Study On Group Classification, Conservation Laws And Exact Solutions Of Second Order Nonlinear Evolution Equations With Variable Coefficients

It is focused on differential equations as a major tool for mathematical modelling on natural and social problems. Solutions of differential equations always help us to look into the considering problem. Most of mathematical models in physics, engineering sciences, biomathematics, etc. lead to nonlinear differential equations. Therefore, to find solutions of differential equations, especially nonlinear differential equations, in a sense, is one of the most important ways to study the nature and society.The French genius mathematicians Galois realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial; similarly, Sophus Lie observed that those special methods, which designed to solve certain particular, seemingly unrelated types of equation, such as separable, homogeneous, or exact equations, were, in fact, all special cases of a general integration procedure based on the invariance of the differential equation under a continuous group of symmetries, the Lie groupUsing the Lie group method, one can obtain the symmetry of the system of differential equations; then the knowledge of the symmetry group of differential equations allow one to reduce the order of the differential equations. This method can deal with not only the differential equations with constant coefficient but with variable coefficients.The key tool of the Lie group method is the infinitesimal generator and its connecting symmetry. In1918, E.Noether constructed a one-to-one correspondence between a nontriv-ial generalized variational symmetry of some functionals and a nontrivial conservation law. In2007, N.H.Ibragimov proved that any Lie point, Lie-Backlund, and nonlocal symmetry of any system of differential equations, provided that the number of equations is equal to the number of dependent variables, provides a conservation law. However, his form of con-servation law involves certain of "non-physical" variables. To obtain the local conservation laws, one need considering the self-adjointness of the original equation. If the equation has a property of self-adjointness, one can get local conservation laws by removing the "non- physical" variables. Otherwise, the local conservation law does not exist. However, the nonlocal conservation law still reflects the symmetry property of the equation.By using the symmetry obtained, one can reduce the order of the system of differential equations. The reduction of order can help us solve the differential equations. Sometimes, one can obtain some particular solutions of a system of differential equations directly by nontrivial conservation laws.This work is devoted to discussing three kinds of scalar partial differential equations with variable coefficient. The contents are as follows.In Chapter1, firstly, the historical background, some basic concepts, and corresponding research are summarized. Then the general relation between symmetry and conservation law is presented. Finally, the main results of this work are introduced briefly.Chapter2is devoted to introducing the method of constructing conservation law of the system of differential equation through its symmetry provided by N.H.Ibragimov and another methods to solve the particular partial differential equations.In Chapter3, we find the Lie point symmetries of a class of second-order nonlinear diffusion-convection-reaction equations containing an un-specified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlin-early self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinear-ly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves. Finally, based on the method of conservation laws for constructing solutions to system of partial differential equation, with the aid of three local nontrivial conserva-tion laws obtained, we find some particular solutions to some certain kinds of the system considered.In Chapter4, a space dependent reaction-diffusion equation, i.e. a second-order reaction-diffusion equation with a variable coefficient b(x) is considered. Firstly, we find the Lie point symmetries of the equation and meantime classify the model into three kinds. Then, since the equation does not have any self-adjointness, we establish conservation laws of the si-multaneous system-the original equation and its adjoint equation-corresponding to every Lie symmetry obtained. For the original equation, these are nonlocal conservation laws. Also, we proved that the local conservation law of the original equation does not exist. In addition, some exact solutions are constructed.In Chapter5, another variable-coefficient reaction-diffusion equation is studied. We classify it into three kinds by different restraints imposed on the variable coefficient b(x) in the process of solving the determining equations of Lie groups. Then, for each kind, the conservation laws corresponding to the symmetries obtained are considered. Finally, some exact solutions are constructed.In Chapter6, the summary of this dissertation and the prospect of future study are given.