Study On Darboux Transformation And Analytic Solutions Of Nonlinear Evolution Equations With Symbolic Computation

Nonlinear evolution equations play an essential role in meteorology,physics and even engineering,and are also the key research problem in the field of nonlinear science.Using nonlinear evolution equations to get a mathematical model is an important means of understanding and characterizing complex physical phenomena.Studying the analytic solution of nonlinear evolution equations based on the symbolic computation can help people gain insights of the internal structure of the system and the relationship between different quantities so as to effectively broaden the application scope of the nonlinear evolution equation.Solving nonlinear evolution equations generally involve massive complex calculations and derivations,which poses a huge challenge to traditional research manners.With the rapid development of computer software technologies,the emergence and vigorous development of various high-performance symbolic computing softwares has greatly improved people's ability and research level in dealing with complex and tedious symbolic calculations,and also facilitated and promoted the development of nonlinear science.Based on symbolic computing software Maple,this paper mainly focuses on the construction algorithms and the mechanization regarding the Dardoux transformation and analytic solutions of the nonlinear evolution equations,specially including the following two parts:In the first part,based on the Dardoux transformation theory,the Dardoux transformation and the various types of analytic solutions with respect to several nonlinear evolution equations are constructed with the help of Maple.Based on the classical Dardoux transformation,the N-th Dardoux transformation and various types of analytic solutions of(2+1)-dimensional nonlocal NLS equations were studied.The computational difficulty in the calculation of higher order classical Dardoux transformation rapidly increases with the increasement of the number of iterations,while the generalized Dardoux transformation is able to overcome this limitation.By virtue of the generalized Dardoux transformation,the N-th generalized Dardoux transformation and different types of analytic solutions of the FNLS equation are constructed.However,there may be no Darboux transformation in the differential from for some nonlinear evolution equations.Therefore,the binary Darboux transformation is born,in which integral calculation is involved.We further study the binary Dardoux transformation for nonlocal DS II equation,and obtain its breather,lump solutions and so on.In the second part,based on the simple Hirota method,the long limit method and the undetermined coefficient method,we study mechanization algorithms to construct higher-order lump and the interactive wave solutions of high-dimensional nonlinear evolution equations,and the corresponding software LumpSolver was developed in the Maple platform.Recent years,the construction of lump solutions,rogue wave solutions and interactive wave solutions is one of the hot topics in the research of nonlinear mathematical and physical equations.From the relevant research results in recent years,there are mainly two approaches for constructing lump solutions of nonlinear evolution equations,namely the direct algebra method and the long limit method.The idea of direct algebra method is simple,but the solving of nonlinear algebra equations generated in the solving procedure appears to be a computational bottleneck.In addition,it is difficult to construct higher-order lump solutions.The long limit method is based on the soliton solutions of nonlinear evolution equations to construct lump solutions.In this paper,the long limit method is applied to construct higher-order lump solutions of nonlinear evolution equations.Through careful analysis,we learnt that the famous N-soliton solution formula is only valid for integrable equations,and generally does not work for non-integrable equations.In this paper,the N-soliton solution formula is extended to non-integrable equations.Through repeated analysis and testing,a class of constraints for the N-soliton solution formula of non-integrable equations is given.Based on the foregoing work,the mechanized algorithm to construct lump and interactive wave solutions of the nonlinear evolution equations is developed,and the corresponding software LumpSolver is developed,which provides an effective tool for solving nonlinear evolution equations.