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Soliton Solutions and Group Analysis of a New
Coupled (2 + 1)-Dimensional Burgers Equations
Gangwei Wang
Kamran Fakhar
Kwantlen Polytechnic University

A. H. Kara
University of the Witwatersrand

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Original Publication Citation
Wang, G., K. Fakhar, and A. H. Kara. (2015) Soliton solutions and group analysis of a new coupled (2+1)-dimensional Burgers'
equations. Acta Physica Polonica B 46.5 http:\\DOI:10.5506/APhysPolB.46.923

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Vol. 46 (2015)

ACTA PHYSICA POLONICA B

No 5

SOLITON SOLUTIONS AND GROUP ANALYSIS
OF A NEW COUPLED (2 + 1)-DIMENSIONAL
BURGERS EQUATIONS∗
Gangwei Wanga,b,† , K. Fakharb,c , A.H. Karad
a

School of Mathematics and Statistics, Beijing Institute of Technology
Beijing 100081, P.R. China
b
Department of Mathematics, University of British Columbia
Vancouver V6T 1Z2, Canada
c
Department of Mathematics, Faculty of Science and Horticulture
Kwantlen Polytechnic University, Surrey, British Columbia V3W 2MB Canada
d
School of Mathematics, University of the Witwatersrand
Private Bag 3, Wits 2050, Johannesburg, South Africa
(Received December 23, 2014; revised version received March 4, 2015)
This paper focuses on a new coupled (2 + 1)-dimensional Burgers equations. The shock wave solution is obtained by the aid of Ansatz method.
There are several constraint conditions which guarantee the existence of
the derived solutions. Subsequently, the simplified Hirota bilinear method,
established by Hereman, is applied to construct soliton solutions to the
equation. Finally, the classic Lie symmetry analysis is employed to generate a class of new solutions to the equation based on the solutions obtained
earlier by Ansatz and simplified Hirota bilinear methods.
DOI:10.5506/APhysPolB.46.923
PACS numbers: 02.30.Jr, 05.45.Yv, 02.30.Ik

1. Introduction
Nonlinear phenomena come up in a variety of scientific fields, such as
solid-state physics, plasma physics, fluid dynamics, mathematical biology,
chemical kinetics, etc. Nonlinear evolution equations can model nonlinear
phenomena that appear in nature quite frequently. Therefore, searching for
∗

Presented at the Research Project of China Scholarship Council (No. 201406030057),
National Natural Science Foundation of China (NNSFC) (Grant No. 11171022), Graduate Student Science and Technology Innovation Activities of Beijing Institute of
Technology (No. 2014cx10037).
†
Corresponding author: pukai1121@163.com

(923)

924

G.W. Wang, K. Fakhar, A.H. Kara

exact solutions of these equations is of paramount importance. A great deal
of literature concerning these equations and methods to obtain solutions is
available in the text. Amongst the several methods which are in use to handle these equations, the well known are: the inverse scattering method, the
Baklund transformation method, Darboux transformation, Painleve analysis
method, Exp-function method, the Hirota bilinear method, simplified Hirota
bilinear method, Lie group method and, solitary wave Ansatz method. In
addition, the computer symbolic systems, such as Maple and Mathematica,
help us to deal with complicated and tedious calculations [1–10].
It is well known that traveling waves appear in solitary waves theory
in distinct features such as solitons, kinks, etc. The soliton theory is an
important part of the nonlinear science. Soliton-like particles can travel
over long-distances without attenuation and changes of wave shapes due to
the balance of the interplay between dispersion and nonlinearity ([11] and
papers cited therein).
It is generally known that the Burgers equation describes the coupling
between diffusion uxx and the convection process uux . The Burger’s types of
equations have been applied to various physical phenomena, such as: fluid
dynamics, gas dynamics, traffic flow, etc. The Burgers equation (BE) is
one of the fundamental model equations that is in existence for a very long
time. Many researchers have been investigated these types of equations with
different configurations [12–21].
In this work, our aim is to investigate a new coupled (2 + 1)-dimensional
Burgers equations given through
ut − 2uux − uy v − uvy − uxx − uyy = 0 ,
vt − 2vvy − ux v − uvx − vxx − vyy = 0 ,

(1)

where a mixed partial derivative terms uvy and ux v are added to generalize
coupled (2 + 1)-dimensional Burgers equations to a more common situation.
The two additional terms, one in each equation, represent divergences with
respect to y and x, respectively. Thus, it is expected that the symmetry
group and the conservation laws would take different forms.
Here, we will study Eq. (1) via Ansatz method, simplified Hirota bilinear
method and Lie symmetry analysis. The layout of the paper is as follows.
In Section 2, the Ansatz method is applied to get the topological soliton
solutions of Eq. (1). In Section 3, Eq. (1) will be studied using simplified
Hirota bilinear method. In Section 4, we perform Lie symmetry analysis on
Eq. (1), whereas last section is for conclusions.

