Research On Mechanization Algorithm For Constructing Higher Order Wave Solutions Of Nonlinear Evolution Equations

Nonlinear evolution equation(NLEE)is a type of significant mathematical model to describe numerous complicated phenomena in nature.For a long time,the research in this direction plays an important role in many subject domains,such as mathematics,physics and so on.With the rapid development of computer science and the continuous improvement of computer algebra system,the symbolic computation has become a powerful tool and means to solve exact solutions of NLEEs.Under the guidance of mathematical mechanization,this paper studies the mechanization algorithm for constructing higher order wave solutions of NLEEs based on symbolic computation and computer algebra system Maple.Our research work mainly includes the following two aspects:Firstly,by redefining the balance constraints,more abundant finite series solutions of nonlinear evolution equations are obtained.The birth of Wu's elimination method provides a very effective tool and means for solving nonlinear algebraic equations.Especially with the mechanized realization of this method,there is an upsurge in the study of direct algebraic methods for constructing exact solutions of NLEEs.The most typical methods are tanh function method and its various generalizations and transformations.The orders of solutions are determined by balancing the highest nonlinear terms and the highest derivative terms.However,if there are more than one the highest orders in the balance equation,by the traditional way we usually ignore the repeated orders,which is bound to miss some homogeneous balance conditions and some possible finite series solutions.In this paper,the balance constraints are redefined,the original order balance condition is extended,and three kinds of balance points are proposed.Because of the derivation,the orders of the solutions will not only appear in the exponent of the highest order terms in the equation,but also in the coefficients of these terms.In this paper,the exponent and coefficients of the highest order terms are considered comprehensively.By considering more possibilities of homogeneous balance conditions,new orders of solutions are determined to obtain a series of new higher order solutions for the required equations.By applying our newly defined homogeneous balance conditions and algorithm to a series of sub-equation methods,we obtained different types of higher order finite series solutions for the considered nonlinear evolution equations.Secondly,with the direct algebraic method,many specific types of higher order wave solutions and interaction solutions between different waves for NLEEs are constructed.The direct algebraic method is widely applied to solve NLEEs.By assuming the required solutions as specific forms,the direct algebraic method simplifies the problem of solving NLEEs into the problem of solving nonlinear algebraic equations.The main difficulty in the process of this method is that the scale of the obtained nonlinear algebraic equations is usually very large,and it is usually impossible to solve them directly by Maple in limited time and limited memory.Combining the group parallel computing and inheritance solving strategies with the direct algebraic method,we can effectively reduce the scale of the obtained nonlinear algebraic equations,and then solve it efficiently,so as to obtain higher order or more complex interaction solutions.On this basis,this paper develops a software package DAM based on software platform Maple,which can derive the specific types of exact solutions for NLEEs.The software provides several friendly interfaces for calculation and plotting.In order to illustrate the effectiveness of DAM,this paper applies it to solve several classical NLEEs,and successfully constructs different kinds of higher order wave solutions and their interaction solutions.