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In this paper, based on classical Lie group method, we study a multidimensional double dispersion equation, and get its infinitesimal generator, symmetry group and similarity reductions. In particular, similarity solutions and travelling wave solutions of the multidimensional double dispersion equation are derived from the reduction equations.

The double dispersion Equation (1) was introduced as a mathematical model of nonlinear dispersive waves in various contexts (see [

u t t − u x x + a u x x x x − b u x x t t = f ( u ) x x , (1)

where u = u ( x , t ) is a real-valued function, a , b are positive real constants with a ≥ b . It also presents the plots of the instability/stability regions of travelling waves for various values of p, where f ( u ) = | u | p − 1 u , p > 1 . The present paper provides an overview of results obtained in [

Considering the possibility of energy exchange through lateral surfaces of the waveguide in the physical study of nonlinear wave propagation in waveguide, the longitudinal displacement u ( x , t ) of the rod satisfies the following double dispersion equation (DDE) (see [

u t t − u x x = 1 4 ( 6 u 2 + a u t t − b u x x ) x x , (2)

and the general cubic DDE (CDDE)

u t t − u x x = 1 4 ( c u 3 + 6 u 2 + a u t t − b u x x + d u t ) x x , (3)

where a , b and c are positive constants. The Equations (2) and (3) were studied in some literatures, the travelling wave solutions, depending upon the phase variable z = x ± c t were studied by Samsonov in [

In [

u t t − u x x − a u x x t t + b u x x x x − d u x x t = f ( u ) x x , (4)

where a > 0 , b > 0 and d are constants.

Recently in [

u t t − Δ u − Δ u t t + Δ 2 u − k Δ u t = Δ f ( u ) , x ∈ R n , t > 0 , (5)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ R n , (6)

where k is an arbitrary real constant. The authors gave the existence of local solution and the existence of global solution. And in [

Symmetry reductions have several important applications in the context of differential equations. Since solutions of partial differential equations asymptotically tend to solutions of equations obtained by symmetry reduction, some of these special solutions will illustrate important physical phenomena (see [

In [

u x x x x + f ( u ) x x = u t t , (7)

In [

u t t − u x x + u x x x x − u x x t t − ( u n ) x x = 0 , (8)

and they obtained exact solutions which can be expressed by various single and combined nondegenerative Jacobi elliptic function solutions.

In [

u t t − u x x − a u x x t t + b u x x x x − d u x x t − f ( u ) x x = 0 , (9)

where a > 0 , b > 0 and d are constants. They study the functional forms f ( u ) for which Equation (9) with a , b ≠ 0 admits classical symmetries.

In this paper, we consider the following multidimensional double dispersion equation

u t t − Δ u − Δ u t t + Δ 2 u + k Δ u t = Δ f ( u ) , x ∈ R n , t > 0 , (10)

where n = 3 , f ( u ) = | u | p , p > 1 or f ( u ) = u 2 k , k = 1 , 2 , ⋅ ⋅ ⋅ ,

In this paper, the symmetry group of the n-dimensional double dispersion Equation (10) is obtained by using the classical method in Section 2. In Section 3, we discuss the Lie symmetry group of Equation (10). Finally, we obtain similarity solutions or travelling wave solutions of Equation (10) by using similarity variables to obtain reduction equations, and solving the reduction equations in Section 4.

In this section, we perform Lie symmetry analysis for Equation (10), and obtain its infinitesimal generator.

Theorem 1. [

V _ = ∑ i = 1 p ξ i ( x , u ) ∂ ∂ x i + ∑ α = 1 p ϕ α ( x , u ) ∂ ∂ x α ,

be a vector field on X × U , where X = ( x 1 , x 2 , ⋯ , x p ) , U = ( u 1 , u 2 , ⋯ , u q ) . Then its n-st prolongation is defined a vector on X × U

p r ( n ) V _ = V _ + ∑ α = 1 p ∑ J = 1 ϕ α J ( x , u ( n ) ) ∂ ∂ u J α ,

where, by definition

ϕ α J ( x , u ( n ) ) = D J Q α + ∑ i = 1 p ξ i u J i α , J = ( j 1 , ⋯ , j l ) , 0 ≤ k ≤ n , 1 ≤ j k ≤ p ,

where, Q α = ϕ α ( x , u ) − ∑ i = 1 p ξ i ( x , u ) ∂ ∂ x i u i α , α = 1 , ⋯ , q , Q ( x , u ) = ( Q 1 , ⋯ , Q q )

is referred to as the characteristic of the vector field V _ .

