Conservation Laws Of Nonlinear Evolution Equations, A Number Of Constructor Method And Its Applications

Conservation regards the nature moves in cycles and circulation infinitely. In essence,conservation originates from symmetry.It also makes human the understanding of symmetry and conservation deepen from external into the interior. Symmetry and conservation also experience a long developing process from seperation to integration.In the study of differential equations,conservation laws have many significant uses,particularly with regard to integrability and linearization, constant of motion,analysis of solutions,and numerical solution methods.Consequently, an important problem is how to calculate all of the conservation laws for given differential equations.As is known that Noether theorem plays a vital role in the constructing conservation laws of differential equations,profoundly revealed the connection between symmetries and conserved vectors of Mechanic system.Therefore Noether theorem has been a classical approach for conservation laws.However,the restriction to Euler-Lagrange equation reduces application of Noether theorem significantly.For example,Noether theorem is not applicable to evolution equations,to differential equations of an odd order,etc. In order to overcome these problems,various generalization of Noether theorem have been discussed,such as,characteristic method,variational method,direct construction method,partial Noether approach,and adjoint equation approach and so on.This paper is organized as follows:In section 1,we introduce the elementary knowledge of conservation laws and notations involved in the paper.In section 2,we show nine methods of conservation laws construction.In section 3,we discuss the conserved vector of several differential equations——nonlinear Schr(o|¨)dinger equation,Fujimoto-Watanabe equation,Maxwellian model equa- tion,Black-Scholes equation,fifth-order diffusion equation,and so on.Finally, we construct new conservation laws for those equations via different methods. The result which we obtained can give more useful information to the corresponding equations.The last part is the conclusions and the study in the future.