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Three (2 + 1)-dimensional equations—Burgers equation, cylindrical Burgers equation and spherical Burgers equation, have been reduced to the classical Burgers equation by different transformation of variables respectively. The decay mode solutions of the Burgers equation have been obtained by using the extended -expansion method, substituting the solutions obtained into the corresponding transformation of variables, the decay mode solutions of the three (2 + 1)-dimensional equations have been obtained successfully.

Many famous nonlinear evolution equations such as Korteweg-de Vries (KdV), Modified KdV (mKdV), Kadomstev-Perviashvili (KP), Coupled KP and Zakharov-Kuznetsov (ZK) have been obtained by using the standard reductive perturbation method in nonlinear propagation of dust-acoustic wave, especially, the dust-acoustic solitary wave (DASW) in space and laboratory plasma [

Following the extension sense of the KP equation Ref. [

( u t + u u x + v u x x ) x + λ u y y = 0 , (1)

where v is a constant that defines the kinematic viscosity, λ is a constant. If surface tension is weak compared to gravitational forces, then λ > 0 is used. However if surface tension is strong, then λ < 0 is used. The kink solutions and periodic solutions were obtained by using the tanh-coth method, N-soliton solutions were established by applying the powerful Hirota’s bilinear method in Ref. [

In this work, following the extension sense of (2 + 1)-Burgers equation [

( u t + u u x + v u x x + 1 2 t u ) x + λ t 2 u y y = 0 , (2)

where v is a constant that defines the kinematic viscosity, λ is a constant.

Following the (2 + 1)-Burgers equation and spherical KP equation [

( u t + u u x + v u x x + 1 t u ) x + λ t 2 ( u y y + 1 y u y ) = 0 , (3)

where v is a constant that defines the kinematic viscosity, λ is a constant.

The nonlinear evolution equation can describe various motions. So it is important to study their exact solutions. There exist many kinds of solutions to some integrable equations such as soliton, complexiton, negaton, rational and periodic solutions [

In the present paper, the aim is to study the decay mode solutions of Equations (1), (2) and (3). The paper is organized as follows: In Section 2, making transformation of variables, by which reduction of (2 + 1)-dimensional Burgers equation, cylindrical Burgers equation and spherical Burgers equation to the classical Burgers equation; In Section 3, the decay mode solutions of the classical

Burgers equation are obtained by using the extended ( G ′ G ) -expansion method (the original ( G ′ G ) -expansion method can be found in Ref. [

using the results obtained in Section 3, the decay mode solutions of Equations (1), (2) and (3) can be obtained by using different transformations of variables, respectively; In Section 5, some conclusions are made.

In Equation (1), assume that

u = w ( ξ , t ) , ξ = x − q ( y , t ) , (4)

where q = q ( y , t ) is to be determined later. Substituting Equation (4) into Equation (1), yields an equation as follows

∂ ∂ ξ ( w t + w w ξ + v w ξ ξ ) − λ q y y w ξ + ( λ q y 2 − q t ) w ξ ξ = 0. (5)

Setting the coefficients of w ξ and w ξ ξ to zero, yields

− λ q y y = 0 , λ q y 2 − q t = 0. (6)

the system (6) admits a solution:

q ( y , t ) = μ y + λ μ 2 t , (7)

where μ is a nonzero arbitrary constant. Using Equation (7) the expression (4) becomes

u = w ( ξ , t ) , ξ = x − ( μ y + λ μ 2 t ) , (8)

and after integrating Equation (5) with respect to ξ once and taking the constant of integration to zero, Equation (5) becomes the classical Burgers equation for w = w ( ξ , t )

w t + w w ξ + v w ξ ξ = 0. (9)

From the discussion above, the conclusion can be made that the (2 + 1)-di- mensional Burgers Equation (1) for u = u ( x , y , t ) is reduced to the Burgers Equation (9) for w = w ( ξ , t ) by using the transformation of variables (8), if w ( ξ , t ) is a solution of Burgers Equation (9), substituting it into Equation (8), then the exact solution of the (2 + 1)-Burgers equation can be obtained.

Similarly, the (2 + 1)-dimensional cylindrical Burgers Equation (2) for u = u ( x , y , t ) is reduced to Equation (9) for w = w ( ξ , t ) by using the transformation of variables

u = w ( ξ , t ) , ξ = x − ( 1 4 λ y 2 t + μ y t + λ μ 2 t ) , (10)

where μ is a nonzero arbitrary constant, if w ( ξ , t ) is a solution of Equation (9), and substituting it into Equation (10), the exact solution of Equation (2) can be obtained.

The conclusion can be made that the (2 + 1)-dimensional spherical Burgers Equation (3) for u = u ( x , y , t ) is reduced to Equation (9) for w = w ( ξ , t ) by using the transformation of variables

u = w ( ξ , t ) , ξ = x − ( 1 4 λ y 2 t + μ ) , (11)

where μ is a nonzero arbitrary constant, if w ( ξ , t ) is a solution of Equation (9), and substituting it into Equation (11), the exact solution of Equation (3) can be obtained.

Considering the homogeneous balance between w w ξ and w ξ ξ in Equation (9) ( 2 m + 1 = m + 2 → m = 1 ), according to the extended ( G ′ G ) -expansion method,

a suppose can be made that the solution of Equation (9) is of the form

w = v 1 ( ξ , t ) G ′ ( φ ) G ( φ ) + v 0 ( ξ , t ) , (12)

where v 1 = v 1 ( ξ , t ) , φ = φ ( ξ , t ) , v 0 = v 0 ( ξ , t ) are to be determined later (Noticed that in the original ( G ′ G ) -expansion, v 1 , v 0 are constants, φ = ξ = x − v t is traveling waves), and G = G ( φ ) satisfies the second order LODE

G ″ + δ G = 0 , (13)

where δ is a constant to be determined later.