Here are four independent variables x , y , z being spatial coordinates and t the time. According to the method of determining the infinitesimal generator of nonlinear partial differential equation [

V _ = ξ ( x , y , z , t , u ) ∂ ∂ x + η ( x , y , z , t , u ) ∂ ∂ y + ζ ( x , y , z , t , u ) ∂ ∂ z + τ ( x , y , z , t , u ) ∂ ∂ t + ϕ ( x , y , z , t , u ) ∂ ∂ u , (11)

be a vector field on X × U . Where ξ , η , ζ , τ , ϕ are coefficient functions of the infinitesimal generator to be determined. We wish to determine all possible coe- fficient functions ξ , η , ζ , τ and ϕ so that the corresponding one-parameter group exp ( ε V _ ) is a symmetry group of the double dispersion equation. Applying the forth prolongation of V _ to Equation (10), we find the invariance condition p r ( 4 ) V _ ( Δ ) | Δ = 0 , where Δ is u t t − Δ u − Δ u t t + Δ 2 u + k Δ u t − Δ ( u p ) and with help of Maple software, we find the following system of symmetry equations

ϕ t t − ( ϕ x x + ϕ y y + ϕ z z ) − ( ϕ x x t t + ϕ y y t t + ϕ z z t t ) + ( ϕ x x x x + ϕ y y y y + ϕ z z z z ) + k ( ϕ x x t + ϕ y y t + ϕ z z t ) = p ( p − 1 ) ( p − 2 ) u p − 3 ( u x 2 + u y 2 + u z 2 ) ϕ + p ( p − 1 ) u p − 2 ( 2 ∇ u ϕ x + Δ u ϕ ) + p u p − 1 ( ϕ x x + ϕ y y + ϕ z z ) , (12)

which must be satisfied whenever u satisfy Equation (10). Here ϕ t t , ϕ x x , etc. are

the coefficients of the second order derivatives ∂ ∂ u t t , ∂ ∂ u x x , etc. appearing in p r ( 4 ) V _ .

According to Th. 1, ϕ α J ( x , u ( n ) ) = D J ( ϕ α − Σ ξ i u i α ) + Σ ξ i u J , i α , we have

ϕ x x = D x 2 ϕ − u x D x 2 ξ − u y D x 2 η − u z D x 2 ς − u t D x 2 τ − 2 u x x D x ξ − 2 u x y D x η − 2 u x z D x ς − 2 u x t D x τ .

Similarly, we can get ϕ t t , ϕ x x t , ϕ x x x x , etc. we find the determining equations for the symmetry group of Equation (10) to be the following:

{ ξ u = η u = ζ u = 0 , ξ t = η t = ζ t = 0 , ϕ u u = 0 , τ x = τ y = τ z = τ u = 0 , ( ξ x + η y + ζ z ) − τ t = 0 , 2 ∇ ϕ u − 3 ( ξ x x + η y y + ζ z z ) = 0 , 2 ∇ ϕ u − ( ξ x x + η y y + ζ z z ) = 0 , Δ ϕ u − 2 ( ξ x + η y + ζ z ) = 0 , 2 ϕ t u − k τ t − τ t t = 0 , ∇ ϕ t u + k ( ξ x x + η y y + ζ z z ) − 2 k ∇ ϕ u = 0 , k Δ ϕ u − 2 Δ ϕ t + 2 ϕ t u − τ t t = 0 , u p − 3 [ u ϕ u + ( p − 2 ) ϕ + 2 τ t u ] = 0 , − 6 Δ ϕ u + ϕ t t u − k ϕ t u + p ( p − 1 ) u p − 2 η + 2 p u p − 1 τ t = 0 , Δ 2 ϕ − p u p − 1 Δ ϕ − Δ ϕ − Δ ϕ t t + ϕ t t + k Δ ϕ t = 0 , 2 p ( p − 1 ) u p − 2 ∇ ϕ − 4 ∇ Δ ϕ u + 2 p u p − 1 ∇ ϕ u + 2 ∇ ϕ u + 2 ∇ ϕ t t u − 2 k ∇ ϕ t u − p u p − 1 ( ξ x x + η y y + ζ z z ) + ( ξ x x x x + η y y y y + ζ z z z z ) − ( ξ x x + η y y + ζ z z ) = 0 ,

