| Based on symbolic computation,this dissertation studies the symmetry reduction,differential invariants,group foliation and soliton solutions of some important mathematic models in nonlinear science.The main work is classified into four aspects: The symmetry analysis and exact solutions of the 2D unsteady incompressible Boundary-Layer equations are investigated by using the Lie group method;Based on the method of equivariant moving frames,the differential invariants and group foliation of the(2+1)-dimensional breaking soliton equation are performed;A Maple package named as GFSMF is developed to perform group foliation of nonlinear systems with respect to scaling symmetry;N-dark soliton solutions and bright-dark mixed N-soliton solutions of the multi-component Mel'nikov system are investigated by virtue of the KP hierarchy reduction method.The main work and innovations are:Chapter 1 is an introduction of the research background and current status of symmetry reduction,moving frames,KP hierarchy reduction method and symbolic computation.The main results of this dissertation are also presented.In chapter 2,we study the symmetry reduction of a boundary value problem.To find intrinsically different symmetry reductions and inequivalent group invariant solutions of the 2D unsteady incompressible boundary-layer equations,a two-dimensional optimal system is constructed which attributed to the classification of the corresponding Lie subalgebras.Then by virtue of the optimal system obtained,the boundary-layer equations are directly reduced to a system of ordinary differential equations(ODEs)by only one step.It has been shown that not only do we recover many of the known results but also find some new reductions and explicit solutions,which may be previously unknown.In chapter 3,by virtue of the equivariant moving frames,we construct the differential invariants of Lie symmetry pseudo-groups of the(2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra,their syzygies and recurrence relations are classified.In addition,group foliation of the equation is also performed within the theory of equivariant moving frames.The infinite-dimensional part of the admitted symmetry group is utilized to produce a foliation of its entire solution space,so the resolving system inherits only the finite-dimensional part of the symmetry group.Some explicit solutions of the equation are obtained which are closed with respect to the regarding infinite-dimensional subgroup.The procedure of the method is completely symbolic and algorithmic.In addition,a Maple package named as GFSMF is developed.On the Maple platform,we develop a program to automatically perform group foliation with respect to scaling symmetry.The effectiveness of the package is verified by some examples.In chapter 4,based on the KP hierarchy reduction method,the soliton solutions of the multi-component Mel'nikov system are studied in detail.Firstly,a general form of N-dark soliton solution to the two-component Mel'nikov system are presented.The dynamic analysis shows that the collisions of dark-dark solitons are elastic and energies of the solitons in different components completely transmit through.In particular,it may be the first time that we find the moving dark-dark soliton bound state can exist when nonlinearity coefficients are both negative.Secondly,the general bright-dark mixed N-soliton solution of the multi-component Mel'nikov system are also constructed.The formula obtained unifies the all-bright,all-dark and bright-dark mixed N-soliton solutions.For the collision of two solitons,the asymptotic analysis shows that for a M-component Mel'nikov system with M ? 3,inelastic collision takes place,resulting in energy exchange among the short-wave components supporting bright solitons only if the bright solitons appear at least in two short-wave components.In chapter 5,the summary of the whole dissertation is presented,and the prospect of future work is also discussed. |
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