Invariant Subspaces and Exact Solutions to the Generalized Strongly Dispersive DGH Equation

In this paper, the invariant subspaces of the generalized strongly dispersive DGH equation are given, and the exact solutions of the strongly dispersive DGH equation are obtained. Firstly, transform nonlinear partial differential Equation (PDE) into ordinary differential Equation (ODE) systems by using the invariant subspace method. Secondly, combining with the dynamical system method, we use the invariant subspaces which have been obtained to construct the exact solutions of the equation. In the end, the figures of the exact solutions are given.


Introduction
Nonlinear phenomena occur widely in various scientific fields, such as fluid mechanics, solid state physics, etc. As an important mathematical model to describe nonlinear phenomena, nonlinear partial differential equation has been widely concerned by many scholars in recent years, and the exact solutions of the PDE, including the soliton solution and wave solution, has always been a subject of interest to mathematical physicists. Recently, the methods of solving nonlinear partial differential equations mainly include Backlund-transformation method, non-locally symmetric method, Lie group method, and invariant subspace method, etc.
The invariant subspace method was first proposed by Galaktionov et al., and then extended by many scholars and widely applied. For example, precise solutions of Hunter-Saxton equation and compressible Euler equation were obtained. This method was derived from the Lie symmetry analysis, and it is re-lated to the conditional Lie-Bäcklund symmetry method [1] [2] and the differential constraint method; its key step is to transform a nonlinear PDE into ODE systems based on the invariant subspaces [3]. Then, we use the invariant subspaces we have obtained to construct the exact solutions of the nonlinearly PDE equations. So it is a dynamical system method by nature [3].
The outstanding feature of the invariant subspace method is its wide application range; it is also an algorithm that can construct more solutions and similar solutions for nonlinear partial differential equations.
However, there are still many problems in this method, such as how to construct more exact solutions according to different nonlinear equations. Furthermore, how to improve the efficiency of solving ordinary differential equations, such as with the help of Maple program, should be further studied. Finally, the figures of the exact solutions are given. This paper is organized as follows: In Section 2, the invariant subspace method is introduced briefly. In Section 3, all invariant subspaces and their basis functions are given. In Section 4, by using the results given in Section 3, the rational function solution, trigonometric function solution and exponential function solution of the Equation (1.3) are obtained. In Section 5, the conclusion and prospect are given.
In the following, we use G to represent the m-order differential operator, and by using the invariant subspace method, giving the nonlinear evolution equation [ ] F u determined by the differential operator and its corresponding invariant subspaces, and the basis functions in the invariant subspaces.

Invariant Subspace Method
Firstly, we consider a nonlinear evolution equation as follows: , G is a m-th order differential operator, if have p functions are linearly independent, and the n-dimensional linear space is The invariant subspace p W is allowed by the operator G, so the nonlinear Equation (2.1) have the solution as follows

Invariant Subspace of Equation (1.3)
Firstly, Equation (1.3) can be written in the form of general evolution Equation the nonlinear operator G on the right side of Equation (

p = 3
In this case, the invariant condition of 3 W reads From the invariant subspace obtained when n = 2 and n = 3, we can see that the invariant subspace method is only related to the nonlinear terms of the equation. According to the similar calculation steps above, we can get the invariant subspaces when p = 4, 5, 6. But, when p = 7, there is no solution.

p = 4
The four-dimensional invariant subspace allowed by the differential operator

p = 5
The five-dimensional invariant subspace allowed by the differential operator

p = 6
The six-dimensional invariant subspace allowed by the differential operator

Exact Solutions and Figures of Equation (1.3)
In this section, combining the invariant subspace method with the dynamical system method, by using the results obtained in Section 3, construing the exact

Polynomial Solution of the Equation (1.3)
where 1 2 , , c c γ are arbitrary constants, from 22 So, we can suppose that Equation (4.1) has the solution as follows: Substituting (4.2) into Equation The Polynomial solution of Equation (4.1) are obtained See Figure 1(a) and Figure 1(b).   [ ] ( ) substituting (     And symmetry analysis is an invariance analysis to some extent. The above theory provides a direction for future research. Moreover, the results in the present paper are verified by the maple procedure.