Since we have now satisfied all the determining equations, we conclude that general infinitesimal symmetry of Equation (10) has coefficient functions of the following form:

ξ = c 1 ,

η = c 2 ,

ζ = c 3 ,

τ = c 4 ,

where c 1 , ⋯ , c 4 are arbitrary constants. Thus the Lie-algebra of infinitesimal of the double dispersion equation is spanned by four vector fields:

V 1 = ∂ ∂ x ,

V 2 = ∂ ∂ y ,

V 3 = ∂ ∂ z ,

V 4 = ∂ ∂ t ,

so we have

V _ = c 1 V 1 + c 2 V 2 + c 3 V 3 + c 4 V 4 .

In this section, in order to get some exact solutions from a known solution of Equation (10), we should find the one-parameter symmetry groups g i : ( x , y , z , y , u ) → ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) of corresponding infinitesimal generators. To get the Lie symmetry groups, we should solve the following initial problems of ordinary differential equations:

{ d ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) d ε = ( ξ , η , ζ , τ , ϕ ) ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) | ε = 0 = ( x , y , z , t , u ) , (13)

where

ξ = ξ ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) ,

η = η ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) ,

ζ = ζ ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) ,

τ = τ ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) ,

ϕ = ϕ ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ )

and ε is a group parameter.

For the infinitesimal generator V _ = c 1 V 1 + c 2 V 2 + c 3 V 3 + c 4 V 4 , we will take the following different values to obtain the corresponding infinitesimal generators:

Case 1. C 1 = 1 , C 2 = C 3 = C 4 = 0 , the infinitesimal generator is V 1 = ∂ ∂ x ,

Case 2. C 2 = 1 , C 1 = C 3 = C 4 = 0 , the infinitesimal generator is V 2 = ∂ ∂ y ,

Case 3. C 3 = 1 , C 1 = C 2 = C 4 = 0 , the infinitesimal generator is V 3 = ∂ ∂ z ,

Case 4. C 4 = 1 , C 1 = C 2 = C 3 = 0 , the infinitesimal generator is V 4 = ∂ ∂ t ,

Case 5. C 1 = C 2 = C 3 = 1 , C 4 = 0 , the infinitesimal generator is V 5 = V 1 + V 2 + V 3 = ∂ ∂ x + ∂ ∂ y + ∂ ∂ z ,

Case 6. C 1 = C 2 = C 3 = C 4 = 1 , the infinitesimal generator is V 6 = V 1 + V 2 + V 3 + V 4 = ∂ ∂ x + ∂ ∂ y + ∂ ∂ z + ∂ ∂ t ,

Case 7. C 1 = C 2 = C 3 = λ , C 4 = β , the infinitesimal generator is V 7 = V 1 + V 2 + V 3 + V 4 = λ ( ∂ ∂ x + ∂ ∂ y + ∂ ∂ z ) + β ∂ ∂ t ,

The one-parameter groups G i generated by the V i . The entries give the transformed point exp ( ε V i ) ( x , y , z , t , u ) = ( x ¯ , y ¯ , z ¯ , t ¯ , u ¯ ) :

G 1 : ( x + ε , y , z , t , u ) ,

G 2 : ( x , y + ε , z , t , u ) ,

G 3 : ( x , y , z + ε , t , u ) ,

G 4 : ( x , y , z , t + ε , u ) ,

G 5 : ( x + ε , y + ε , z + ε , t , u ) ,

G 6 : ( x + ε , y + ε , z + ε , t + ε , u ) ,

G 7 : ( x + λ ε , y + λ ε , z + λ ε , t + β ε , u ) ,

where G 1 , G 2 , G 3 are space translations, G 4 is a time translation. ε is an arbitrary constant.

Theorem 2. If u = f ( x , y , z , t ) is a known solution of Equation (10), then by using the symmetry groups G i ( i = 1 , 2 , 3 , 4 ) , so are the functions

u 1 = f ( x − ε , y , z , t ) ,

u 2 = f ( x , y − ε , z , t ) ,

u 3 = f ( x , y , z − ε , t ) ,

u 4 = f ( x , y , z , t − ε ) ,

u 5 = f ( x − ε , y − ε , z − ε , t ) ,

u 6 = f ( x − ε , y − ε , z − ε , t − ε ) ,

u 7 = f ( x − λ ε , y − λ ε , z − λ ε , t − λ ε ) ,

where ε is any real constant.

