Broer-Kau-Kupershmidt equations andthe (2+1)-dimensional Painleve Integrable Burgers equations. Using the symmetries, wefind some classical similarity reductions of the four equations. Many kinds of exactsolutions are derived by solving the reduced equations.In Chapter one, by using the classical Lie group method, we obtain the classical Liepoint symmetry of the (2+1)-dimensional Kadomtsov-Petviashvili-Joseph-Egri (KP-JE forshort) equation. Using the symmetries, we find some classical similarity reductions ofKP-JE equation. Many kinds of exact solutions of the KP-JE equation are derived bysolving the reduced equations, including the elliptic functions, the rational functions，thehyperbolic functions, the trigonometric functions, and the power series solutions. To showthe kinetic property of the derived solutions intuitively, we draw the wave sectionaldrawing at last.In Chapter two, using the compatibility method, we find the nonclassical symmetry,reductions and new exact solutions of mKdV-KP equation including the hyperbolicfunctions, the trigonometric functions, the rational functions, the elliptic function solutionsand so on.In Chapter three, using the modified Clarkson–Kruskal direct method, the symmetriesand the reductions of (2+1) dimensional Broer-Kau-Kupe-rshmidt (BKK for short) areobtained. At the same time, a great many of solutions are derived by solving the reductionequations, including the rational functions, hyperbolic functions, the trigonometricfunctions, the power series solution and the Airy function solution.In Chapter four, by using the modified CK direct method, the relationship is derivedbetween the new solutions and the old ones of the (2+1)-dimensional Painleve IntegrableBurgers (PIB) equations. Based on the relationship, some new solutions are obtained by using the given solutions of the equation. At the same time, the symmetry of PIB equationsis also obtained. Using the symmetry, the similarity reductions and some new exactsolutions are also presented.In sum up, the main feature of this paper include choosing a proper transformation toeffectively reduce and solve the nonlinear evolution equations and building the relationshipbetween the new solutions and the old ones by using the modified CK direct reductionmethod.