1+1 - Çankaya University Journal of Science and Engineering

1+1 - Çankaya University Journal of Science and Engineering

Çankaya University Journal of Science and Engineering
Volume 8 (2011), No. 2, 205–223
Generalized Jacobi Elliptic Function Method for
Periodic Wave Solutions of SRLW Equation and
(1+1)-Dimensional Dispersive Long Wave
Equation
Yavuz Uğurlu1,∗ , Doğan Kaya2 and İbrahim E. İnan3
1
2
Fırat University, Department of Mathematics, 23119 Elazığ, Turkey
İstanbul Ticaret University, Department of Mathematics, İstanbul, Turkey
3
Fırat University, Faculty of Education, 23119 Elazığ, Turkey
∗
Corresponding author: matematikci [email protected]
Özet. Biz bu çalışmada SRLW ve (1+1)-boyutlu dispersive uzun dalga denkleminin periyodik dalga çözümlerini elde etmek için genelleştirilmiş Jacobi eliptik fonksiyon metodunu
sunarız.
Anahtar Kelimeler. SRLW denklemi, (1+1)-boyutlu dispersive uzun dalga denklemi,
genelleştirilmiş Jakobi eliptik fonksiyon metot, periyodik çözümler, hareket eden dalga
çözümler.
Abstract. We implement the generalized Jacobi elliptic function method with symbolic computation to construct periodic solutions for the symmetric regularized long wave
(SRLW) equation and (1+1)-dimensional dispersive long wave equation.
Keywords. SRLW equation, (1+1)-dimensional dispersive long wave equation, generalized Jacobi elliptic function method, periodic solutions, traveling wave solutions.
1. Introduction
The mathematical modeling of events in nature can be explained by differential
equations. These equations are mathematical models of complex physical occurrences that arise in engineering, chemistry, biology, mechanics and physics. The
theory of nonlinear dispersive wave motion has recently been the subject of much
study. The solutions of nonlinear equations play a crucial role in applied mathematics and physics, because; solutions of nonlinear partial differential equations make a
very significant contribution to people’s knowledge about the nature of physical phenomenon. Furthermore, when an original nonlinear equation is directly calculated,
Received March 24, 2011; accepted October 2, 2011.
ISSN 1309 - 6788 © 2011 Çankaya University
206
Uğurlu et al.
the solution will preserve the actual physical characters of solutions. Therefore, interest in the solution of nonlinear partial differential equations has never decreased.
Because of this interest, many techniques and methods have been developed [1-10].
Recently, there has been intensive study to obtain analytical solutions of nonlinear partial differential equations. Interest has focused on obtaining exact solutions
of nonlinear partial differential equations through using symbolical computer programs, such as Maple, Matlab, and Mathematica that facilitate complex and tedious
algebraic computations. Various methods are presented which are used by scientists
to obtain exact and analytic solutions of nonlinear partial differential equations with
help of symbolical computer programs. Most of these methods are based on finding