Soliton Solutions and Group Analysis of a New Coupled (2+1)-dimensional . . . 925

2. Ansatz method
In this section, we will investigate the new coupled BE given by Eq. (1)
using the Ansatz approach. The shock wave solutions that are also known
as topological solitons in theoretical physics are derived. In order to get the
shock wave solution, we first give the following hypothesis [10, 19, 22, 23]
u(x, y, t) = A1 tanhp1 τ

(2)

v(x, y, t) = A2 tanhp2 τ ,

(3)

τ = B1 x + B2 y − ct ,

(4)

and

where

and Aj , Bj for j = 1, 2 are free parameters and c is the speed of the shock
wave. Meanwhile, the unknown exponent p1 and p2 will be further fixed.
Thus, substituting (2) and (3) into (1) yields the following equations:




p1 A1 c tanhp1 +1 τ − tanhp1 −1 τ − 2p1 A21 B1 tanh2p1 −1 τ − tanh2p1 +1 τ




−(p1 + p2 )A1 A2 B2 tanhp1 +p2 −1 τ − tanhp1 +p2 +1 τ − A1 p1 B12 + B22

× (p1 − 1) tanhp1 −2 τ − 2p1 tanhp1 τ + (p1 + 1) tanhp1 +2 τ = 0 ,
(5)
and




p2 A2 c tanhp2 +1 τ − tanhp2 −1 τ − 2p2 A22 B2 tanh2p2 −1 τ − tanh2p2 +1 τ




−(p1 + p2 )A1 A2 B2 tanhp1 +p2 −1 τ − tanhp1 +p2 +1 τ − A2 p2 B12 + B22

× (p2 − 1) tanhp2 −2 τ − 2p2 tanhp2 τ + (p2 + 1) tanhp2 +2 τ = 0 .
(6)
From (5), based on balancing principle, one can get
2p1 − 1 = p1 + p2 − 1 = p1 ,

(7)

p1 = p2 = 1 .

(8)

in other words,

Also, from (6), one can get the same conclusion as in (8). Now, from (5)
and (6), assuming that the coefficients of the linearly independent functions
pj and pj+2 , for j = 1, 2, equal to zero, one can arrive at
c = 0,

(9)

926

G.W. Wang, K. Fakhar, A.H. Kara

and
A1 B1 + A2 B2 = B12 + B22 .

(10)

This implies that the shock wave solution of the equation is a stationary
shock wave, which is given by
u(x, y, t) = A1 tanh (B1 x + B2 y) ,

(11)

v(x, y, t) = A2 tanh (B1 x + B2 y) ,

(12)

and
where A1 , A2 and B1 , B2 are given by (10).
Remark 1: It is worth noting that the terms uvy and ux v are added,
the obtained results are the same as in [19].
3. The simplified Hereman–Nuseir method
In this section, we will deal with (1) by using the simplified Hereman–
Nuseir method. The main steps of this method can be found in [7, 8, 11, 12].
Here, we adopt the procedure outlined in [12]. Since the calculations are
straightforward, therefore by omitting the details of the calculations, we
directly write the N -kink solutions for equation (1) as
u =

ki (x+y)+2ki2 t
i=1 ki e
,
P
ki (x+y)+2ki2 t
1+ N
i=1 e

v =

ki (x+y)+2ki2 t
i=1 ki e
.
P
ki (x+y)+2ki2 t
1+ N
e
i=1

PN

PN

(13)

Similarly, the N -singular kink solutions are
ki (x+y)+2ki2 t
i=1 ki e
,
P
ki (x+y)+2ki2 t
1− N
e
i=1

PN

u = −

ki (x+y)+2ki2 t
i=1 ki e
−
.
P
ki (x+y)+2ki2 t
1− N
i=1 e

PN

v =

(14)

For the details of calculations, the reader is referred to [12].
Remark 2: It should be noted that uvy and ux v are added, but the
obtained results are the same as in [12].

