The Conservation Laws Of Partial Differential Equations And Related Variational Approaches

With the rapid development of nonlinear science,a large number of nonlinear partial differential equations(PDEs)have emerged in the fields of physics,mechanics,economics,and engineering etc.As well known,solving nonlinear PDEs is important research significance in theory with practice.In most situations,the solving for non-linear PDEs can only rely on numerical methods,but the numerical solutions have many limitations.Therefore,various rational analysis of solving analytical solutions and it's related properties for PDEs show the important theoretical significance and application value.There is a close relationship between PDEs and various variational methods.In particular,He-variational method,variational iteration method,and variational deriva-tive method can play a positive effects in related the relevant properties of PDEs.The He-variational method is a simple and effective method to solving PDEs.And it based on the traveling wave transform,semi-inverse method techniques and variational func-tional tactics,then constructs a solitary wave solution by substituting a assuming solution into the variational formula and observing its stationary point.Variational iteration method can effectively solve a variety of linear,nonlinear,and initial bound-ary value conditions problems,and it is an effective method to gradually improve the accuracy of approximate solutions.This method obtained the approximate solution-s or exact solutions of the given equations by iteration formula with the halp of a correction functional and Lagrangian multipliers.The variational derivative method is a direct method to solve many problems of PDEs.It mainly uses the variational derivative(Euler operator)to action on the corresponding differential operators,and then derives the multipliers and conservation Integrals,Lagrangian functions and other important conclusions.Symmetries and conservation laws are two important attributes of PDEs.Symme-tries response the structure rule of nonlinear PDEs,and conservation laws reflect the motion characteristics of nonlinear PDEs.The conservation laws have the important applications in the development and research for PDEs integrability,properties of so-lutions and numerical solutions of PDEs.Therefore,construction of conservation laws of equations is very important.In this paper,the variational derivative method and the symmetry/adjoint symmetry pair method are introduced.The variational deriva-tive method determines the multiplier sets based on the variational derivative(Euler operator),and then use the multiplier strategy to obtain the conservation laws of the given equations.The symmetry/adjoint symmetry pair method derives the symmetry characteristic forms and adjoint symmetries of PDEs via Frechet derivatives and ad-joint Frechet derivatives,and then use a bilinear skew symmetric identity to generate the conservation laws of the given PDEs.From the above,we give play to the techniques of variational method and ideas of conservation laws,obtained related properties of nonlinear PDEs are expected.The specific research arrangement for this paper is as follows:In Chapter 1,we briefly introduce the solitons variational-like methods,Lie sym-metry and conservation laws in the field of PDEs and the overviews of this paper.In Chapter 2,we introduced the main ideas and algorithm frameworks of He-variational method,variational iteration method,variational derivative method and the symmetry/adjoint symmetry pair method.In Chapter 3,we use the above methods that described in Chapter 2 and the computer algebra system(Maple,Mathematica)to solve solutions and conservation laws of several important PDEs.(1)The solitary wave solutions of the cubic nonlinear Schodinger equation and Dave-Stewartson equations are constructed by using the variational method,then the optical soliton solutions of the generalized Zakaharov equations are also obtained.(2)The traveling wave solutions of the Whitham-Broer-Kaup equations and the mKdV equation are numerically simulated by using the variational iteration method;(3)The infinite multipliers and infinite conservation laws of the nonlinear Com-pacton ZK equation are derived based on the Wu's method and the variational deriva-tive method.(4)The conservation laws of the telegraph system and the dispersive long wave equations are constructed using the symmetry/adjoint symmetry pair method with the help of the Frechet derivatives and it's adjoint Frechet.In Chapter 4,a brief summary of the whole work is given,and provided the extended research direction and work of future.