c
Applied Mathematics E-Notes, 10(2010), 235-245 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/
ISSN 1607-2510
The Extended Tanh-Method For Finding Traveling
Wave Solutions Of Nonlinear Evolution Equations∗
Elsayed M. E. Zayed†, Hanan M. Abdel Rahman‡
Received 10 September 2009
Abstract
In this article, we find traveling wave solutions of the coupled (2+1)-dimensional
Nizhnik-Novikov-Veselov and the (1+1)-dimensional Jaulent-Miodek (JM) equations. Based on the extended tanh method, an efficient method is proposed to
obtain the exact solutions to the coupled nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave
equations.
1
Introduction
The investigation of the traveling wave solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. Nonlinear
wave phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics,
chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations.
In recent years, new exact solutions may help us find new phenomena. A variety of
powerful methods, such as the inverse scattering method [1, 13], bilinear transformation [7], tanh-sech method [10, 11], extended tanh method [5, 10], homogeneous balance
method [5] and Jacobi elliptic function method [15] were used to develop nonlinear dispersive and dissipative problems. The pioneer work of Malfiet in [10, 11] introduced
the powerful tanh method for reliable treatment of the nonlinear wave equations. The
useful tanh method is widely used by many authors such as [17–20] and the references
therein. Later, the extended tanh method, developed by Wazwaz [21, 22], is a direct and
effective algebraic method for handling nonlinear equations. Various extensions of the
method were developed as well. The next interest is in the determination of the exact
traveling wave solutions for the coupled (2+1)-dimensional Nizhnik-Novikov-Veselov
and the (1+1)-dimensional Jaulent-Miodek (JM) equations. Searching for the exact
solutions of nonlinear problems has attracted a considerable amount of research work
where computer symbolic systems facilitate the computational work. We implement
∗ Mathematics
Subject Classifications: 35K99,35P05,35P99.
of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
‡ Department of Basic Sciences, Higher Technological Institute, Tenth Of Ramadan City, Egypt
† Department
235
236
Traveling Wave Solutions of Nonlinear Evolution Equations
the proposed method for the (2+1)-dimensional Nizhnik-Novikov-Veselov equations
[16]
ut + kuxxx + ruyyy + sux + quy = 3k(uv)x + 3r(uw)y ,
(1)
ux = vy , uy = wx,
and the (1+1)-dimensional Jaulent-Miodek (JM) equations
ut + uxxx + 32 vvxxx + 92 vx vxx − 6uux − 6uvvx − 23 uxv2 = 0,
vt + vxxx − 6ux v − 6uvx − 15
v v2 = 0,
2 x
(2)
where k, r, s and q are arbitrary constants. In the past years, many people studied
the Nizhnik-Novikov-Veselov equations. For instance, Pempinelli et al. [2] solved NNV
equations via the inverse scattering transformation, Zhang et al. [14] and Zhang et
al. [23] obtained the Jacobi elliptic function solution of the NNV equations by the
sinh-cosh method. Lou [9] analyzed the coherent structures of the NNV equation
by separation of variables approach. The coupled system of equations (2) associates
with the JM spectral problem [8], the relation between this system and Euler-Darboux
equation was found by Matsuno [12]. In recent years, much work associated with
the JM spectral problems has been done [24, 25]. Fan [4] has investigated the exact
solution of (2) using the unified algebraic method. Our first interest in the present work
is in implementing the extended tanh method to stress its power in handling nonlinear
equations so that one can apply it to models of various types of nonlinearity such as
(1) and (2).
2
The Extended Tanh Method
Wazwaz has summarized the use of the extended tanh method. A PDE
P (u, ut , ux, uxx, ...) = 0,
(3)
can be converted to the following ODE
Q(U, U 0 , U 00 , U 000, ...) = 0,
(4)
by means of a wave variable ξ = x − βt so that u(x, t) = U (ξ) and using the following
change of variables (in the derivatives)
∂
d
∂
d
∂2
d2
= −β ,
=
,
= 2 , ... .