In the previous sections, we obtained the infinitesimal generators V i ( i = 1 , 2 , ⋯ , 7 ) . In this section, we will get similarity variables and its reduction equations, and we obtain similarity solutions and travelling wave solutions of Equation (10) by solving the reduction equations.

Case 1. For the infinitesimal generator V 1 = ∂ ∂ x , the similarity variables are r = t , F ( r ) = 1 3 x − t u , and the group-invariant solution is u = 1 3 x − F ( r ) t , subs-

tituting the group-invariant solution into Equation (10), we obtain the following reduction equation

F r = 0 , (14)

Obviously, F = c 1 is a solution of Equation (14), where c 1 is an arbitrary constant. Therefore, Equation (10) has a similarity solution as follows:

u = 1 3 x − c 1 t (15)

where p = 3 .

Case 2. For the infinitesimal generator V 2 = ∂ ∂ y , the similarity variables are r = t , F ( r ) = 1 3 y − t u , and the group-invariant solution is u = 1 3 y − F ( r ) t , subs-

tituting the group-invariant solution into Equation (10), we obtain the following reduction equation

F r = 0 , (16)

Obviously, F = c 2 is a solution of Equation (16), where c 2 is an arbitrary constant. Therefore, Equation (10) has a similarity solution as follows:

u = 1 3 y − c 2 t (17)

where p = 3 .

Case 3. For the infinitesimal generator V 3 = ∂ ∂ z , the similarity variables are r = t , F ( r ) = 1 3 z − t u , and the group-invariant solution is u = 1 3 z − F ( r ) t , subs-

tituting the group-invariant solution into Equation (10), we obtain the following reduction equation

F r = 0 , (18)

Obviously, F = c 3 is a solution of Equation (18), where c 3 is an arbitrary constant. Therefore, Equation (10) has a similarity solution as follows:

u = 1 3 z − c 1 t (19)

where p = 3 .

Case 4. For the infinitesimal generator V 4 = ∂ ∂ t , the similarity variables are

r = x + y + z , F ( r ) = u , and the group-invariant solution is u = F ( r ) , substituting the group-invariant solution into Equation (10), we obtain the following reduction equation

Δ F ( r ) − Δ 2 F ( r ) + Δ ( F ( r ) p ) = 0 , (20)

Case 5. For the infinitesimal generator V 5 = ∂ ∂ x + ∂ ∂ y + ∂ ∂ z , the similarity variables are F ( r ) = 1 3 ( x + y + z ) − t u , and the group-invariant solution is u = 1 3 ( x + y + z ) − F ( r ) t , substituting the group-invariant solution into Equation (10), we obtain the following reduction equation

F r = 0 , (21)

Obviously, F = c 5 is a solution of Equation (21), where c 5 is an arbitrary constant. Therefore, Equation (10) has a similarity solution as follows:

u = 1 3 ( x + y + z ) − c 5 t (22)

where p = 3 .

Case 6. For the infinitesimal generator V 6 = ∂ ∂ x + ∂ ∂ y + ∂ ∂ z + ∂ ∂ t , the similarity

variables are r = x + y + z − λ t , F ( r ) = u , and the group-invariant solution is u = F ( r ) , If we assume k = 0 of Equation (10), substituting the group-invariant solution into Equation (10), we obtain the following reduction equation

( 1 − λ 2 ) F r r r r − p ( p − 1 ) F p − 2 F r 2 − p F p − 1 F r r + ( λ 2 − 1 ) F r r = 0 , (23)

So that Equation (23) is solvable in terms of Jacobi elliptic function, following the method described in [

If λ 2 = 4 3 c 6 5 + 1 , F ( r ) = 5 c 6 d n 2 ( r , 8 5 ) , where d n ( r , 8 5 ) is the Jacobi

elliptic of the third kind function. Therefore, Equation (10) has a travelling wave solution as follows:

u ( x , y , z , t ) = 5 c 6 d n 2 ( x + y + z − 4 3 c 6 5 + 1 t , 8 5 ) (24)

where p = 2 , c 6 is an arbitrary constant.

