Part Of The Lagrangian Method To Construct A Number Of Physical Models To Approximate Conservation Laws

With the rapid development of science and technology, scientific research core has from the original linear into modern nonlinear direction. Nonlinear phenomenon arise in many fields of science and engineering technology, many nonlinear science issues of research, can use the nonlinear equation of this mathe-matics physics model to succinct and accurate description, and solving equations has been long physicists and mathematicians the key. The notion of conserva-tion laws plays an important role in the solution process. How to determine conservation laws of the differential equation of system is often the first step towards finding the solution. Conservation laws plays a vital role in the study of differential equation and have many significant uses in the study of the analysis of stability and global behavior of solutions and in the integrality, linearization, and in developing numerical methods for DEs arise in nonlinear science. There-fore, how to decide partial differential equations of the conservation laws became our main research subject.A systematic for the determination of conservation laws associated with variational symmetries for systems of Euler-Lagrange-type equations is given by the well known Noether theorem. There are some differential equations which do not possess Lagrangians e.g. parabolic partial differential equations, scalar evolution equations etc. The notion of a partial Lagrangian was introduced and the formula to construct partial Noether operators associated with a partial La-grangian was discovered. These partial Noether operators are not symmetries of the equations under consideration but are very useful in constructing conser-vation laws. In engineering science, mathematical physics and the mechanics of many physical models, a number of differential equations depend on a small pa- rameter, which are usually called perturbed differential equations. Perturbation theory was consequently developed and plays an essential role in nonlinear sci-ence,especially in finding approximate analytical solutions to perturbed partial differential equations. The definition of approximate conservation laws of the partial differential equations was introduced by Baikov and Ibragimov on the basis of the notion of conservation law. Kara extended the partial Noether ap-proach of constructing conservation laws to which of construction of approximate conservation laws for perturbed equations.The long wave in shallow water wave phenomena generated extension can be used with a convection term perturbed Boussinesq equation model to describe. Kuramoto-Sovashonsky equation is one of the most prominent and generic equa-tions that arise in nonequilibrium systerms, such as hydrodynamics and moving interfaces. A number of special cases of the equation has been used as math-ematical models in physically significant nonlinear problems in mathematical physics, nonlinear dynamics and plasma physics. In the following KdV equa-tion with weak damping represented including the the study of real phenomena such as the coastal waves in ocean, liquid drops and bubbles in the context of atmospheric blocking phenomenon particularly in the aspects of dipole blocking. Our main work is to construct approximate conservation laws for perturbed par-tial differential equations, mainly utilizing the partial Lagrangian approach to construct approximate Noether symmetry operators, we obtain complete clas-sifications of approximate conservation laws and conserved vector for different forms of perturbed equations.The layout of the paper is as follows. The first chapter is a brief review of the history of researches on the conservation laws and approximate conser-vation laws of the background and significance of the subject relating to the topics of this article, we also outline the main results of this paper. In Sec. 2, we present the theory necessarily relating to approximate conservation laws. In Sec.3,4,5, respectively, we introduce the background of each model, and construct the approximate conservation laws and conserved vectors for some per-turbed PDEs in terms of our new definitions of partial Lagrangian and partial Euler-Lagrange-type equation. These equations include the perturbed Boussi-nesq equation, the generalized perturbed Kuramoto-Sovashonsky equation and the K(n,1)-type KdV equation with weak damping. We obtain complete classi-fications of their approximate conservation laws.