2
∂t
dξ ∂x
dξ ∂x
dξ
(5)
Eq. (4) is then integrated as long as all terms contain derivatives where integration
constants are considered zeros. Introducing a new independent variable
Y = tanh(ξ),
(6)
leads to a change in the derivatives
d3
dξ 3
d
= (1 − Y 2 ) dY
,
d
d2
= (1 − Y ){−2Y dY + (1 − Y 2 ) dY
2 },
2
d
d3
2
2 d
= (1 − Y ){(6Y − 2) dY − 6Y (1 − Y ) dY 2 + (1 − Y 2 )2 dY
3 },
d2
dξ 2
2
d
dξ
2
(7)
237
E. M. E. Zayed and H. M. Rahman
and the remaining derivatives were derived similarly. The extended tanh method [19]
admits the use of finite expansion
U (ξ) = S(Y ) = a0 +
m
X
[ak Y k + a−k Y −k ],
(8)
k=1
where m is a positive integer which will be determined. The parameter m is usually
obtained by balancing the highest order derivatives with the nonlinear terms in (4).
Substituting (8) into (4) results in algebraic equations in powers of Y , that will lead
to the determination of the parameters ak , (k = 0, 1, 2, 3, ...m), a−k , (k = 1, 2, 3, ..., m)
and β.
3
The (2+1)-Dimensional Nizhnik-Novikov-Veselov
Equations
In order to present some new types of the exact solutions to (1), we use the extended
tanh method. On using the traveling wave transformations
Pm
u(x, y, t) = U (ξ) = a0 + Pk=1 [ak Y k + a−k Y −k ],
v(x, y, t) = V (ξ) = b0 + nk=1 [bk Y k + b−k Y −k ],
(9)
Pl
w(x, y, t) = Z(ξ) = c0 + k=1 [ck Y k + c−k Y −k ],
where ξ = αx + λy − βt, (1) becomes
−βU 0 + kα3 U 000 + rλ3 U 000 + sαU 0 + qλU 0 − 3kα(U V 0 + U 0 V ) − 3rλ(U Z 0 + U 0 Z) = 0,
αU 0 − λV 0 = 0,
λU 0 − αZ 0 = 0.
(10)
Balancing the U 000 term with the U Z 0 in the first equation and U 0 term with V 0 or U 0
term with Z 0 in the third equation in (10) gives
m + 3 = m + n + 1, m + 3 = m + l + 1, m + 1 = n + 1, m + 1 = l + 1,
(11)
so that m = n = l = 2. The extended tanh method admits the use of the finite
expansion
U (ξ) = a0 + a1 Y + a2 Y 2 + aY−1 + aY−22 ,
b
b
(12)
V (ξ) = b0 + b1 Y + b2 Y 2 + Y−1 + Y−22 ,
c−1
c−2
2
Z(ξ) = c0 + c1 Y + c2 Y + Y + Y 2 .
Substituting (12) into (10) and equating the coefficient of the powers of Y to zero, we
obtain the following system of algebraic equations
0 =
−βa1 + 3a1 b0 αk + 3a0 b1 αk + 2a1 α3 k − βa−1 + 3b0 αa−1 + 3b2 αka−1
+2α3 ka−1 + 3a0 λc1 r + 3b1 αka−2 + 3a0 αkb−1 + 3a2 αkb−1
+3a1 αkb−2 − a1 λq − λa−1 q + 2a1 λ3 r + 3a1 λc0 r
+3a0 λc−1 r + 3a2 λc−1 r + 3a1 λc−2 r + 2λ3 a−1 r + 3λc0 a−1 r
238
0 =
Traveling Wave Solutions of Nonlinear Evolution Equations