the balance term with balancing of the highest order linear and nonlinear term.
These methods can be only applied to nonlinear partial differential equations.
In this study, we analyze the generalized Jacobi elliptic function method [11] and
we obtain periodic wave solutions of SRLW equations [12] and (1+1)-dimensional
dispersive long wave equations [13] by using the generalized Jacobi elliptic function method. Then, we show three-dimensional and two-dimensional periodic wave
graphics for the SRLW equation and (1+1)-dimensional dispersive long wave equation by using a solution of these equations.
A symmetric regularized long wave equation (SRLWE)
utt + uxx + uuxt + ux ut + uxxtt = 0,
(1)
had been investigated in [14-16]. Eq. (1) is used to describe nonlinear ion-acoustic
wave and space-charge waves. This equation is symmetrical with respect to x,
and t. It arises in many nonlinear problems of mathematical physics and applied
mathematics. Periodic wave solutions of SRLW have been given by using the Exp
function method [17] and (G0 /G)-expansion method [18].
The (1+1)-dimensional dispersive long wave equation
ut + uux + vx = 0
1
(2)
vt + ux v + uvx + uxxx = 0
3
has been studied in [4]. Eq. (2) is one of the basic equations of fluid dynamics. v − 1
shows the height of water wave and u denotes the velocity water’s surface along the
x axis. This equation is used to model in the coastal edge waves.
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2. An Analysis of the Method and Applications
Before starting to give a generalized Jacobi elliptic function method [11], we will
give a simple description of the generalized Jacobi elliptic function method. For
doing this, it is considered in a two variables general form of nonlinear PDE
Q(u, ut , ux , uxx , . . .) = 0,
(3)
with u(x, t) = u(ξ), ξ = kx + wt. Eq. (3) converts to nonlinear ODE for u(ξ) as
follows, where k, w are constants.
Q0 (u0 , u00 , u000 , . . .) = 0.
(4)
The solution of the Eq. (4) is supposed in the form
n
X
u(ξ) = a0 +
ai F i (ξ) + bi F −i (ξ) ,
(5)
i=1
where n is a positive integer that can be determined by balancing the highest order
derivate and the highest nonlinear terms in equation, k, w, a0 , ai , bi and ξ can be
determined. Substituting solution (5) into Eq. (4) yields a set of algebraic equations
√
j
for F i
A + BF 2 + CF 4 , (j = 0, 1) and (i = 0, 1, 2, . . .) then, all coefficients of
√
j
Fi
A + BF 2 + CF 4 have to vanish. After this separated algebraic equation, we
can find coefficients and k, w, a0 , ai , bi and ξ.
In this work, we study the solution of the SRLW equation and (1+1)-dimensional dispersive long wave equation by using the generalized Jacobi elliptic function method
which is introduced by [11]. The fundamental idea of their method is to take full advantage of the elliptic equation and use its solutions F . The desired elliptic equation
is given as
F 02 = A + BF 2 + CF 4 ,
(6)
dF
where F 0 =
and A, B, C are constants. Some solutions of Eq. (6) are given in
dξ
paper [11].
