Study On Nonlocal Symmetry And Interaction Solution Of Nonlinear Differential System With Symbolic Computation

Nonlinear systems occupy an important role in many research fields,and their significance is not limited to the field of mathematics,but also fully reflected in the engineering applications of modern science.Nonlinear differential systems are used to describe various phenomena in nature,which are important mathematical models for studying physical phenomena.The research on symmetries and exact solutions of non-linear differential systems is helpful to understand the movement of matter and plays an important role in scientific interpretation of the corresponding physical phenomena and engineering applications.In the study of nonlinear differential systems,the analysis of symmetries and the construction of exact solutions often involve tedious symbolic rea-soning and calculations,which are often unmanageable in practice.In recent years,the rapid developments and wide applications of high performance computers have greatly accelerated the symbolic computation of nonlinear systems.Symbolic computation has become an extremely powerful tool for solving nonlinear problems.This dissertation mainly studies symmetries and interaction solutions of nonlinear differential systems with the aid of symbolic computation.Our work consists of the following two parts.The first part is devoted to studying nonlocal symmetries and constructing exact solutions of nonlinear differential systems with the aid of symmetry transformations.The infinitesimal variables of nonlocal symmetries depend on global behavior,which involve the integral forms of the dependent variables.The nonlocal symmetries of non-linear differential systems can be constructed by truncated Painlevéexpansion,potential function,pseudopotential function,Darboux transformation,B?cklund transformation and recursive operator,etc.Different from the classical Lie point symmetry,in general,the prolongation does not close neither for local nor for nonlocal variables and the sym-metry transformations cannot be calculated.Therefore,it is necessary to introduce new auxiliary variables to localize the nonlocal symmetries to Lie point symmetries of the prolonged systems.Nonlocal symmetries enrich the applications of symmetry methods in nonlinear systems and promote the developments of nonlinear science.It was considered that it is difficult to construct N^thfinite symmetry transforma-tions of nonlinear differential systems from symmetries.Inspired by the work[1,2]and from different forms of nonlocal symmetries,we construct N^thsymmetry transfor-mations for nonlinear differential systems.Based on N^thsymmetry transformations,N-soliton solutions of nonlinear differential systems can be obtained.We focus on how to find nonlocal symmetries and localization processes of given nonlinear differential systems.Firstly,the nonlocal residual symmetries are obtained through the truncated Painlevéexpansion and based on Lax pair and conservation law,the nonlocal potential symmetries and nonlocal pseudopotential symmetries are constructed.Secondly,some necessary auxiliary variables are introduced to localize nonlocal symmetries,which are equivalent to Lie point symmetries of closed prolongation.Due to the arbitrariness of the spectral parameter?appeared in Schwarzian equation,Lax pair and conserva-tion law,we have infinitely many nonlocal symmetries,then extend the once symmetry transformation to N^thsymmetry transformations,which can be used to construct exact solutions of nonlinear systems.The second part focuses on the applications of symbolic computation in consistent Riccati expansion method.The usual Riccati equation method of constructing exact solutions might lose some essential information of the original nonlinear differential system,and consequently,only some quite special solutions can be obtained.Consis-tent Riccati expansion generalized the Riccati expansion method to find not only var-ious interaction solutions between different types of excitations but also possible new integrable systems.The consistent Riccati expansion method converts the problem of finding inter-action solutions for nonlinear differential systems to the problem of solving nonlinear algebraic equations,which involves tedious calculations.Based on Wu's character-istic sets method and consistent Riccati expansion,a Maple package,named CRE,is developed for constructing interaction solutions of nonlinear differential system.The package CRE has a wide range of applications,and there is no limit to dimensions of nonlinear differential systems.The CRE can automatically calculate interaction so-lutions of nonlinear differential systems and output the parameters constraints.Some examples are given to illustrate practicality and effectiveness of the algorithm.