The Symmetries,Conservation Laws And Solutions To Several Nonlinear Systems

By using symmetry theory,this dissertation studies the nonlocal sym-metry,conditional Lie-Backlund symmetry and approximate conditional Lie-Backlund symmetry of several nonlinear systems.Meanwhile,it,with the adjoint equation and related theories concerning conservation laws,also focus-es on the conservation laws of the several nonlinear systems.The dissertation is mainly arranged as follows:Chapter 1 serves as an introduction to the research background,the cur-rent situation of symmetry theory,conservation laws,approximate symmetry,and also the main work of this dissertation.In chapter 2,nonlocal symmetries and interaction solution of several non-linear systems axe studied.Firstly,the application of truncated Painleve ex?pansion method,we derive nonlocal residual symmetries of(2+1)-dimensional dispersive long-wave(DLW)system and higer-order Broer-Kaup(HBK)sys-tem,and the introducing of appropriate auxiliary dependent variable localizes the nonlocal symmetries to Lie symmetry in a new closed system;Second-ly,the consistent Riccati expansion(CRE)solvability and consistent tanh ex-pansion(CTE)solvability of the following systens,such as(2+1)-dimensional DLW system,HBK system and modified dispersive water-wave system,are proved.In addition,the corresponding images of the above solutions are given for good understanding their properties.In chapter 3,the Lie symmetry analysis of modified Boussinesq system,HBK system,MDWW system and(2+1)-dimensional DLW is discussed,the optimal system for modified Boussinesq system and HBK system are given,and the nonlinear self-adjointness of the above systems is proved,which can be used to convert the adjoint system to equivalent form to original system.Furthermore,the conservation laws of the above systems are investigated with Lie point symmetries and Ibragimov theorem.In chapter 4,the conditional Lie-Backlund symmetry(CLBS)approach is applied to study the Reaction-Diffusion system.It is shown that the system of equation admit a class of invariant subspaces,which is equivalent to a kind of higher-order conditional Lie-Backlund symmetries of the system.As a result,generalized separable variable solutions defined on the polynomial,trigono-metric and exponential invariant subspace are constructed for some resulting equation.Chapter 5,expanding the conditional Lie-Backlund symmetry(CLBS)and invariant subspace approach to the perturbed equation and proposing the concept of perturbed invariant subspace(PIS),studies the approximate generalized functional variable separation solution for the nonlinear diffusion-convection equations with weak source by the approximate generalized con-ditional symmetries(AGCSs)related to the PISs.Therefore,complete classi-fication of the perturbed equations which admit the approximate generalized conditional symmetries(AGCSs)is obtained.As a consequence,some AGFSSs to the resulting equations are explicitly constructed by way of examples.Chapter 6 is the summary and discussion of this dissertation,and the outlook of future work.