Construction Algorithm Of Polynomial Conservation Laws Of Nonlinear Evolution Equations And Realization

Some nonlinear evolution equations having strong physical background are worthy to be studied by us. Nonlinear partial differential equations (PDEs) arising in many disciplines of applied science admmit conservation laws. In modern nonlinear science, the study of the exact solution of nonlinear equations and the integrability property are two important research topics for a vast number of researchers. Seeking for the infinite many of conservation laws for PDEs is one of the most effective means to check whether a non-linear differential equation is integrable or not. The study on the conservation laws (such as energy conservation, quality conservation) has always been a central topic in the research field of physics. In this dissertation, by learning the integrable theory of the differential equations from the conservation laws view of point and drawing on the theories and ideas of scientists and scholars, we make an improvement on the construction algorithm of conservation laws. Then, taking nonlinear evolution equations as the object of study and with the help of symbolic computation system Maple which is a powerful tool, an automated Maple software package which is used for constructing polynomial conservation laws of nonlinear evolution equations is developed. In this dissertation, we study from the following parts:First, we introduce the important meaning of conservation laws, the research status about conservation integrable theory both from domestic and abroad, and then its future trends.In Chapter 2, we mainly introduce the theory of conservation laws and some methods of constructing conservation laws, which can be obtained in the previous study.In Chapter 3, by the description of construction algorithm of conservation laws, we explain the basic principles of the construction algorithm of polynomial conservation laws for the nonlinear evolution equations, which makes one know how to construct the conservation laws for a certain nonlinear evolution equation. With the principle of the construction algorithm of conservation laws, we improve the existing construction algorithms of conservation laws, and with the help of symbolic computation system Maple, we develop a software package for constructing the conservation laws automatically.In Chapter 4, we describe the automated software package POCLAW, and load POCLAW in the platform of Maple to obtain polynomial conservation laws for a number of significant non-linear models. Then we analysis the results of the property of conservation integrbaility and give the exact forms of conservation laws. Importantly, for parameterized equations, we aim to obtain the constraint conditions of integrability, and then take these parameters constraints as starting point to search for new integrable systems.Chapter 5 summarizes the full text.