Difference Equation Invariant Subspace

Nonlinear partial differential equations, describing the mathematics model, appear in physics, chemical, biology, information science, space science, geogra-phy science and environmental science. The method to solve the equations and solution property are important parts of nonlinear science. There are many key theoretical research and application research directions with regard to nonlinear partial differential equations, one of which is reduction to obtain exact solutions. At present, there are a number of methods to find exact solutions of equations, such as homogeneous balance method, the trail function method, Sine-cosine method, hyperbolic tangent expansion method, Jacobi elliptic function expan-sion method, the nonlinear transform method, invariant subspace method etc.This paper solves the short wave equations, and compares the polynomial subspace and trigonometric subspace of operators by invariant subspace method of difference equations. The first chapter introduces the research background and invariant subspace method of continuous equations. In the second chapter, we give the basic theory of discrete equations. The first part of the third chapter solves the short wave equation obtain new exact solutions, and verifies the following conclusion:for polynomial subspaces, if all the degrees of components in invariant subspace admitted by an operator are smaller than three, the dynamical system obtained by reducing the continuous equations is identical to the one by reducing the difference equa-tion, and the solutions are the same; if the highest degree is three or more, the conclusion doesn't hold. The second part discusses the trigonometric subspace, uses them to study the quasilinear evolution equation and system verifies the conclusion:if differential operator admits a trigonometric subspace, the dynamical system under the corresponding difference equation is differential from the one under the continuous equation, which has nothing to do with the degree.