Several Classes Of Nonlinear Evolution Equations Of The Solution Of The Symmetry Group And Constant Study

In this paper, by applying the classical Lie group method, nonclassical Lie groupmethod, the direct symmetry method and the improved CK direct reduction method, westudy some partial differential equations such as fifth-order KdV equation with variablecoefficients,(2+1)-dimensional Burgers equations, new fifth-order nonlinear integrableequation. And then by using the Lie symmetry to reduce and solve the above equations, weget some new exact solutions of the above equations. In addition, we study the mKdVdifference equation of invariance. Lie group method is applied to the study of differencescheme is so less, thus the Lie group method is used to investigate difference mKdVequation invariance is the innovation points in this paper.In Chapter one, by applying the classical Lie group method, we studies the fifth-orderKdV equation with variable coefficients, and then we obtain the corresponding Lie algebra,optimal system and the similarity reductions. Then we give some exact solutions for somespecial forms of the equations.In Chapter two, by using the direct symmetry method, namely undeterminedcoefficient method, we obtain the Lie symmetry and similarity reductions of (2+1)-dimensional Burgers equations. Using the derived Lie symmetry, the correspondingreduced equations of (2+1)-dimensional Burgers equations are derived. Then we obtainsome new exact solutions of (2+1)-dimensional Burgers equations through the aid of theauxiliary function method to solve the reduced equations.In Chapter three, by applying Lie symmetry method and the direct symmetry method,we get the corresponding Lie algebra and similarity reductions of a new fifth-order nonlinear integrable equation. At the same time, the explicit and exact analytic solutionsare obtained by means of the power series method. At last, we also give the conservationlaws.In Chapter four, by applying the improved CK direct reduction method, we derive anequivalence transformation of fifth-order KdV equation with variable coefficients. Also therelationship between the solutions of fifth-order KdV equation with variable coefficientsand those of the corresponding constant coefficients fifth-order KdV equation withvariable coefficients is found. Then we reduce into and solve the constant coefficientsfifth-order KdV equation by using the classical Lie group method, nonclassical Lie groupmethod. According to the derived relationship of solutions, we obtain some new exactsolutions of fifth-order KdV equation with variable coefficients.In Chapter five, by means of Lie point symmetry method, we have studied thedifference equation of the mKdV, and then get the Lie algebra infinitesimal operators. Wefind out that Lie point symmetry not only keep mKdV symmetry of the differentialequation invariance, but also keep the difference equations with unified orthogonal meshinvariance.In a word, first, the unique feature of this paper is the Lie point symmetry applying toequations which arises to in mathematical physics partial differential equations (PDEs),especially in difference equations, and then we reduce and solve them; obtaining anequivalence transformation of equations with variable coefficients by using the improvedCK direct reduction method, according to the derived equivalence transformation, weobtain some new exact solutions of the variable coefficients equations. At last, the Liepoint symmetry group is applied to difference equation, and we study the invarianceproperties of them. It is important to study the practical problem. This is one of the biggestinnovations in this paper.