Example 1. Let’s consider SRLW equation,
utt + uxx + uuxt + ux ut + uxxtt = 0,
(7)
with u(x, t) = u(ξ), ξ = kx+wt, and then Eq. (7) converts to an ordinary differential
equation as follows
w2 u00 + k 2 u00 + kwuu00 + kw(u0 )2 + w2 k 2 u(4) = 0,
(8)
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Uğurlu et al.
and integrating (8) yields, we find the following equation
w2 u0 + k 2 u0 + kwuu0 + k 2 w2 u000 = 0,
(9)
where the integration constant is taken as zero. The balance term is obtained as
n = 2 by balancing uu0 with u000 in Eq. (9). Therefore, we may choose to give the
solution of Eq. (9) as follows
b1
b2
+ 2.
(10)
F
F
Substituting (10) into Eq. (9) yields a set of algebraic equations for a0 , a1 , a2 , b1 , b2 , k, w.
The algebraic equation system is obtained as
u = a0 + a1 F + a2 F 2 +
a1 k 2 + a0 a1 kw + a2 b1 kw + a1 w2 + a1 Bk 2 w2 = 0,
−2b22 kB − 24Ab2 k 2 w2 = 0,
−3b1 b2 kw − 6Ab1 k 2 w2 − 2b2 k 2 − b21 kw − 2a0 b2 kw − 2b2 w2 − 8Bb2 k 2 w2 = 0,
−b1 k 2 − a0 b1 kw − a1 b2 kw − b1 w2 − Bb1 k 2 w2 = 0, (11)
2a2 k 2 + a21 kw + 2a0 a2 kw + 2a2 w2 + 8a2 Bk 2 w2 = 0,
3a1 a2 kw + 6a1 Ck 2 w2 = 0,
2a22 kw + 24a2 Ck 2 w2 = 0.
If the system of algebraic equations is solved with the help of Mathematica, we have
w
k
− − 4Bkw, a1 = 0, a2 = −12Ckw,
w
k
b1 = 0, b2 = −12Akw, k 6= 0, w 6= 0.
a0 = −
(12)
Substituting (12) into (10), we have obtained analytic solutions of equation (7) as
follows
(i) A = 1, B = −(1 + m2 ), C = m2 ,
k
w
1
2
u1 = − − − 4Bkw − 12Ckw sn (kx + wt) − 12Akw
. (13)
w
k
sn2 (kx + wt)
2
2
k
w
cn(kx + wt)
dn(kx + wt)
u2 = − − − 4Bkw−12Ckw
−12Akw
. (14)
w
k
dn(kx + wt)
cn(kx + wt)
(ii) A = 1 − m2 , B = 2m2 − 1, C = −m2 ,
k
w
u3 = − − − 4Bkw − 12Ckw cn2 (kx + wt) − 12Akw
w
k
1
2
cn (kx + wt)
. (15)
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(iii) A = m2 − 1, B = 2 − m2 , C = −1,
k
w
u4 = − − − 4Bkw − 12Ckw dn2 (kx + wt) − 12Akw
w
k
1
2
dn (kx + wt)
. (16)
(iv) A = −m2 (1 − m2 ), B = 2m2 − 1, C = 1,
2
2
k w
dn(kx + wt)
sn(kx + wt)
u5 = − − − 4Bkw − 12Ckw
− 12Akw
. (17)
w k
sn(kx + wt)
dn(kx + wt)
(v) A = 1 − m2 , B = 2 − m2 , C = 1,
2
2
k w
cn(kx + wt)
sn(kx + wt)
u6 = − − − 4Bkw − 12Ckw
− 12Akw
. (18)
w k
sn(kx + wt)
cn(kx + wt)
m2 − 2
m2
1
,C=
,
(vi) A = , B =
4
2
4
2
k
w
sn(kx + wt)
u7 = − − − 4Bkw − 12Ckw
w
k
1 ± dn(kx + wt)
2
1 ± dn(kx + wt)
− 12Akw
(19)
sn(kx + wt)
(vii) A =
u8 = −
m2
m2 − 2
m2
,B=
,C=
,
4
2
4
k
w
− − 4Bkw − 12Ckw(sn(kx + wt) ± i cn(kx + wt))2
w
k
2
1
− 12Akw
. (20)
sn(kx + wt) ± i cn(kx + wt)
2
w
dn(kx + wt)
k
u9 = − − − 4Bkw − 12Ckw √
w
k
i 1 − m2 sn(kx + wt) ± cn(kx + wt)
√
2
i 1 − m2 sn(kx + wt) ± cn(kx + wt)
− 12Akw
. (21)
dn(kx + wt)
1
1 − 2m2
1
(viii) A = , B =
,C= ,
4
2
4
u10 = −
k
w
− − 4Bkw − 12Ckw(m sn(kx + wt) ± i dn(kx + wt))2
w
k
2
1
− 12Akw
. (22)
m sn(kx + wt) ± i dn(kx + wt)
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Uğurlu et al.
2
dn(kx + wt)
√
m cn(kx + wt) ± i 1 − m2
√
2
m cn(kx + wt) ± i 1 − m2
. (23)
− 12Akw
dn(kx + wt)
u11
k
w
= − − − 4Bkw − 12Ckw
w
k
u12
k
w
= − − − 4Bkw − 12Ckw
w
k
sn(kx + wt)