Soliton Solutions and Group Analysis of a New Coupled (2+1)-dimensional . . . 927

4. Lie symmetry analysis of (1)
In this section, we will handle Eq. (1) using Lie symmetry analysis.
On the basis of Lie group theory, a one-parameter Lie group of point
transformations are given by [19]:




t∗ = t + τ (x, y, t, u, v) + O 2 ,
x∗ = x + ξ(x, y, t, u, v) + O 2 ,




y ∗ = y + η(x, y, t, u, v) + O 2 ,
u∗ = u + φ(x, y, t, u, v) + O 2 ,


v ∗ = v + ψ(x, y, t, u, v) + O 2 ,
(15)
the associated vector field is of the form
∂
∂
∂
+ ξ(x, y, t, u, v)
+ η(x, y, t, u, v)
∂t
∂x
∂y
∂
∂
+φ(x, y, t, u, v)
+ ψ(x, y, t, u, v) .
∂u
∂v

V = τ (x, y, t, u, v)

(16)

Here, the coefficient functions τ (x, y, t, u, v), ξ(x, y, t, u, v), η(x, y, t, u, v),
φ(x, y, t, u, v), and ψ(x, y, t, u, v) are to be further fixed.
If the vector field (16) has to generate a symmetry of the Eq. (1), then
V need to satisfy the following condition
pr(2) V (∆1 , ∆2 )|∆1 =0,∆2 =0 = 0 .

(17)

Here, ∆1 = ut − 2uux − uy v − uvy − uxx − uyy , and ∆2 = vt − 2vvy − ux v −
uvx − vxx − vyy and pr(2) V is the second prolongation of the vector field. In
other words,
φt − 2ux φ − 2uφx − φy v − uy ψ − uφy − φvy − φxx − φyy = 0 ,
ψ t − 2vy ψ − 2vψ y − vx φ − uψ x − φx v − ux ψ − ψ xx − ψ yy = 0 . (18)
In (18), we only solve the following coefficients functions [19]:
φt = Dt (φ) − ux Dt (ξ) − uy Dt (η) − ut Dt (τ ) ,
φx = Dx (φ) − ux Dx (ξ) − uy Dx (η) − ut Dx (τ ) ,
φy = Dy (φ) − ux Dy (ξ) − uy Dy (η) − ut Dy (τ ) ,
ψ t = Dt (ψ) − vx Dt (ξ) − vy Dt (η) − vt Dt (τ ) ,
ψ x = Dx (ψ) − vx Dx (ξ) − vy Dx (η) − vt Dx (τ ) ,
ψ y = Dy (ψ) − vx Dy (ξ) − vy Dy (η) − vt Dy (τ ) ,
φxx = Dx (φx ) − uxt Dx (τ ) − uxx Dx (ξ) − uxy Dx (η) ,
φyy = Dy (φy ) − uyt Dy (τ ) − uxy Dy (ξ) − uyy Dy (η) ,
ψ xx = Dx (ψ x ) − vxt Dx (τ ) − vxx Dx (ξ) − vxy Dx (η) ,
ψ yy = Dy (ψ y ) − vyt Dy (τ ) − vxy Dy (ξ) − vyy Dy (η) .

(19)

928

G.W. Wang, K. Fakhar, A.H. Kara

Here, Di denotes the operators of total differentiation with respect to x, y
and t, respectively
Di =

∂
∂
∂
+ ... ,
+ upi
+ upij
i
∂x
∂u
∂uj

i = 1, 2, 3 ,

p = 1, 2 ,

(20)

and (x1 , x2 , x3 ) = (t, x, y), (u1 , u2 ) = (u, v).
Then, considering the Lie symmetry analysis method, one can obtain
ξ = 12 c1 x − c3 y + c5 ,

τ = c1 t + c2 ,

φ = − 12 c1 u − c3 v ,

ψ = c3 u − 12 c1 v ,

η = 12 c1 y + c3 x + c4 ,
(21)

where ci (i = 1, 2 . . . 5) are arbitrary constants. Thus, the five vector fields
are given by
∂
∂
∂
,
V2 =
,
V3 =
,
∂x
∂y
∂t
∂
∂
∂
∂
∂
V4 = x
+ 2t + y
−u
−v
,
∂x
∂t
∂y
∂u
∂v
∂
∂
∂
∂
V5 = −y
+x
−v
+u .
∂x
∂y
∂u
∂v
V1 =

