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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.

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 related to the conditional Lie-Bäcklund symmetry method [

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.

In 1993, Camassa and Holm obtained a new class of completely integrable nonlinear shallow water wave equations Camassa-Holm (CH) equation

u t + 2 a u x − u x x t + 3 u u x = 2 u x u x x x + u u x x x (1.1)

Equation (1.1) has a double Hamilton structure and is completely integrable. When a ≠ 0 , it has smooth isolated wave solutions, when a = 0 , it has a peaked solitary wave solution the form of u = c e − | x − c t | (where c is the velocity).

Dullin, Gottwald and Holm derived the DGH equation from a class of shallow water wave equations of non-local asymptotic form [

u t + 2 a u x − u x x t + 3 u u x + γ u x x x = 2 u x u x x + u u x x x (1.2)

where u x x is the linear dispersive term, when γ = 0 , Equation (1.2) is the Camassa-Holm (CH) equation, when ( u − u x x ) x x x is the dispersive term [

u t + 2 c 1 u x − u x x t + c 2 u u x + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x (1.3)

where u = u ( x , t ) , is an unknown function, and c 1 , c 2 , γ are arbitrary constants. Equation (1.3) describes the unidirectional propagation of surface waves in shallow water.

In recent years, many scholars have also done a lot of research on DGH equation. In 2007, Tian Lixin studied the global existence of the solution and Blow-up phenomenon and Hamilton structure of DGH equation. By using the qualitative analysis of planar autonomous system method, Guo Bolin and Liu Zhengrong obtained the spike isolated wave solutions of DGH equation. Compared with other methods, the invariant subspace method can obtain many new exact solutions of the generalized strongly dispersive DGH equation, including rational function solutions, which are different from symmetric reduction solutions and soliton solutions [

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.

Firstly, we consider a nonlinear evolution equation as follows:

u t = G [ u ] ≡ G [ x , u , ∂ u ( x , t ) ∂ x , ⋯ , ∂ n u ( x , t ) ∂ x n ] (2.1)

where u = u ( x , t ) , G is a m-th order differential operator, if have p functions g 1 ( x ) , ⋯ , g p ( x ) are linearly independent, and the n-dimensional linear space is [

W p = ℧ { g i ( x ) , g 2 ( x ) , ⋯ , g p ( x ) } = { ∑ i = 1 p c i ( t ) g i ( x ) , 1 ≤ i ≤ p }

If G satisfies G [ W p ] ⊆ W p , that is, for any constant c i , satisfy

G ( ∑ i = 1 p c i g i ( x ) ) = ∑ i = 1 p G i ( c i , c 2 , ⋯ , c p ) g i ( x )

The invariant subspace W p is allowed by the operator G, so the nonlinear Equation (2.1) have the solution as follows

u ( x , t ) = ∑ i = 1 p c i ( t ) g i ( x )

where c i satisfy n-dimensional dynamical system [

c ′ ( t ) = G ( c 1 ( t ) , c 2 ( t ) , ⋯ , c p ( t ) ) , 1 ≤ i ≤ p

Suppose the subspace W p composed of solutions to the following linear ODE

L [ y ] ≡ y ( p ) + a p − 1 ( x ) y ( p − 1 ) + ⋯ + a 1 y ′ + a 0 y = 0

then the invariant condition with respect to G takes the form

L [ G [ u ] ] | L [ u ] = 0 ≡ 0

Theorem 1. If the subspace W P is invariant under a nonlinear differential operator G of order m, then p ≤ 2 m + 1 .

Firstly, Equation (1.3) can be written in the form of general evolution Equation (2.1)

u t − u x x t = F [ u ] = 2 u x u x x + u u x x x − 2 c 1 u x − c 2 u u x − γ ( u − u x x ) x x x (3.1)

the nonlinear operator G on the right side of Equation (3.1) is

G [ u ] = 2 u x u x x + u u x x x − 2 c 1 u x − c 2 u u x (3.2)

then, by the invariant subspace theorem 1, we have p ≤ 7 , in what follows, the invariant subspaces of Equation (1.3) are given.

In this case, the invariant condition of W 2 reads

L 2 [ G [ u ] ] | L 2 [ u ] = 0 = D 2 G + a 1 D G + a 0 G | L 2 [ u ] = 0 = 0 (3.3)

where L 2 [ u ] = u ″ + a 1 u ′ + a 0 u = 0 , substituting Equation (3.2) into Equation (3.3), we have a polynomial in u x , u

( − 6 a 1 3 + 10 a 0 a 1 + 2 a 1 c 1 ) u x 2 + ( − 11 a 0 a 1 2 + 9 a 0 2 + 3 a 0 c 2 ) u u x − 5 a 0 2 a 1 u 2 = 0.

take the coefficients of the above polynomial to be zero, we obtain the overdetermined equations for solving a i , c i

u x 2 : − 6 a 1 3 + 10 a 0 a 1 + 2 a 1 c 1 = 0 u u x : − 11 a 0 a 1 2 + 9 a 0 2 + 3 a 0 c 2 = 0 u 2 : 5 a 0 2 a 1 = 0

By solving the above equations, we can get three sets of solutions as follows:

1) a 0 = 0 , a 1 = a 1 , c 1 = c 1 , c 2 = 3 a 1 2 ;

2) a 0 = 0 , a 1 = 0 , c 1 = c 1 , c 2 = c 2 ;

3) a 0 = a 0 , a 1 = 0 , c 1 = c 1 , c 2 = − 3 a 0 .