1 ± cn(kx + wt)
2
− 12Akw
u13
1 ± cn(kx + wt)
sn(kx + wt)
2
. (24)
2
cn(kx + wt)
√
1 − m2 sn(kx + wt) ± dn(kx + wt)
√
2
1 − m2 sn(kx + wt) ± dn(kx + wt)
(25)
− 12Akw
cn(kx + wt)
k
w
= − − − 4Bkw − 12Ckw
w
k
m2 − 1
m2 + 1
m2 − 1
,B=
,C=
,
4
2
4
2
k
w
dn(kx + wt)
= − − − 4Bkw − 12Ckw
w
k
1 ± m sn(kx + wt)
2
1 ± m sn(kx + wt)
. (26)
− 12Akw
dn(kx + wt)
(ix) A =
u14
1 − m2
m2 + 1
1 − m2
(x) A =
,B=
,C=
,
4
2
4
2
k
w
cn(kx + wt)
u15 = − − − 4Bkw − 12Ckw
w
k
1 ± sn(kx + wt)
2
1 ± sn(kx + wt)
. (27)
− 12Akw
cn(kx + wt)
(xi) A = −
u16 = −
(1 − m2 )2
m2 + 1
1
,B=
,C=− ,
4
2
4
k
w
− − 4Bkw − 12Ckw(m cn(kx + wt) ± dn(kx + wt))2
w
k
2
1
− 12Akw
. (28)
m cn(kx + wt) ± dn(kx + wt)
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1
m2 + 1
(1 − m2 )2
(xii) A = , B =
,C=
,
4
2
4
2
k
w
sn(kx + wt)
u17 = − − − 4Bkw − 12Ckw
w
k
dn(kx + wt) ± cn(kx + wt)
2
dn(kx + wt) ± cn(kx + wt)
. (29)
− 12Akw
sn(kx + wt)
1
m2 − 2
m2
(xiii) A = , B =
,C=
,
4
2
4
u18
k
w
= − − − 4Bkw − 12Ckw
w
k
2
cn(kx + wt)
√
1 − m2 ± dn(kx + wt)
√
2
1 − m2 ± dn(kx + wt)
− 12Akw
. (30)
cn(kx + wt)
Remark. Here sn(ξ, m), cn(ξ, m), dn(ξ, m) are Jacobi elliptic functions and m
shows the modulus of the Jacobi elliptic functions.
If m → 1, then sn ξ → tanh ξ, cn ξ → sech ξ, dn ξ → sech ξ. If m → 0, then
sn ξ → sin ξ, cn ξ → cos ξ, dn ξ → 1.
We can obtain the following periodic solutions of (7) by using solutions (13-30) for
m → 0,
k
w
k
u(x, t) = −
w
k
u(x, t) = −
w
k
u(x, t) = −
w
u(x, t) = −
w
k
w
−
k
w
−
k
w
−
k
−
1
,
sin (kx + wt)
1
+ 4kw − 12kw 2
,
cos (kx + wt)
+ 4kw − 12kw
2
− 8kw − 12kw(cot2 (kx + wt) + tan2 (kx + wt)),
sin2 (kx + wt)
(1 ± cos(kx + wt))2
− 2kw − 3kw
+
,
(1 ± cos(kx + wt))2
sin2 (kx + wt)
k
w
cos2 (kx + wt)
(sin(kx + wt) ± 1)2
u(x, t) = − − − 2kw − 3kw
+
.
w
k
(sin(kx + wt) ± 1)2
cos2 (kx + wt)
(31)
(32)
(33)
(34)
(35)
In Figure 1 and Figure 2 are shown graphics of periodic wave solutions of the SRLW
equation in three and two dimensions, respectively. In Figure 2, the periodic waves
move to the left with time. If ξ = kx − wt, the periodic waves move to the right.
212
Uğurlu et al.
Figure 1. Periodic wave graphic of the SRLW equation for solution
(32) in three dimensions, k = 1, w = 0.5.
(a)
(b)
(c)
(d)
Figure 2. Periodic wave graphic of the SRLW equation for solution
(32) in two dimensions (a) t = 0, (b) t = 0.5, (c) t = 1, (d) t = 2,
(k = 1, w = 0.5).
CUJSE
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Example 2. Let’s consider (1+1)-dimensional dispersive long wave equation
ut + uux + vx = 0,
1
vt + ux v + uvx + uxxx = 0,
(36)
3
with u(x, t) = u(ξ), ξ = kx + wt, and then Eq. (36) converts to an ordinary
differential equation as follows
wu0 + kuu0 + kv 0 = 0,
1
wv 0 + k(uv)0 + k 3 u000 = 0,
3
and integrating (37) yields, we find the following equation
(37)
k
wu + u2 + kv = 0,
2
1
wv + kuv + k 3 u00 = 0,
(38)
3
where the integration constant is taken as zero. Balance terms are found as n1 = 1,
n2 = 2 respectively, by balancing v with y 2 and uv with u00 in Eq. (38). Therefore,
we may choose


