.1 +2- Dimensional Nonlinear Evolution Equation Of One-dimensional Optimal System And Its Symmetry Reduction

ABSTRACTMaster degree dissertationOne-parameter Optimal Systems for the 1+2-dimensional NonlinearEvolution Equations and Symmetry Reduction(Department of Mathematics. Northwest University China,)Liu RuochenDirected by Qu ChangzhengThe symmetry algebras of the 1+2-dimensional nonlinear evolution equations are systemically studied in this paper. The equations are invariant under the sevendimensional Lie point symmetry. The details of the construction for an one-parameter optimal systems of the symmetry algebras are presented with the method developed by Qlver. We develop his ideas by finding some algebraic invariants under the adjoint actions of the group in addition to the Killing form and give a rigorous proof of the optimality of the one-parameter systems for the symmetry group of the given equations. To each element in this one-parameter optimal system, we can obtain the symmetry reduction and some invariant solution of the equations.Firstly, we calculate the Lie point symmetry of 1+2-dimensional nonlinear evolution equation:Which arises from the motion of plane curves in affine geometry. In chapter three, we will establish the one-parameter optimal systems of its symmetry algebras. There are twenty-one elements in this optimal systems, to each element in the system, we will reduce the four-order nonlinear evolution equation to ordinary differential equation (ODE) and get some invariant solutions. In chapter five, we consider the nonlinear evolution equation:Which can be transformed to Sawada-equation. By a straight calculation, we find that the five-order equation admit the similar Lie point symmetry to the above four-order nonlinear evolution equation. therefore we can use the results in chapter three and chapter four to reduce this five-order evolution equation and make it become ODE.