According to the above three solutions, the two-dimensional invariant subspaces allowed by the differential operator G [ u ] of the Equation (3.2) are

G 21 { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x − 3 a 1 2 u u x , L 2 [ y ] = y ″ + a 1 y = 0 , W 21 = ℧ { 1 , exp ( − a 1 x ) } .

G 22 { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x − c 2 u u x , L 2 [ y ] = y ″ = 0 , W 22 = ℧ { 1 , x } .

G 23 { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x + 3 a 0 u u x , L [ y ] = y ″ + a 0 y = 0 , W 21 = ℧ { cos ( a 0 x ) , sin ( a 0 x ) } ( a 0 > 0 ) , W 23 = ℧ { exp ( − − a 0 x ) , exp ( − a 0 x ) } ( a 0 < 0 ) .

In this case, the invariant condition of W 3 reads

L [ G [ u ] ] | L 3 [ u ] = 0 = ( D 3 G + a 2 D 2 G + a 1 D G + a 0 G ) | L 3 [ u ] = 0 = 0

where L 3 [ u ] = u ‴ + a 2 u ″ + a 1 u ′ + a 0 u = 0 , substituting the Equation (3.2) into the Equation (3.3), we have a polynomial in u 2 , u u x , u x 2 , u u x x , u x u x x , u x x 2 , and take its coefficients to be zero, we obtain the overdetermined equations for solving a i , c i

u 2 : 7 a 0 2 = 0 u u x : 18 a 0 a 2 = 0 u x 2 : − a 0 a 2 2 + 16 a 0 a 1 + 4 a 0 c 2 = 0 u u x x : − a 1 a 2 2 + a 0 a 2 + 9 a 1 2 + 3 a 1 c 2 = 0 u x u x x : − a 2 3 + 21 a 1 a 2 + a 2 c 2 − 14 a 0 = 0 u x x 2 : 11 a 2 2 − 9 a 1 − 3 c 2 = 0

By solving the above equations, we get a unique set of solutions as follows

{ a 0 = 0 , a 1 = a 1 , a 2 = 0 , c 1 = c 1 , c 2 = − 3 a 1 }

Thus, we obtain the three-dimensional invariant subspace is

G 3 { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x + 3 a 1 u u x , L [ y ] = y ‴ + a 1 y ′ = 0 , W 31 = ℧ { 1 , cos ( a 1 x ) , sin ( a 1 x ) } ( a 1 > 0 ) , W 32 = ℧ { 1 , exp ( − − a 1 x ) , exp ( − a 1 x ) } ( a 1 < 0 ) .

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.

The four-dimensional invariant subspace allowed by the differential operator G [ u ] of the Equation (3.2) is

G 4 = { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x , L [ y ] = y 4 = 0 W 4 = ℧ { 1 , x , x 2 , x 3 }

The five-dimensional invariant subspace allowed by the differential operator G [ u ] of the Equation (3.2) is

G 5 = { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x , L [ y ] = y 5 = 0 , W 5 = ℧ { 1 , x , x 2 , x 3 , x 4 } .

The six-dimensional invariant subspace allowed by the differential operator G [ u ] of the Equation (3.2) is

G 6 = { F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x + 3 a 1 u u x , L [ y ] = y 6 = 0 , W 6 = ℧ { 1 , x , x 2 , x 3 , x 4 , x 5 } .

In this section, combining the invariant subspace method with the dynamical system method, by using the results obtained in Section 3, construing the exact solution of the Equation (1.3) in invariant subspaces. We can divide the solutions of generalized strongly dispersive DGH equation into three types: polynomial function solutions, exponential function solutions and trigonometric function solutions. Then, the figures of the exact solutions are given.

When n = 2, for

F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x − c 2 u u x (4.1)

where c 1 , c 2 , γ are arbitrary constants, from G 22 in Section 3, we get W 22 = ℧ { 1 , x } .

So, we can suppose that Equation (4.1) has the solution as follows:

u 1 = ϕ 0 ( t ) + ϕ 1 ( t ) x . (4.2)

Substituting (4.2) into Equation (4.1), we get the equation for x n ( n = 0 , 1 ) , and take its coefficients to be zero, we get

{ ϕ ′ 2 ( t ) − ϕ 2 2 ( t ) c 2 = 0 , ϕ ′ 1 ( t ) − ϕ 1 ( t ) ϕ 2 ( t ) c 2 − 2 c 1 ϕ 2 ( t ) = 0.

