Study On Special Wave Solutions For Higher-dimensional Nonlinear Evolution Equations With Symbolic Computation

The universe is nonlinear,so various phenomena in nature are basically nonlinear.The linear equations are only applied to approximately describe nonlinear phenomena.However,with the continuous development of computer and scientific computing techn-ology,people’s computing ability has greatly improved.The nonlinear differential equa-tions have gradually become the main object of research for mathematics and physicists.Compared to the linear differential equations,the study of the properties and solutions of nonlinear differential equations is much more difficult.It is possible to get general solutions for linear differential equations,but nonlinear differential equations cannot usually be solved directly.They are often transformed into solvable differential or algebraic problems through transformations.Thus we can obtain some specific exact solutions of the original equation,especially special wave solutions.In this paper,two different approaches are used to construct higher order wave solutions or complex interaction wave solutions for higher-dimensional nonlinear evolution equations.In the first part of this paper,based on the direct algebraic method,we mainly study the mechanical algorithms for constructing higher order wave solutions and complex interaction solutions for higher-dimensional nonlinear evolution equations.The simplified Hirota method can be used to construct the N-soliton solution for higher-dimensional nonlinear evolution equations.In this paper,combining with the long wave limit method,the generation formula for calculating the interaction solution among solitons and lump waves is given for the first time and the correctness of the formula is proved.Based on this formula,the generation formula for calculating the interaction solution among solitons,breathers and lump waves is further provided by virtue of the parameters conjugated assignment.On this basis,with the help of the symbolic computing system Maple,the multi-wave interaction solutions for a higher-dimensional integrable and a higher-dimensional nonintegrable nonlinear evolution eq-uation are constructed,respectively.Because the special wave solutions for nonlinear evolution equations obtained by different direct algebraic methods usually have specific forms,for example,the soliton solutions can be expressed as simple combinations of exponential functions.Similarly,the lump wave solutions can be expressed as simple combinations of even degree polynomials and so on.On this basis,it is natural to apply the hypothesis method,that is,by assuming that the auxiliary function has a specific form,the problem of solving the original differential equation can be transformed into solving a nonlinear algebraic equations.However,when using it to construct the complex wave solutions for higher-dimensional nonlinear evolution equations,there will be the problem of solvi-ng（super）large-scale nonlinear algebraic equations.The inheritance solving strategy can rapidly reduce the scale of（super）large-scale nonlinear algebraic equations.In this paper,based on the hypothesis method and the inheritance solving strategy,different types of multi-wave interaction solutions for two higher-dimensional nonlinear evolution equations are constructed with different inheritance paths.In the second part,we construct the soliton molecule solutions,the nonlinear super-position or nested solutions of different types of waves for higher-dimensional nonlinear evolution equations by constructing nonlocal symmetries and symmetry transformati-ons.The Lie group method is a universal method for constructing analytical solutions for nonlinear differential equations.Based on the local Lie point symmetry,scholars have constructed many exact solutions of nonlinear differential equations.However,based on the local Lie point symmetry,it is difficult to construct N^thsymmetry transfor-mations for nonlinear systems,and it is also impossible to directly calculate the higher-order wave solutions for nonlinear evolution equations.Based on the nonlocal symmet-ry,any N^thsymmetry transformations for some nonlinear systems can be obtained.Based on higher-order symmetry transformations,the N-soliton solution or other higher-order wave solutions for these equations can be directly calculated,that is,more exact wave solutions can be obtained.However,the construction of nonlocal symmetry and its localization,as well as the construction of symmetry transformations are computatio-nally complex.Based on previous work,this paper constructs potential symmetry,residual symmetry and nonlocal symmetry derived from Darboux transformation for several higher-dimensional nonlinear systems based on Lax pairs,the truncated Painlevéexpansion and Darboux transformation,respectively.Then nonlocal symmetries can be localized into Lie point symmetries of extended systems by introducing localized variables.On this basis,the（higher-order）symmetry transformations can be further constructed.We can apply the obtained transformations and similarity reduction method to construct higher-order wave solutions and complex interaction solutions for these higher-dimensional nonlinear systems,such as multi-soliton solutions,multi-soliton molecular solutions and the interaction solutions among soliton and different periodic waves and so on.