Several Solutions To The Two-dimensional Leznov Equation

In this paper,we mainly study the application of differential difference(D-D)characteristic series method and differential difference(D-D)Lie symmetry method in solving differential difference equations,the characteristic set method will con-vert the zero set of equation to the union of set zero of characteristic series,which is irreducible triangular series.So that the degree and dimension of the equations can be reduced.Lie symmetry provides a systematic approach,To reduce the degree of the equations by studying the symmetry reduction and group invariant solution.Both of the two methods have the same purpose,that is to reduce the equations into some simple equations with Leznov degree and dimension,to reduce the calculation of solving the equations.In the two and three chapter of this paper,we introduce the basic knowledge of D-D characteristic and D-D Lie symmetry method.Firstly,we introduce the concepts such as ascending sequence,feature set and zero set.Then,we introduce the concepts of invariant group,extension and generator of differential difference equations.In the fourth chapter,we study the two-dimensional Leznov lattice equation by using the D-D method and the D-D Lie symmetry method respectively,The main contents are as follows:1)The D-D characteristic method is extended to the application of the two-dimensional differential order Leznov lattice equations.First,the equation is trans-formed and found to an effective D-D equation,And then the D-D characteristics,Finally,the zero set of Leznov lattice equations are decomposed by using the zero decomposition theorem,which is divided into 3 groups of zero characteristic series.2)The two-dimensional Leznov lattice equations are studied by using the D-D Lie symmetry method First,the decision equation is obtained by using the prolonga-tion formulas,Then we use the Maple software to solve the decision equations,and the symmetries of the Leznov lattice equations are obtained.At last,the Leznov lattice equation is reduced to a same lower order equations by using the symmetry.