| The nonlinear science is substantially studied and widely applied in natural sciences such as biology,chemistry,communication and almost all the physical branches like condensed matters,field theory,low temperature physics,hydrodynamics,plasma physics, optics and so on,where a large amount of nonlinear systems emerge.Therefore,a lot of questions are naturally asked:How to solve these nonlinear partial differential equations which describe the nonlinear systems? What kind of characters do the solutions of these nonlinear systems possess? How to symmetry reduction and solve for a coupled nonlinear system? How to construct B(a|¨)cklund transformation for a partial differential equation?Through many efforts of scientists,many methods have been established and developed to solve the nonlinear systems especially those integrable ones.For instance, the Inverse scattering transformation,Darboux transformation,B(a|¨)cklund transformation, Functional variable separation approach,Bilinear method and Multilinear method, Classical and non-classical Lie group approaches,Clarkson-Kruskal's direct method,Deformation mapping method,Truncated Painlevéexpansion,Function expansion method and so on.As the development of soliton theory,some actual problems cannot be simplified to either one-dimention or mono-equation,like high-dimention non-linear system should be considered when studying on monopole in particle physics,or coupling NLS equation group should be concerned when dealing with interaction between coupling optical fiber, or coupling KdV equation group in two-layer fluid model when describing both atmosphere phenomenon and sea phenomenon,etc.Otherwise,its important characteristics will lost. Therefore,study on from mono-dimention to muti-dimention,from mono-solition equation to coupling equation group is a hot object of nowaday's international research.This dissertation is carried out around the nonlinear coupled systems. Symmetry method is one of the most fundamental methods in natural science.In the study of the integrable models,symmetries play an important role because there exist infinitely many symmetries and conservation laws.Through many efforts of mathematicians and physicists,a lot of powerful approaches have been established in the continuous integrable systems.In the traditional studies of the symmetry group of a given nonlinear PDE,one usually restrict himself to find the Lie point symmetry group.In the standard Lie group theory, it is in principle enough to study its infinitesimal form,Lie algebra,because the related Lie group can be uniquely obtained by solving an initial problem related to an ordinary differential equation(ODE) system.However,for a given nonlinear PDE,there are still some serious problems.For instance,(ⅰ) once the Lie algebra is obtained,it is still very difficult to solve the related initial problem to give out the finite transformations,the symmetry group.(ⅱ) In many cases,even if the initial problem can be obtained,the final expressions are very complicated and it is not convenient in the real applications.(ⅲ) In some other cases,the general symmetry groups of the nonlinear systems are not Lie groups at all and there may be some types of more general continuous groups.Obviously, it is very difficult or even impossible to solve all these problem for any nonlinear systems.As is well known,to solve nonlinear systems especially to obtain reduction solutions, there are three fundamental methods,CK's direct method,and classical/nonclassical Lie group approach.The former solves PDEs algebraically and the latter is based on group theory.It is believed that traditional reduction methods are quite perfect and some standard types of reduction solutions are exhausted.So,one has to find some novel ideas to get some new types of symmetry reductions.In chapter 2,many methods have been briefly narrated to solve the nonlinear systems, and basis concept have been introduced for applications of Lip groups to differential equations.In chapter 3,a kind nonlinear systems which has the actuai applied physics background have been introduced.we extend the classical Lie group approaches to coupled nonlinear partial differential system,and then apple it to coupled KdV nonlinear partial differential system,we have obtained symmetry groups,Lie algebras of local Lie groups group-invariant solutions and so on.The especial symmetry be obtained to a classical coupled KdV nonlinear partial differential system by use the classical Lie group approaches,and introducing some potentials for a coupled KdV equation,then we obtain Lie-B(a|¨)cklund transformation for the coupled system.Using the classical Lie group approaches to find the symmetry reductions of nonlinear evolution equations is equal to Lie point symmetry method for the reduction of equation.In chapter 4,we extend the nonclassical Lie group approaches to coupled nonhnear partial differential system,and then apple it to coupled KdV nonlinear partial differential system by strengthening some constraints,that will lead to new results.In chapter 5,we discuss the the modified Clarkson and Kruskal's direct method,and the modified CK direct method is used to reduce the coupled nonlinear partial differential system.The result tells us that the results obtained by the CK direct method contain those obtained by the classical Lie group approach and the results of the nonclassical Lie approach include those of the direct method,and the diect method can have a corresponding group explanation.In chapter 6, abundant exact solutions are obtained such as the bell shaped soliton,the compacton,the peakon,Double solitary wave solution with peakon,singular solitary wave solution,the exponential solution,the periodic travelling wave solution and so on.it is proved that the strong dispersive DGH equation is integrable under the meaning that it possesses Bi-Hamiltonian structure and infinitely many symmetries.The singularity analysis of the strong dispersive DGH equation is performed by using the WTC method.By homogenous balance(HB) method,B(a|¨)cklund transformation for the strong dispersive DGH equation is established.The result tells us that using the truncated Painlevéexpansion method to find the B(a|¨)cklund transformation of nonlinear evolution equations is equal to by homogenous balance(HB) method.The symmetry reduction of this equation is deduced by the HB method.The result tells us that using the homogeneous balance method to find the symmetry reductions of nonlinear evolution equations is equal to CK direct method for solving similarity solutions. This dissertation ends with summary and research prospects.The innovations and features of this dissertation are as follows:(1)New integrability model:we mostly study some new types of coupled KdV equation systems with some arbitrary parameters from a two-layer fluid model.They can describe many physical problem for multiple-layered fluid system,such as the atmospheric blockings, the interactions between the atmosphere and ocean,the oceanic circulations and even hurricanes or typhoons.(2)New ideas:Put forward a new idea of using modified direct method,we have modified the suppositional form of the direct method.It is proposed that one original field can be related to two reduced fields while in the old approach,one original field can only be related to one reduced field.(3)New method:We are enlightened to modify the constraints for the traditional direct method,namely,require parts of the coefficients have one common ratio instead of all and the ratios among different parts should be only a function with respect to space time variables.(4) After introducing some potentials for a coupled KdV equation which is derived from the two layer fluid model,the nonlocal Lie B(a|¨)cklund transformation is obtained. Using the Lie B(a|¨)klund transformation theorem to the trivial zero solution,the single soliton solution is also found.(5)New results:Get conditional similarity reduction solutions of coupled nonlinear systems and give out their corresponding group explanations;obtain the new soliton of the strong dispersive DGH equation.
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