b


 u = a0 + a1 F + 1 ,
F
(39)


d
d

 v = c0 + c1 F + 1 + c2 F 2 + 2 .
F
F2
Substituting (39) into Eq. (38) yields a set of algebraic equations for a0 , a1 , b1 , c0 ,
c1 , c2 , d1 , d2 , k, and w. The algebraic equation system is obtained as
b1 d2 k +
2Ab1 k 3
= 0,
3
b1 d1 k + a0 d2 k + d2 w = 0,
a1 c1 k + a0 c2 k + c2 w = 0,
2
a1 c2 k + a1 Ck 3 = 0,
3
b21 k
+ d2 k = 0,
2
a0 a1 k + c1 k + a1 w = 0,
where C 6= 0, k 6= 0.
a0 c0 k + b1 c1 k + a1 d1 k + c0 w = 0,
Bb1 k 3
b1 c0 k + a0 d1 k + a1 d2 k +
+ d1 w = 0,
3
1
a1 c0 k + a0 c1 k + b1 c2 k + a1 Bk 3 + c1 w = 0,
3
2
a0 k
+ a1 b1 k + c0 k + a0 w = 0,
2
a0 b1 k + d1 k + b1 w = 0,
1 2
a k + c2 k = 0,
2 1
(40)
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Uğurlu et al.
From the solutions of the above system, we can find
√
√
√ √
2Bk 2
2
Ck
2
Ak
a0 = − −
− 4 A Ck 2 , a1 = − √ , b1 = √ ,
3
3
3
√
√
1
2Ck 2
c0 = (−Bk 2 − 2 A Ck 2 ),
c1 = 0,
c2 = −
,
(41)
3
3
r
√ √
2Ak 2
2Bk 2
d1 = 0,
d2 = −
, w=k −
− 4 A Ck 2 .
3
3
r
Substituting (41) into (39), we have obtained analytic solutions of equation (36) as
follows
r
√ √
2Bk 2
− 4 A Ck 2 t,
(i) A = 1, B = −(1 + m2 ), C = m2 , ξ = kx + k −
3

r
√
√

2
√ √

2Bk
2
Ck
Ak
2
1


− 4 A Ck 2 − √ sn(ξ) + √
 u1 = − −
3
sn(ξ)
3
3
(42)
2

2
2
√
√

1
2Ck
2Ak
1


−Bk 2 − 2 A Ck 2 −
sn2 (ξ) −
 v1 =
3
3
3
sn(ξ)

r
√
√

2
√ √

Ck
Ak
2Bk
2
2
cn(ξ)
dn(ξ)


− 4 A Ck 2 − √
+ √
 u2 = − −
3
dn(ξ)
cn(ξ)
3
3
(43)
2

2
2
√
√

1
2Ck
cn(ξ)
2Ak
dn(ξ)


−Bk 2 − 2 A Ck 2 −
−
 v2 =
3
3
dn(ξ)
3
cn(ξ)
r
√ √
2Bk 2
− 4 A Ck 2 t,
(ii) A = (1 − m2 ), B = 2m2 − 1, C = −m2 , ξ = kx + k −
3

r
√
√

√ √

2Bk 2
2 Ck
2 Ak
1

2

− 4 A Ck − √ cn(ξ) + √
 u3 = − −
3
cn(ξ)
3
3
(44)
2

2Ck 2
2
√
√

2Ak
1
1


−Bk 2 − 2 A Ck 2 −
cn2 (ξ) −
 v3 =
3
3
3
cn(ξ)
r
√ √
2Bk 2
(iii) A = (m2 − 1), B = (2 − m2 ), C = −1, ξ = kx + k −
− 4 A Ck 2 t,
3

r
√
√

2
√ √

2Bk
2
Ck
2
Ak
1


− 4 A Ck 2 − √ dn(ξ) + √
 u4 = − −
3
dn(ξ)
3
3
(45)
2

2
2
√
√

1
2Ck
2Ak
1


−Bk 2 − 2 A Ck 2 −
dn2 (ξ) −
 v4 =
3
3
3
dn(ξ)
CUJSE
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r
√ √
2Bk 2
(iv) A = −m2 (1 − m2 ), B = 2m2 − 1, C = 1, ξ = kx + k −
− 4 A Ck 2 t,
3

r
√
√

2
√ √

dn(ξ)
2
sn(ξ)
2Bk
2
Ck
Ak


− 4 A Ck 2 − √
+ √
 u5 = − −
3
sn(ξ)
dn(ξ)
3
3
(46)
2
2

2
2
√
√

1
dn(ξ)
2Ak
sn(ξ)
2Ck


−Bk 2 − 2 A Ck 2 −
−
 v5 =
3
3
sn(ξ)
3
dn(ξ)
r
√ √
2Bk 2
(v) A = (1 − m2 ), B = (2 − m2 ), C = 1, ξ = kx + k −
− 4 A Ck 2 t,
3

r
√
√

√ √

2Bk 2
2 Ck cn(ξ)
2 Ak sn(ξ)