When δ < 0 , ODE (13) has general solution

G ( φ ) = A cosh ( − δ φ ) + B sinh ( − δ φ ) , A , B are constants

then

G ′ ( φ ) G ( φ ) = − δ A sinh ( − δ φ ) + B cosh ( − δ φ ) A cosh ( − δ φ ) + B sinh ( − δ φ ) . (14)

Choose B = 0 , then

G ′ ( φ ) G ( φ ) = − δ tanh ( − δ φ ) . (15)

Now our main goal is to determine v 1 = v 1 ( ξ , t ) , φ = φ ( ξ , t ) , v 0 = v 0 ( ξ , t ) and constant δ , such that expression (12) satisfies Equation (9).

Substituting Equation (12) into the left side of Equation (9), collecting the coefficients of each power of ( G ′ G ) i ,( i = 0 , 1 , 2 , 3 ), setting each coefficient to zero, the PDEs can be obtained for v 1 = v 1 ( ξ , t ) , φ = φ ( ξ , t ) , v 0 = v 0 ( ξ , t ) as follows

( G ′ G ) 3 : − v 1 φ ξ ( v 1 − 2 v φ ξ ) = 0 , (16)

( G ′ G ) 2 : − 2 v φ ξ v 1 ξ − v 1 ( φ t + v 0 φ ξ − v 1 ξ + v φ ξ ξ ) = 0 , (17)

( G ′ G ) : v 1 t − δ v 1 2 φ ξ + v 1 ( 2 v δ φ ξ 2 + v 0 ξ ) + v 0 v 1 ξ + v v 1 ξ ξ = 0 , (18)

( G ′ G ) 0 : v 0 t + v 0 v 0 ξ − 2 v δ φ ξ v 1 ξ − δ v 1 ( φ t + v 0 φ ξ + v φ ξ ξ ) + v v 0 ξ ξ = 0. (19)

Simplifying Equations (16)-(19), then

v 1 = 2 v φ ξ , (20)

− 2 v φ ξ ( φ t + v 0 φ ξ + v φ ξ ξ ) = 0 , (21)

2 v ∂ ∂ ξ ( φ t + v 0 φ ξ + v φ ξ ξ ) = 0 , (22)

( v 0 t + v 0 v 0 ξ + v v 0 ξ ξ ) − 2 δ v φ ξ ( φ t + v 0 φ ξ − 3 v φ ξ ξ ) = 0. (23)

Noticed that in Equations (20)-(23), if

φ = v 0 = ξ t , then v 1 = 2 v φ ξ = 2 v t , (24)

Equations (20)-(23) are satisfied completely for arbitrary constant δ < 0 .

Substituting Equation (15) and Equation (24) into Equation (12), the decay mode solution for Equation (9) can be expressed as follows

w ( ξ , t ) = 2 v − δ t tanh ( − δ ξ t ) + ξ t , δ < 0. (25)

As far as we know, the solution (25) has never seen in early literatures.

Substituting Equation (25) into Equation (8), the decay mode solution for Equation (1) can be obtained as follows

u ( x , y , t ) = 2 v − δ t tanh ( − δ ξ t ) + ξ t , (26)

where δ < 0 and ξ = x − ( μ y + λ μ 2 t ) .

Substituting Equation (25) into Equation (10), the decay mode solution for Equation (2) can be obtained as follows

u ( x , y , t ) = 2 v − δ t tanh ( − δ ξ t ) + ξ t , (27)

where δ < 0 and ξ = x − ( 1 4 λ y 2 t + μ y t + λ μ 2 t ) .

Substituting Equation (25) into Equation (11), the decay mode solution for Equation (3) can be obtained as follows

u ( x , y , t ) = 2 v − δ t tanh ( − δ ξ t ) + ξ t , (28)

where δ < 0 and ξ = x − ( 1 4 λ y 2 t + μ ) .

In this paper, by making corresponding transformation of variables, the (2 + 1)- dimensional Burgers equation, (2 + 1)-dimensional cylindrical Burgers equation and (2 + 1)-dimensional spherical Burgers equation are all reduced to the clas-

sical Burgers equation, which can be solved by using extended ( G ′ G ) -expansion

method to obtain a novel type of decay mode solution. Substituting the novel solution of the Burgers equation into the corresponding transformation of variables, the decay mode solutions of the (2 + 1)-dimensional Burgers equation, (2 + 1)-dimensional cylindrical Burgers equation and (2 + 1)-dimensional spherical Burgers equation have been obtained for the first time, respectively. The analysis may be extended to other works to make further progress.

The authors would like to thank the Editorand anonymous referees for their helpful suggestions and valuable comments. The project supported in part by the National Natural Science Foundation of China (Grant No. 11301153, 11601225) and The Doctoral Foundation of Henan University of Science and Technology (Grant No. 09001562) and The Science and Technology Innovation Platform of Henan University of Science and Technology(Grant No.2015XPT001).

Li, X.Z., Zhang, J.L. and Wang, M.L. (2017) Decay Mode Solutions to (2 + 1)-Dimensional Burgers Equation, Cylindrical Burgers Equation and Spherical Burgers Equation. Journal of Applied Mathematics and Physics, 5, 1009- 1015. https://doi.org/10.4236/jamp.2017.55088