+3λc2 a−1 r + 3λc1 a−2 r − a1 αs − αa−1 s,
24α3 ka−2 − c−2 αka−2 b−2 + 24λ3 a−2 r − 12λc−2 a−2 r,
0 =
0 =
6α3 ka−1 − 9αka−2b−1 − 9 αka−1 b−2 + 6λ3 a−1 r − 9λc−2 a−1 r − 9λc−1 a−2 r,
2βa−2 − 6b0 αka−2 − 40α3 ka−2 − 6αka−1 b−1 − 6a0 αkb−2
+c−2 αka−2 b−2 + 2λa−2 q − 6a0 λc−2 r − 6λc−1 a−1 r − 40λ3 a−2 r
−6λc0 a−2 r + 12λc−2 a−2 r + 2αa−2 s,
0 =
αa−1 − 3b0 αka−1 − 8α3 ka−1 − 3b1 αka−2 − 3a0 αkb−1 + 9αka−2 b−1
−3a1 αkb−2 + 9αka−1 b−2 + λa−1 q − 3a0 λc−1 r − 3a1 λc−2 r
−8λ3 a−1 r − 3λc0 a−1 r + 9λc−2 a−1 r − 3λc1 a−2 r + 9λc−1 a−2 r + αa−1 s,
0 =
−2βa−2 + 6b0 αka−2 + 16α3 ka−2 + 6αka−1 b−1 + 6a0 αkb−2 − 2λa−2 q
+6a0 λc−2 r + 6λc1 a−1 r + 16λ3 a−2 r + 6λc0 a−2 r − 2αa−2 s,
−2βa2 + 6a2 b0 αk + 6a1 b1 αk + 6a0 b2 αk + 16a2 α3 k − 2a2 λq
+16a2 λ3 r + 6a2 λc0 r + 6a1 λc1 r + 6a0 λc2 r − 2a2 αs,
0 =
0 =
βc−1 − 3a1 b0 αk − 3a0 b1 αk + 9a2 b1 αk + 9a1 b2 αk − 8a1 α3 k
−3b2 c2 ka−1 − 3a2 b−1 + a1 λq − 8a1 λ3 r − 3a1 λc0 r − 3a0 λc1 r
+9a2 λc1 r + 9a1 λc2 r − 3a2 λl1 r − 3λc2 a−1 r + a1 αs,
2βa2 − 6a2 b0 αk − 6a1 b1 αk − 6a0 b2 αk + 12a2 b2 αk − 40a2 α3 k
0 =
0 =
+2a2 λq − 40a2 λ3 r − 6a2 λc0 r − 6a1 λc1 r − 6a0 λc2 r + 12a2 λc2 r + 2a2 αs,
−9a2 b1 αk − 9a1 b2 αk + 6a1 α3 k + 6a1 λ3 r − 9a2 λc1 r − 9a1 λc2 r,
−12a2 b2 αk + 24a2 α3 k + 24a2 b3 r − 12a2 λc2 r, −λb1 + a1 α + αa−1 − λb−1 ,
0 =
0
0
0
0
=
=
=
=
−2αa−2 + 2λb−2 ,
αc−1 − λa−1 ,
−2αc−2 + 2λa−2 ,
−2λb2 + 2a2 α,
0 =
0 =
0 =
2λb2 − 2a2 α
−2a2 λ + 2αc2 ,
−a1 λ + αc1 ,
0
0
0
0
=
=
=
=
λb1 − a1 α,
2a2 λ − 2αc2 ,
−2a2 λ + 2αc2 ,
−a1 λ + αc1 ,
0
0
0
0
=
=
=
=
λb1 − a1 α,
2a2 λ − 2αc2 ,
−2αc−2 + 2λa−2
αc−1 − λ a−1 ,
0 =
2αc−2 − 2λa−2 ,
E. M. E. Zayed and H. M. Rahman
0 =
239
a1 λ − αc1 − αc−1 + λa−1 ,
These algebraic equations can be solved by Mathematica and give the following sets
of solutions. The first set is
b2 = b−2 = c−2 = c2 = a−2 = a2 = 0,
1
b0 =
{β + λq + αs − 3λc0 r},
3αk
λ2 a−1 r
a1 λ2 r
a1 λ
λa−1
b−1 =
,
b
=
, c1 =
, c−1 =
.
1
kα2
kα2
α
α
The second set is
b2 = b−2 = c−2 = c2 = b1 = c1 = a−2 = a2 = a1 = 0,
1
b0 =
{β + λq + αs − 3λc0 r},
3αk
λa−1
λ2 a−1 r
, c−1 =
.
b−1 =
α2 k
α
The third set is
b−1 = b1 = c1 = c−1 = a−1 = a1 = 0,
1
b0 =
{βλα4 k − 3a0 α6 k 2 − 8λα7 k 2 + λ2 α4 kq
2
3
λα k(3α k + 3λ3 r)
+βλ4 αr − 6 a0 λ3 α3 kr − 16λ4 α4 kr − 3λ2 α4 c0 kr + λ5 αqr − 3a0 λ6 r 2
−8λ7 αr 2 − 3λ5 αc0 r 2 + λα5 ks + λ4 α2 rs},
b−2 = 2α2 , b2 = 2α2 , c2 = 2λ2 , c−2 = 2λ2 , a2 = 2αλ, a−2 = 2αλ.