2

− 4 A Ck − √
+ √
 u6 = − −
3
sn(ξ)
cn(ξ)
3
3
(47)
2
2

2Ck 2 cn(ξ)
2
√
√

1
sn(ξ)
2Ak


−Bk 2 − 2 A Ck 2 −
−
 v6 =
3
3
sn(ξ)
3
cn(ξ)
r
√ √
m2 − 2
m2
2Bk 2
1
,C=
, ξ = kx + k −
− 4 A Ck 2 t,
(vi) A = , B =
4
2
4
3

r
√
√

2
√ √

2Bk
2
Ck
sn(ξ)
2
Ak
1
±
dn(ξ)

2

− 4 A Ck − √
+ √
 u7 = − −
3
1 ± dn(ξ)
sn(ξ)
3
3

2
√ √ 2

1
2Ck 2
sn(ξ)
2Ak 2 1 ± dn(ξ) 2

2

v
=
A
Ck
−
−Bk
−
2
−
 7
3
3
1 ± dn(ξ)
3
sn(ξ)
m2
m2 − 2
(vii) A = C =
,B=
, ξ = kx + k
4
2
r
−
(48)
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√

2
√ √

2Bk
2
Ck


u8 = − −
− 4 A Ck 2 − √ (sn(ξ) ± i cn(ξ))



3
3


√



2
Ak
1


√
+

(sn(ξ) ± i cn(ξ))
3

√ √ 2 2Ck 2

1

2

v
=
−Bk
−
2
A Ck −
(sn(ξ) ± i cn(ξ))2

8


3
3



2


2Ak 2
1


−

3
(sn(ξ) ± i cn(ξ))
(49)
216
Uğurlu et al.

r
√

√ √

2Bk 2
2 Ck
dn(ξ)

2

√
− 4 A Ck − √
u9 = − −


3

3
i 1 − m2 sn(ξ) ± cn(ξ)


!
√
√


2 sn(ξ) ± cn(ξ)

2
Ak
i
1
−
m


+ √


dn(ξ)
3
2

√ √ 2 2Ck 2

1
dn(ξ)

2

√
v9 =
−Bk − 2 A Ck −



3
3
i
1 − m2 sn(ξ) ± cn(ξ)


!2

√

2

2 sn(ξ) ± cn(ξ)

1
−
m
i
2Ak


−

3
dn(ξ)
1 − 2m2
1
, ξ = kx + k
(viii) A = C = , B =
4
2
r
−
(50)
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√

√ √

2Bk 2
2 Ck

2

− 4 A Ck − √ (m sn(ξ) ± i dn(ξ))
u10 = − −



3
3


√



2 Ak
1


+ √

(m sn(ξ) ± i dn(ξ))
3

2
√ √

2Ck
1


v10 =
−Bk 2 − 2 A Ck 2 −
(m sn(ξ) ± i dn(ξ))2



3
3



2


2Ak 2
1


−

3
(m sn(ξ) ± i dn(ξ))
(51)

r
√

√ √

2Bk 2
2
Ck
dn(ξ)

2− √

√
u
=
−
−
4
−
A
Ck
11


3

3
m cn(ξ) ± i 1 − m2


!
√
√


2

2
Ak
m
cn(ξ)
±
i
1
−
m


+ √


dn(ξ)
3
2

√ √ 2 2Ck 2

1
dn(ξ)

2

√
v11 =
−Bk − 2 A Ck −



3
3
m cn(ξ) ± i 1 − m2


!2

√



2Ak 2 m cn(ξ) ± i 1 − m2


−

3
dn(ξ)
(52)

r
√
√

2
√ √

2Bk
2
Ck
sn(ξ)
2
Ak 1 ± cn(ξ)