By solving Equation (4.3), we get

ϕ 1 ( t ) = 2 c 1 t ± C 2 − c 2 t ± C 1 , ϕ 2 ( t ) = 1 − c 2 t ± C 1

The Polynomial solution of Equation (4.1) are obtained

u 1 = 2 c 1 t + C 2 − c 2 t + C 1 + 1 − c 2 t + C 1 x (4.3)

See

When n = 3, for

F [ u ] = u t − u x x t + γ ( u − u x x ) x x x = 2 u x u x x + u u x x x − 2 c 1 u x + 3 a 1 u u x (4.4)

where c 1 , c 2 , γ are arbitrary constants. Suppose that Equation (4.4) has exact solutions in the form of trigonometric and exponential functions:

u 2 , 1 = ϕ 0 ( t ) + ϕ 1 ( t ) cos ( a 1 x ) + ϕ 2 ( t ) sin ( a 1 x ) , a 1 > 0 , (4.5)

u 2 , 2 = ϕ 0 ( t ) + ϕ 1 ( t ) exp ( − a 1 x ) + ϕ 2 ( t ) exp ( − − a 1 x ) , a 1 < 0. (4.6)

substituting (4.5) into Equation (4.4), we get the equation for cos ( a 1 x ) , sin ( a 1 x ) , and take its coefficients to be zero, we get

{ ( 1 + a 1 ) ϕ ′ 2 ( t ) + γ a 1 5 2 ϕ 1 ( t ) − 2 a 1 3 2 ϕ 0 ( t ) ϕ 1 ( t ) − γ a 1 3 2 ϕ 1 ( t ) − 2 a 1 c 1 ϕ 1 ( t ) = 0 ( 1 + a 1 ) ϕ ′ 1 ( t ) − γ a 1 5 2 ϕ 2 ( t ) + 2 a 1 3 2 ϕ 0 ( t ) ϕ 2 ( t ) − γ a 1 3 2 ϕ 2 ( t ) − 2 a 1 c 1 ϕ 2 ( t ) = 0 ϕ ′ 0 ( t ) = 0

By solving Equation (4.3), we get

ϕ 0 ( t ) = C 1 , ϕ 1 ( t ) = C 2 sin ( A t ) + C 3 cos ( A t ) , ϕ 2 ( t ) = C 2 cos ( A t ) − C 3 sin ( A t )

So, the solution in form of the trigonometric function of Equation (4.4) is

u 2 , 1 = C 1 + [ C 2 sin ( A t ) + C 3 cos ( A t ) ] cos ( a 1 x ) + [ C 2 cos ( A t ) − C 3 sin ( A t ) ] sin ( a 1 x ) .

where C 1 , C 2 , C 3 are arbitrary constants, and A = ( a 1 2 γ − 2 C 1 a 1 + a 1 γ + c 1 ) a 1 t a 1 + 1 .

Substituting (4.6) into Equation (4.4), we get the equation for exp ( − a 1 x ) , exp ( − − a 1 x ) , and take its coefficients to be zero, we get

{ γ − a 1 ϕ 1 a 1 2 − 2 ϕ 0 − a 1 ϕ 1 a 1 + γ − a 1 ϕ 1 a 1 + 2 − a 1 c 1 ϕ 1 + ( a 1 + 1 ) ϕ ′ 1 = 0 − γ − a 1 ϕ 2 a 1 2 + 2 ϕ 0 − a 1 ϕ 1 a 1 − γ − a 1 ϕ 2 a 1 − 2 − a 1 c 1 ϕ 2 + ( a 1 + 1 ) ϕ ′ 2 = 0 ϕ ′ 0 = 0

By solving Equation (4.8), we get

ϕ 0 ( t ) = C 1 , ϕ 1 ( t ) = C 2 exp ( B t ) exp ( D t ) , ϕ 2 ( t ) = C 3 exp ( D t ) exp ( B t ) .

So, the trigonometric solution of Equation (4.4) is

u 2 , 2 = C 1 + C 2 exp ( B t ) exp ( D t ) exp ( − a 1 x ) + C 3 exp ( D t ) exp ( B t ) exp ( − − a 1 x ) .

where C 1 , C 2 , C 3 are arbitrary constants, and B = 2 C 1 a 1 − a 1 a 1 + 1 , D = ( a 1 2 γ + a 1 γ + 2 c 1 ) − a 1 a 1 + 1 .

See

Firstly, by using the invariant subspace method, the invariant subspaces of the generalized strongly dispersive DGH equation are obtained. Secondly, we selected the partial invariant subspaces, and we get an exact solution other than the soliton of the equation, such as the solutions in form of polynomial function, trigonometric function and exponential function of the generalized strongly dispersive DGH equation. Meanwhile, the invariant subspace method derived from symmetry analysis, this method also can be used to solve the nonlinear PDEs. 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.

The authors would like to thank the Editor and Reviewers for their valuable comments.

The authors declare no conflicts of interest regarding the publication of this paper.

Li, X.X., Liu, H.Z. and Chang, L.N. (2020) Invariant Subspaces and Exact Solutions to the Generalized Strongly Dispersive DGH Equation. Journal of Applied Mathematics and Physics, 8, 1654-1663. https://doi.org/10.4236/jamp.2020.88126