The fourth set is
b2 = c2 = b−1 = b1 = c1 = c−1 = 0 = a2 = a−1 = a1 = 0,
1
b0 =
{βλα4 k − 3a0 α6 k 2 − 8λα7 k 2 + λ2 α4 kq
λα2 k(3α3 k + 3λ3 r)
+βλ4 αr − 6 a0 λ3 α3 kr − 16λ4 α4 kr − 3λ2 α4 c0 kr + λ5 αqr − 3a0 λ6 r 2
−8λ7 αr 2 − 3λ5 αc0 r 2 + λα5 ks + λ4 α2 rs},
b−2 = 2α2 , c−2 = 2λ2 , a−2 = 2αλ.
The fifth set is
b−2 = c−2 = b−1 = b1 = c1 = c−1 = a−2 = a−1 = a1 = 0,
1
b0 =
{βλα4 k − 3a0 α6 k 2 − 8λα7 k 2 + λ2 α4 kq
2
3
λα k(3α k + 3λ3 r)
+βλ4 αr − 6 a0 λ3 α3 kr − 16λ4 α4 kr − 3λ2 α4 c0 kr + λ5 αqr − 3a0 λ6 r 2
−8λ7 αr 2 − 3λ5 αc0 r 2 + λα5 ks + λ4 α2 rs},
b2 = 2α2 , c2 = 2λ2 , a2 = 2λα.
240
Traveling Wave Solutions of Nonlinear Evolution Equations
In view of these we obtain the following kinds of solutions
u1 (x, y, t) = a0 + a1 tanh ξ + a−1 coth ξ,
1
a1 λ2 r
λ2 a−1 r
v1 (x, y, t) =
{β + λq + αs − 3λc0 r} +
tanh ξ +
coth ξ,
2
3αk
kα
kα2
a1 λ
λa−1
tanh ξ +
coth ξ,
w1 (x, y, t) = c0 +
α
α
u2 (x, y, t) = a−1 coth ξ,
1
λ2 a−1 r
v2 (x, y, t) =
{β + λq + αs − 3λc0 r} +
coth ξ,
3αk
α2 k
λa−1
w2 (x, y, t) = c0 +
coth ξ,
α
u3 (x, y, t) = a0 + 2λα{tanh2 ξ + coth2 ξ},
v3 (x, y, t) = b0 + 2α2 {tanh2 ξ + coth2 ξ},
w3 (x, y, t) = c0 + 2λ2 {tanh2 ξ + coth2 ξ},
u4 (x, y, t) = a0 + 2λα coth2 ξ,
v4 (x, y, t) = b0 + 2α2 coth2 ξ ,
w4 (x, y, t) = c0 + 2λ2 coth2 ξ,
and
u5 (x, y, t) = a0 + 2λα tanh2 ξ,
v5 (x, y, t) = b0 + 2α2 tanh2 ξ,
w5 (x, y, t) = c0 + 2λ2 tanh2 ξ,
where ξ = αx + λy − βt, a0 , a1 , a−1 and c0 are arbitrary constants, b0 defined in the
fifth set.
4
The (1+1)-Dimensional Jaulent-Miodek (JM) Equations
In this section, we will use the extended tanh method to handle (2). Let
Pm
u(x, t) = U (ξ) = a0 + Pk=1 [ak Y k + a−k Y −k ],
n
v(x, t) = V (ξ) = b0 + k=1 [bk Y k + b−k Y −k ],
(13)
where ξ = α(x + βt). Then (2) becomes
3
3
αβU 0 + α3 U 000 + 3α2 V V 000 + 9α2 V 0 V 00 − 6αU U 0 − 6αU V V 0 − 3α
U 0 V 2 = 0,
2
0 2
αβV 0 + α3 V 000 − 6αU 0 V − 6αU V 0 − 15α
V
V
=
0.