2

− 4 A Ck − √
+ √
 u12 = − −
3
1 ± cn(ξ)
sn(ξ)
3
3
2

2
2
√
√

1
2Ck
sn(ξ)
2Ak
1 ± cn(ξ) 2

2
2

−Bk − 2 A Ck −
−
 v12 =
3
3
1 ± cn(ξ)
3
sn(ξ)
(53)
CUJSE
8 (2011), No. 2
217

r
√

√ √

2Bk 2
2 Ck
cn(ξ)

2

√
− 4 A Ck − √
u13 = − −


3

3
1 − m2 sn(ξ) ± dn(ξ)


!
√
√


2 sn(ξ) ± dn(ξ)

2
Ak
1
−
m


+ √


cn(ξ)
3
2

√ √ 2 2Ck 2

1
cn(ξ)

2

√
v13 =
−Bk − 2 A Ck −



3
3
1 − m2 sn(ξ) ± dn(ξ)


!2

√

2

2 sn(ξ) ± dn(ξ)

1
−
m
2Ak


−

3
cn(ξ)
m2 + 1
m2 − 1
,B=
, ξ = kx + k
(ix) A = C =
4
2
r
−
(54)
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√
√

2
√ √

2Bk
2
Ck
dn(ξ)
2
Ak
1
±
m
sn(ξ)


− 4 A Ck 2 − √
+ √
 u14 = − −
3
1 ± m sn(ξ)
dn(ξ)
3
3

2
√ √ 2 2Ck 2

dn(ξ)
1
2Ak 2 1 ± m sn(ξ) 2

2

−Bk − 2 A Ck −
−
 v14 =
3
3
1 ± m sn(ξ)
3
dn(ξ)
(55)
r
(x) A = C =
1 − m2
1 + m2
,B=
, ξ = kx + k
4
2
−
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√
√

√ √

2Bk 2
2 Ck
cn(ξ)
2 Ak 1 ± sn(ξ)

2

− 4 A Ck − √
+ √
 u15 = − −
3
1 ± sn(ξ)
cn(ξ)
3
3
2

2
2
√
√

1
cn(ξ)
2Ak
1 ± sn(ξ) 2
2Ck

2
2

−Bk − 2 A Ck −
−
 v15 =
3
3
1 ± sn(ξ)
3
cn(ξ)
(1 − m2 )2
1 + m2
1
(xi) A = −
,B=
, C = − , ξ = kx + k
4
2
4
r
−
(56)
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√

2
√ √

2Bk
2
Ck


u16 = − −
− 4 A Ck 2 − √ (m cn(ξ) ± dn(ξ))



3
3


√



2
Ak
1


√
+

(m cn(ξ) ± dn(ξ))
3

√ √ 2 2Ck 2

1

2

v
=
−Bk
−
2
A Ck −
(m cn(ξ) ± dn(ξ))2

16


3
3



2


2Ak 2
1


−

3
(m cn(ξ) ± dn(ξ))
(57)
218
Uğurlu et al.
1
1 + m2
(1 − m2 )2
(xii) A = , B =
,C=
, ξ = kx + k
4
2
4
r
−
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√

√ √

2Bk 2
2 Ck
sn(ξ)

2

− 4 A Ck − √
u17 = − −



3
dn(ξ) ± cn(ξ)
3


√



2 Ak dn(ξ) ± cn(ξ)


+ √

sn(ξ)
3
2

2
√ √

2Ck
sn(ξ)
1


−Bk 2 − 2 A Ck 2 −
v17 =



3
3
dn(ξ) ± cn(ξ)





2Ak 2 dn(ξ) ± cn(ξ) 2


−

3
sn(ξ)
1
m2 − 2
m2
(xiii) A = , B =
,C=
, ξ = kx + k
4
2
4
r
−
(58)
√ √
2Bk 2
− 4 A Ck 2 t,
3

r
√

√ √

2Bk 2
2
Ck
cn(ξ)

2

√
− 4 A Ck − √
u18 = − −


3

3
1 − m2 ± dn(ξ)


!
√
√


2 ± dn(ξ)

