Several Types Of Nonlinear Partial Differential Cheng Shouheng Development Construction Law

With more and more attention being paid to nonlinear phenomena by the scholars in various fields, the problems in other disciplines are converted to the corresponding nonlinear evolution partial differential equations by mathemati-cians for the further study. As a basic concept in physics, the conservation laws plays a vital role in the study of various properties of nonlinear partial differ-ential equations. Due to the one-to-one corresponding relationship between the symmetry and the conservation laws of differential equations, the symmetry has been proved to be a significant tool in constructing conservation laws for the re-lated mathematicians. On this basis,the mathematician also put forward a series of different methods in constructing the conservation laws of nonlinear partial differential equations.Three kinds of methods for constructing the conservation laws of nonlinear evolution partial differential equations are concretely introduced in this paper. Firstly, We let the variational symmetry act on the conservation laws of dif-ferential equations while the characteristic form has been put forward,which is called calculus of variation; Secondly, Noether theorem and the partial Noether method for the perturbed partial differential equations with relatively small parameters are concretely described; Finally, the key point to this paper is to introduce the method of constructing the extended equations in studying the conservation laws of the nonlinear evolution partial differential equations whose related Lagrangian functions can not be constructed.The content of this paper is divided into the following four parts:In the first chapter, we briefly introduce the background of this paper, significance of the research and the importance of conservation laws to the study on nonlinear science.In the second chapter, the related theoretical knowledge and definition re-quired to solve the problems in this paper are concretely described.In the third chapter, three different kinds of methods and main steps in constructing conservation laws are given in detail.In the fourth chapter, the conservation laws of a series of nonlinear evolution partial differential equations are constructed. Firstly, we apply the calculus of variation to construct the conservation laws of Brusselator equations and Nonlinear telegraph equationsSecondly, we apply the partial Noether method to construct the conser-vation laws of (2+1)-dimensional perturbed Boussinesq equation with small parametersFinally, we apply the method of constructing the extended equations to construct the conservation laws of (2+1)-dimensional BBM-Burgers equation which can not be matched with the exact Lagrangian function ut+ux+uy+uux+uuy-auxx-βuxxt-αuyy-βuyyt＝0...