(22)

Remark 3: It is clear that the vector fields are narrower than the vector
fields in [19], the reason is that added terms affect the properties.
It is to be noted that the symmetry generators found in (22) form a
closed and five-dimensional Lie algebra. Here, they are omitted for the sake
of brevity. In order to obtain the Lie symmetry group, the following initial
problems need to be considered
d
(x̄, ȳ, t̄, ū, v̄) = σ (x̄, ȳ, t̄, ū, v̄) , (x̄, ȳ, t̄, ū, v̄) |ε=0 = (x, y, t, u, v) ,
dε
here, ε is a parameter and
σ = ξux + τ ut + ηuy + φu + ψv .

(23)

(24)

Therefore, we can get the following Lie symmetry group:
g : (x, y, t, u, v) → (x̄, ȳ, t̄, ū, v̄) .

(25)

Thus, one can get the following group:
g1 : (x + ε, y, t, u, v) ,
g2 : (x, y + ε, t, u, v) ,
g3 : (x, y, t + ε, u, v) ,
g4 : (eε x, eε y, e2ε t, e−ε u, e−ε v) ,
g5 : (x cos ε − y sin ε, x sin ε + y cos ε, t, v cos ε − u sin ε, v sin ε + u cos ε) .
(26)

Soliton Solutions and Group Analysis of a New Coupled (2+1)-dimensional . . . 929

Consequently, many new solutions can be derived by applying the above
groups gi (i = 1, . . . 5):
u1 = f1 (x − ε, y, t) ,
v1 = h1 (x − ε, y, t) ,
u2 = f2 (x, y − ε, t) ,
v2 = h2 (x, y − ε, t) ,
u3 = f3 (x, y, t − ε) ,
v3 = h3 (x, y, t − ε) ,




−ε
−ε
−ε
−2ε
u4 = e f4 e x, e y, e t ,
v4 = e−ε h4 e−ε x, e−ε y, e−2ε t ,
u5 = f5 (x cos ε − y sin ε, x sin ε + y cos ε, t, v cos ε − u sin ε) ,
v5 = h5 (x cos ε − y sin ε, x sin ε + y cos ε, t, v sin ε + u cos ε) ,
(27)
where ε is an arbitrary constant.
If taking the shock wave solution of Eq. (1) given through (11) and (12),
one can get new exact solutions of Eq. (1) by applying the scaling symmetry
group g4 as follows:


u = e−ε A1 tanh B1 e−ε x + B2 e−ε y ,


v = e−ε A2 tanh B1 e−ε x + B2 e−ε y .
(28)
Also, from (13) and (14), one can get new explicit solutions of Eq. (1) by
applying the scaling symmetry group g4 as follows:
PN
ki e−ε (x+y)+2ki2 e−2ε t
−ε
i=1 ki e
,
u = e
P
ki e−ε (x+y)+2ki2 e−2ε t
1+ N
i=1 e
PN
ki e−ε (x+y)+2ki2 e−2ε t
−ε
i=1 ki e
v = e
,
(29)
P
ki e−ε (x+y)+2ki2 e−2ε t
e
1+ N
i=1
u = −e

−ε

v = −e

−ε

ki e−ε (x+y)+2ki2 e−2ε t
i=1 ki e
,
P
ki e−ε (x+y)+2ki2 e−2ε t
1− N
i=1 e

PN

ki e−ε (x+y)+2ki2 e−2ε t
i=1 ki e
.
P
ki e−ε (x+y)+2ki2 e−2ε t
1− N
i=1 e

PN

(30)

Remark 4: A class of new invariant solutions can be found by utilizing
the different groups of Eq. (1).
5. Conclusions
This paper addresses a new coupled (2 + 1)-dimensional Burgers equations. By using the Ansatz method, the topological 1-soliton solution is
derived for this equation for the first time. It is also shown that the shock

930

G.W. Wang, K. Fakhar, A.H. Kara

wave solution of the coupled BE is a stationary shock wave. Then, the
equation was investigated for multiple-soliton solutions and multiple singular soliton solutions. The simplified form of the Hirota’s method is used to
obtain these solutions. At last, the Lie symmetry analysis have been applied
to give an additional display of solutions. These solutions may be useful to
further investigate the complicated nonlinear physical phenomena.
The authors are thankful for the referee’s useful suggestions, which have
generally improved the manuscript.
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