Ak
1
−
m
2


+ √


cn(ξ)
3
2

√ √ 2 2Ck 2

cn(ξ)
1

2

√
−Bk − 2 A Ck −
v18 =



3
3
1 − m2 ± dn(ξ)


!2

√

2

2 ± dn(ξ)

1
−
m
2Ak


−

3
cn(ξ)
(59)
We can obtain the following periodic solutions of Eq. (36) by using solutions (42-59)
for m → 0,

r

2k 2
2k
1



u(x,
t)
=
−
−√
q


3
2
3

2k

sin kx + k
t

3


k 2 2k 2
1


v(x,
t)
=
−
2

q

3
3

2

2k

sin kx + k
t
3
(60)
CUJSE
8 (2011), No. 2

!
r
r
2
2

16k
2k
16k


− √ cot kx + k −
t
u(x, t) = − −



3
3
3


!
r


2

2k
16k


+ √ tan kx + k −
t


3
3 !
r
2

2k
1
16k 2

2
2

(−4k
)
−
cot
kx
+
k
t
v(x,
t)
=
−



3
3
3


!
r


2
2

2k
16k


−
tan2 kx + k −
t


3
3



q
2

4k
r

sin kx + k − 3 t



4k 2
k 




−√ 
u(x, t) = − −

q



3
2
3


1 ± cos kx + k − 4k3 t






q


2
4k


1 ± cos kx + k − 3 t 


k 




+√ 

q



2
3
4k


sin kx + k − 3 t

2

q

4k2

−
sin
kx
+
k
t

3



1 2
k2 




v(x,
t)
=
−
(k
)
−
q




3
6

4k2

1
±
cos
kx
+
k
−
t

3



2


q



4k2

1 ± cos kx + k − 3 t 


k2 




−

q



6 
2
4k


sin kx + k − 3 t



q
2

4k
r

cos kx + k − 3 t



k 
4k 2




√
u(x,
t)
=
−
−
−

q




3
2
3
4k


1 ± sin kx + k − 3 t






q


2
4k


1 ± sin kx + k − 3 t 


k 




√
+

q




2
3
4k


cos kx + k − 3 t

2

q
2

4k

cos kx + k − 3 t




1 2
k2 


 v(x, t) = − (k ) −

q




3
6

4k2

1
±
sin
kx
+
k
−
t

3



2


q



4k2

1 ± sin kx + k − 3 t 


k2 




−

q



6 
2
4k


cos kx + k − 3 t
219
(61)
(62)
(63)
220
Uğurlu et al.
Figure 3. Periodic wave graphics of the (1+1)-dimensional dispersive long wave equation for u(x, t) and v(x, t) solutions of (60) in three
dimensions, respectively (k = 1).
(a)
(b)
(c)
(d)
Figure 4. Periodic wave graphic of the (1+1)-dimensional dispersive
long wave equation for u(x, t) of solution (60) in two dimensions (a)
t = 0, (b) t = 0.5, (c) t = 1, (d) t = 2, (k = 1).
CUJSE
8 (2011), No. 2
221
(a)
(b)
(c)
(d)
Figure 5. Periodic wave graphic of the (1+1)-dimensional dispersive
long wave equation for v(x, t) of solution (60) in two dimensions (a)
t = 0, (b) t = 0.5, (c) t = 1, (d) t = 2, (k = 1).
In Figure 3, Figure 4 and Figure 5, are shown graphics of periodic wave solutions of
the (1+1)-dimensional dispersive long wave equation in three and two dimensions,
respectively. In Figure 5, the periodic waves move to the left with time. If ξ =
kx − wt, the periodic waves move to the right.
3. Conclusions
In this paper, we present the generalized Jacobi elliptic function method [11] by using
ansatz (5) and, with aid of Mathematica, implement it in a computer algebraic
system. An implementation of the method is given by applying it to the SRLW
equation and (1+1)-dimensional dispersive long wave equation. We obtain some
periodic solutions of these equations at the same time. The method can be used
for many other nonlinear equations or coupled ones. In addition, this method is
also computerizable, which allows us to perform complicated and tedious algebraic
calculations on a computer.
222
Uğurlu et al.
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