The Conservation Laws And Exact Solutions Of Partial Differential Equations

With the rapid development of sciences and technologies,a large number of nonlinear partial differential equations(PDEs)have appeared in many fields,such as physics,(elasticity,solid,fluid)mechanics,engineering,meteorology,economics and sociology,etc.,and many natural phenomena and engineering problems can be described by PDEs,thus,study of nonlinear PDEs becomes one of the mainstream of the nonlinear science.More than one hundred years,the researchers were studying mainly the structural characteristics,motion laws and stability of solutions for PDEs,which has laid a good theoretical and applied foundations.Symmetries responses the peculiarities of the structure for PDEs;Conservation laws reflect the characteristics of motion changes of PDEs;All kinds of solutions depict the physical properties of PDEs.Symmetry group theory and method is a powerful tool to analyze the problems of PDEs,and it has many important functions,such as reductions,linearization,solving numerical solutions,constructing conservation laws,analyzing stabilities for nonlinear PDEs,and so on.Conservation law of PDEs is a extension of the number of basic physical conservation variables,such as mass,energy and momentum.It provides effective conditions and methods for the integrabilities,linearization,stabilities of the numerical solutions,existences of global solutions,asymptotic behaviors and nonlocalities of nonlinear PDEs.Therefore,with the combinations of symmetries and conservation laws as a breakthrough point,the study of nonlinear PDEs is important significance.From literatures,the comparative study between of conservation laws methods for nonlinear PDEs,breaking of the complex calculations of symmetries of some lower order PDEs problems and studying of n coupled nonlinear integrable systems have been lagged relatively,and fusions of researching these problems is to be expected.To these,we with the help of symbolic computation(Maple or Mathematica)technologies and symmetry theories,we have attempted to use the symmetry/adjoint symmetry pair method and the Ibragimov's new conservation theorem to some nonlinear PDEs,then the internal relations between two methods have revealed by through comparison.We have brooked the bottleneck of the overdetermining equations for symmetries of some PDEs by using the strategy of power series for infinitesimals,then its corresponding symmetries,subalgebra optimal systems and similarity reductions have presented.We have proposed the constructing mechanism of n coupled nonlinear integrable system,and then conservation laws and soliton solutions of 2—coupled Kd V system have introduced.In conclusions,it is expected that the related properties of nonlinear PDEs can be deduced under framework for symmetries and conservation laws of PDEs.The work of this paper is arranged as following:In Chapter 1:The research backgrounds and current situations of Lie symmetry,conservation laws,power series strategy,similarity reductions and exact solutions are briefly introduced.In Chapter 2:Firstly,the basic idea of symmetry/adjoint symmetry pair method and ibragimov's new conservation theorem are introduced.Then the conservation laws of Chaplygin gas equations and dispersive long wave equations are constructed by these two methods.Finally,we have compared above two methods using the obtained conservation laws.In Chapter 3:The basic idea of power series form method for solving Lie symmetry of PDEs is introduced.Then the symmetries of shallow water wave equations have been constructed by using this method.Finally,the sub Lie algebra and its optimal system,similarity reductions and similarity solutions are carried out for shallow water wave equations.In Chapter 4:We introduced the basic idea of n coupled integrable system was obtained from the nonlinear PDEs.Then 2 coupled Kd V system and its conservation laws and the soliton solutions were discussed.In Chapter 5: We briefly summarized the research work of this article,and some extend research directions and future works were proposed.