If λ 2 = 4 3 c 6 5 + 1 , F ( r ) = 5 c 6 c n 2 ( r , 8 5 ) , where c n ( r , 8 5 ) is the Jacobi

elliptic cosine function. Therefore, Equation (10) has a travelling wave solution as follows:

u ( x , y , z , t ) = 5 c 6 c n 2 ( x + y + z − 4 3 c 6 5 + 1 t , 8 5 ) (25)

where p = 2 , c 6 is an arbitrary constant.

Case 7. For the infinitesimal generator V 7 = ( λ ∂ ∂ x + ∂ ∂ x + ∂ ∂ x ) + β ∂ ∂ t , following

the method described in [

( λ 2 − β 2 ) β 2 F r r r r + k λ β 2 F r r r − ( λ 2 − β 2 ) β 2 F r r − β 2 p F p − 1 F r r − β 2 p ( p − 1 ) F p − 2 F r 2 = 0 , (26)

Integrating twice with respect to r, we get

( λ 2 − β 2 ) β 2 F r r + k λ β 2 F r − ( λ 2 − β 2 ) F + β 2 F p = 0 , (27)

Let us assume that Equation (27) has solution of the form

F = c 7 H c 8 ( r ) , (28)

where c 7 , c 8 are arbitrary constants, and H ( r ) is a solution of the Jacobi equation

H r 2 = μ + r H 2 + H 4 , (29)

If H ( r ) = s n ( r , m ) , where s n ( r , m ) is Jacobi elliptic sine function, we obtain F ( r ) = r s n ( r , m ) is a solution of Equation (27). By substituting F ( r ) = r s n ( r , m ) into (27), we obtain the equation

− β 2 r d n 2 ( r , m ) λ 2 − m β 2 r c n 2 ( r , m ) s n 2 ( r , m ) λ 2 − r s n ( r , m ) λ 2 + β 2 r c n 2 ( r , m ) d n 2 ( r , m ) s n − 1 ( r , m ) λ 2 − β 2 r c n 2 ( r , m ) d n 2 ( r , m ) s n − 1 ( r , m ) λ 2 + k β 2 r c n ( r , m ) d n ( r , m ) + β 2 r s n ( r , m ) + m β 4 r c n 2 ( r , m ) n ( r , m ) − β 4 r c n 2 ( r , m ) d n 2 ( r , m ) s n − 1 ( r , m ) + β 4 r c n 2 ( r , m ) d n 2 ( r , m ) s n − 1 ( r , m ) + β 2 r s n 2 ( r , m ) = 0 , (30)

If λ 2 = 2 β 2 , Equation (10) has a travelling wave solution as follows:

u ( x , y , z , t ) = [ β ( x + y + z ) − λ t ] s n ( β ( x + y + z ) − λ t , − 1 ) , (31)

where m = − 1 , p = 2 . As s n ( β ( x + y + z ) − λ t ,1 ) = t a n h ( β ( x + y + z ) − λ t ) , we obtain F ( r ) = 1 4 t a n h ( r ) is a solution of Equation (27), yb substituting F ( r ) = 1 4 t a n h ( r ) into Equation (27), we obtain a travelling wave solution as follows:

u ( x , y , z , t ) = 1 4 t a n h ( x + y + z − t 2 ) , (32)

where k = 8 , p = 3

If F ( r ) = r c n ( r , m ) is a solution of Equation (27). By substituting F ( r ) = r c n ( r , m ) into (27), when λ 2 = 2 β 2 , Equation (10) has a travelling wave solution as follows:

u ( x , y , z , t ) = [ β ( x + y + z ) − λ t ] c n ( β ( x + y + z ) − λ t , − 1 ) , (33)

where m = 1 , p = 2

In this paper, we study the symmetry reductions and explicit solutions of a multidimensional double dispersion equation by means of classical Lie group method. First, we get the symmetry groups and the infinitesimal generators of Equation (10). Then, we discuss the Lie symmetry groups of the multidimensional double dispersion equation and obtain the group-invariant solution. Finally, we obtain similarity solutions and travelling wave solutions of Equation (10) using similarity variables.

This work was supported by the Natural Science Foundation of Education of Guizhou Province: KY[

Yu, J.L., Li, F.Z. and She, L.B. (2017) Lie Symmetry Reductions and Exact Solutions of a Multidimensional Double Dispersion Equation. Applied Mathematics, 8, 712-723. https://doi.org/10.4236/am.2017.85056