2
(14)
241
E. M. E. Zayed and H. M. Rahman
Balancing the highest derivatives term with highest nonlinear terms in (14) gives
m + 3 = 2n + 3 ⇒ m = 2n, n + 3 = 3n + 1,
(15)
so that m = 2, n = 1. The extended tanh method admits the use of the finite expansion
a
U (ξ) = a0 + a1 Y + a2 Y 2 + Y−1 +
V (ξ) = b0 + b1 Y + bY−1 .
a−2
Y2 ,
(16)
Substituting (16) into (14) and equating the coefficient of the powers of Y to zero, we
obtain the following system of algebraic equations
0 =
3
βa1 α − 6a0 a1 α − a1 b20 α − 6a0 b0 b1 α − 2a1 α3 − 3b0 b1 α3 + βαa−1 − 6a0 αa−1
2
3 2
9
−6a2 αa−1 − b0 αa−1 − b21 αa−1 − 2α3 a−1 − 6a1 αa−2 − 6a0 b0 αb−1
2
2
9
3
−3a1 b1 αb−1 − 3b0 α b−1 − 3b1 αa−1 b−1 − a1 αb2−1
2
−24α3 a−2 + 12αa2−2 − 18α3 b2−1 + 9αa−2 b2−1 ,
0 =
−6α3 a−1 + 18αa−1a−2 − 9b0 α3 b−1 + 12b0 αa−2 b−1 +
0 =
0 =
15αa−1 b2−1
,
2
2
2
3
2
6αa−1 − 2a1 αa−2 + 12a0 αa−2 + 3b0 αa−2 + 40α a−2 − 12αa−2
+9b0 αa−1 b−1 + 6b1 αa−2 b−1 + 6a0 αb2−1 + 30 α3 b2−1 − 9αa−2 b2−1 ,
0 =
0 =
0 =
0 =
0 =
0 =
0 =
0 =
3b20 αa−1
+ 8α3 a−1 + 6a1 αa−2 − 18αa−1a−2 + 6a0 b0 αb−1
2
15αa−1 b2−1
9a1 αb2−1
−
,
+12b0 α3 b−1 + 3b1 αa−1 b−1 − 12 b0 αa−2 b−1 +
2
2
−3b0 b1 αa−1 − 6αa2−1 + 2βαa−2 − 12a0 αa−2 − 3 b20 αa−2 − 3 b21 αa−2 − 16 α3 a−2
+3 a1 b0 αb−1 − 9 b0 αa−1 b−1 − 6b1 αa−2 b−1 − 6 a0 αb2−1 + 3 a2 αb2−1 − 12α3b2−1 ,
−6a21 α + 2βa2 α − 12 a0 a2 α − 3 a2 b20 α − 9 a1 b0 b1 α − 6 a0 b21 α − 16 a2 α3
−12 b21 α3 + 3 b2 b1 αa−1 + 3b21 αa−2 − 3a1 b0 αb−1 − 6a2 b1 αb−1 − 3a2 αb2−1 ,
3
−βa1 α + 6 a0 a1 α − 18a1 a2 α + a1 b20 α + 6a0 b0 b1 α − 12a2 b0 b1 α
2
15
9
2
3
3
− a1 b1 α + 8a1 α + 12b0 b1 α + 6a2 αa−1 + b21 αa−1 + 3a1 b1 αb−1 ,
2
2
6a21 α − 2βa2 α + 12a0 a2 α − 12a22 α + 3a2 b20 α + 9a1 b0 b1 α
−βαa−1 + 6 a0 αa−1 +
+6 a0 b21 α − 9 a2 b21 α + 40a2 α3 + 30 b21 α3 + 6a2 b1 αb−1 ,
15
18a1 a2 α + 12a2 b0 b1 α + a1 b21 α − 6a1 α3 − 9b0 b1 α3 ,
2
12a22 α + 9a2 b21 α − 24a2 α3 − 18b21 α3 ,
15
−6a1 b0 α + βb1 α − 6a0 b1 α − b20 b1 α − 2b1 α3 − 6b0 αa−1 − 6 b1 αa−2 + αβb−1
2
15 2
15
15
−6a0 αb−1 − 6a2 αb−1 − b0 αb−1 − b21 αb−1 − 2α3 b−1 − b1 αb2−1 ,
2
2
2
242
Traveling Wave Solutions of Nonlinear Evolution Equations
0 =
0 =
15αb3−1
,
2
12b0 αa−2 + 12αa−1 b−1 + 15b0 αb2−1 = 0, 6b0 αa−1 + 6b1 αa−2 − βαb−1 + 6a0 αb−1
−6α3 b−1 + 18αa−2b−1 +
15b1 αb2−1
15αb3−1
15b20 αb−1
+ 8α3 b−1 − 18αb−2 b−1 +
−
,
2
2
2
−12b0 αa−2 − 12αa−1 b−1 − 15b0 αb2−1 = 0, −12a2 b0 α − 12 a1 b1 α − −15b0 b21 α,
15
15
6a1 b0 α − αb1 α + 6a0 b1 α − 18a21 α + b20 b1 α − b31 α
2
2
15 2
3
+8 b1 α + 6a2 αb−1 + b1 αb−1 ,
2
12a2 b0 α + 12a1 b1 α + 15b0 b21 α,
15
18a2 b1 α + b31 α − 6b1 α3 .
2
+
0 =
0 =
0 =
0 =
These algebraic equations can be solved by Mathematica to yield the following sets
of solutions. The first set is
a1 = a−1 = b0 = a0 = 0, a2 = a−2 = 2α2 , b1 = b−1 = −2iα, β = −16α2 .
The second set is
a1 = a−1 = b0 = b1 = a2 = 0, a−2 = 2α2 , b−1 = −2iα, a0 = −α2 , β = −4α2 .
The third set is
b0 = a1 = a−1 = 0, b1 = b−1 = −iα, = a0 =
3α
α2
, a2 = a−2 =
, β = −4α2 .
2
4
The fourth set is
a−1 = a−2 = b−1 = 0, a0 =
β=
−1 b20
ib0 α
3α
( + α2 ), a1 =
, b1 = −iα, a2 =
,
2 2
2
4
1 2
(6b − 2α2 ).
2 0
The fifth set is
b0 = a1 = a−1 = 0, a0 = −2α2 , a2 = a−2 = 2α2 , b1 = −2iα, b−1 = 2iα, β = 8α2 .
In view of these we obtain the following kinds of solutions
u1 (x, t) = 2α2 {tanh2 (α[x − 16α2 t]) + coth2 (α[x − 16α2 t])},
v1 (x, t) = −2iα{(tanh(α[x − 16α2 t]) + coth(α[x − 16α2 t])},
u2 (x, t) = −α2 + 2α2 coth2 (α[x − 4α2 t]),
v2 (x, t) = −2iα coth(α[x − 4α2 t]),
E. M. E. Zayed and H. M. Rahman
243
3α
α2
+
{tanh2 (α[x − 4α2 t]) + coth2 (α[x − 4α2 t])},
2
4
v3 (x, t) = −iα{(tanh(α[x − 4α2 t]) + coth(α[x − 4α2 t])},
u3 (x, t) =
−1 b20
ib0 α
1
( + α2 ) +
tanh(α[x + (6b20 − 2α2 )t])
2 2
2
2
3α
1 2
2
2
+
tanh (α[x + (6b0 − 2α )t]),
4
2
1
v4 (x, t) = b0 − iα tanh(α[x + (6b20 − 2α2 )t]),
2
u4 (x, t) =
and
u5 (x, t) = −2α2 + 2α2 {tanh2 (α[x + 8α2 t]) + coth2 (α[x + 8α2 t])},
v5 (x, t) = −2iα{(tanh(α[x + 8α2 t]) − coth(α[x + 8α2 t])},
where b0 is arbitrary constant.
5
Conclusions
In this article, the extended tanh method was applied to give the traveling wave
solutions of the coupled (2+1)-dimensional Nizhnik-Novikov-Veselov and the (1+1)dimensional Jaulent-Miodek (JM) equations. The extended tanh method was successfully used to establish these solutions. Many well know nonlinear wave equations were
handled by this method to show the new solutions compared to the solutions obtained
in [4, 16]. The performance of the extended tanh method is reliable and effective and
gives more solutions. The applied method will be used in further works to establish
entirely new solutions for other kinds of nonlinear wave equations.
Acknowledgment. The authors would like to thank the referees for their comments on this paper.
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