Mathematical Problems in Engineering Volume 2012, Article ID 537930,24pages doi:10.1155/2012/537930
Research Article
New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Ito Equations
A. H. Bhrawy,
1, 2M. Sh. Alhuthali,
1and M. A. Abdelkawy
21Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
Correspondence should be addressed to A. H. Bhrawy,alibhrawy@yahoo.co.uk Received 20 March 2012; Revised 14 September 2012; Accepted 14 September 2012 Academic Editor: Massimo Scalia
Copyrightq2012 A. H. Bhrawy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using the extended F-expansion method based on computerized symbolic computation technique, we find several new solutions of11-dimensional and21-dimensional Ito equations. These solutions contain hyperbolic and triangular solutions. It is shown that the power of the extended F-expansion method is its ease of use to determine shock or solitary type of solutions. In addition, as an illustrative sample, the properties for the extended F-expansion solutions of the Ito equations are shown with some figures.
1. Introduction
The nonlinear wave phenomena can be observed in various scientific fields, such as plasma physics, optical fibers, fluid dynamics, and chemical physics. The nonlinear wave phenomena can be obtained in solutions of nonlinear evolution equationsNEEs. The study of NLEEs appear everywhere in applied mathematics and theoretical physics including engineering sciences and biological sciences. These NLEEEs play a key role in describing key scientific phenomena. For example, the nonlinear Schr ¨odinger’s equation describes the dynamics of propagation of solitons through optical fibers. The Korteweg-de Vries equation models the shallow water wave dynamics near ocean shore and beaches. Additionally, the Schr ¨odinger- Hirota equation describes the dispersive soliton propagation through optical fibers. These are just a few examples in the whole wide world of NLEEs and their applications,see, for instance,1–4. While the above mentioned NLEEs are scalar NLEEs, there is a large number of NLEEs that are coupled. Some of them are two-coupled NLEEs such as the Gear-Grimshaw equation2, while there are several others that are three-coupled NLEEs. An example of a three-coupled NLEE is the Wu-Zhang equation4. These coupled NLEEs are also studied in various areas of theoretical physics as well.
The exact solutions of these NEEs play an important role in the understanding of nonlinear phenomena. In the past decades, many methods were developed for finding exact solutions of NEEs such as the inverse scattering method 5,6, improved projective Riccati equations method7,8, Cole-Hopf transformation method9, exp-function method 10–16, bifurcation theory method 17, G/G-expansion method 18, 19, homotopy perturbation method20, tanh function method20–24, and Jacobi and Weierstrass elliptic function method25,26. Although Porubov et al.27–29have obtained some exact periodic solutions to some nonlinear wave equations, they use the Weierstrass elliptic function and involve complicated deducing. A Jacobi elliptic functionJEFexpansion method, which is straightforward and effective, was proposed for constructing periodic wave solutions for some nonlinear evolution equations. The essential idea of this method is similar to the tanh method by replacing the tanh function with some JEFs such as sn, cn, and dn. For example, the Jacobi periodic solution in terms of sn may be obtained by applying the sn-function expansion. Many similarl repetitious calculations have to be done to search for the Jacobi doubly periodic wave solutions in terms of cn and dn30.
Recently, F-expansion method31–34was proposed to obtain periodic wave solutions of NLEEs, which can be thought of as a concentration of JEF expansion since F here stands for every function of JEFs. The objectives of this work are twofold. First, we seek to extend others works to establish new exact solutions of distinct physical structures for the nonlinear equations1.1and1.2. The extended F-expansionEFEmethod will be used to achieve the first goal. The second goal is to show that the power of the EFE method is its ease of use to determine shock or solitary type of solutions. In this paper, we study two well- known PDEs, namely, generalized11-dimensional and generalized21-dimensional Ito equations. Many studies are concerning the 11-dimensional Ito equation and the 21- dimensional Ito equation35–42.
The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870, and, finally, Korteweg and de Vries in 189539. The KdV equation was not studied much after this until Zabusky and Kruskal 1965 40 discovered numerically that its solutions seemed to decompose at large times into a collection of “solitons”: well-separated solitary waves. Ito41,42obtained the well-known generalized11-dimensional and gen- eralized21-dimensional Ito equations by generalization of the bilinear KdV equation as
uttuxxxt3uxutuuxt 3uxx
x
∞utdx0, 1.1 uttuxxxt32uxutuuxt 3uxx
x
∞utdxαuytβuxt0. 1.2 Also Sawada-Kotera-ItoSK-Itoseventh-order equation is the special case of the generalized seventh-order KdV equation as
ut252u2ux63u3x378uuxu2x126u2u3x
63u2xu3x42uxu4x21uu5xu7x0. 1.3 SK-Ito equation is characterized by the presence of three dispersive terms ux,u3x, and u7x, respectively. SK-Ito seventh-order equation is completely integrable and admits of
conservation laws43. Moreover, the Ito-type coupled KdVItcKdVequation44, written in the following form:
utαuuxβvvxγuxxx0; vtβuvx0, 1.4
if we take the special values α −6,β −2, and γ −1. Equation 1.4 describes the interaction process of two internal long waves which has infinitely many conserved quantities45,46.
In this paper, we extend the EFE method with symbolic computation to 1.1 and 1.2for constructing their interesting Jacobi doubly periodic wave solutions. It is shown that soliton solutions and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions. In addition, the algorithm that we use here is also a computerized method, in which we are generating an algebraic system.
2. Extended F-Expansion Method
In this section, we introduce a simple description of the EFE method, for a given partial differential equation as
G
u, ux, uy, uz, uxy, . . .
0. 2.1
We like to know whether travelling wavesor stationary wavesare solutions of2.1. The first step is to unite the independent variablesx,y, andtinto one particular variable through the new variable as
ζxy−νt, u x, y, t
Uζ, 2.2
whereνis wave speed, and reduce2.1to an ordinary differential equationODEas G
U, U, U, U, . . .
0. 2.3
Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE.
For this purpose, let us simply useUas the expansion in the form
u x, y, t
Uζ N
i0
aiFiN
i1
a−iF−i, 2.4
where
F
ABF2CF4, 2.5
Table 1: Relation between values ofA, B, Cand correspondingF.
A B C Fζ
1 −1−m2 m2 snζor cdζ cnζ/dnζ
1−m2 2m2−1 −m2 cnζ
m2−1 2−m2 −1 dnζ
m2 −1−m2 1 nsζ1/snζor dcζ dnζ/cnζ
−m2 2m2−1 1−m2 ncζ 1/cnζ
−1 2−m2 m2−1 ndζ 1/dnζ
1 2−m2 1−m2 scζ snζ/cnζ
1 2m2−1 −m2−1−m2 sdζ snζ/dnζ
1−m2 2−m2 1 csζ cnζ/snζ
−m21−m2 2m2−1 1 dsζ dnζ/snζ
1/4
1−2m2
/2 1/4 nsζ csζ
1−m2
/4 1m2/2
1−m2
/2 ncζ scζ
1/4
m2−2
/2 m2/4 nsζ dsζ
m2/4
m2−2
/2 m2/4 snζ icsζ
the highest degree ofdpU/dζp, is taken as
O dpU
dζp
Np, p1,2,3, . . . , 2.6
O
UqdpU dζp
q1
Np, q0,1,2, . . . , p1,2,3, . . . , 2.7
whereA,B, andCare constants, andNin2.3is a positive integer that can be determined by balancing the nonlinear termsand the highest order derivatives. NormallyNis a positive integer, so that an analytic solution in closed form may be obtained. Substituting2.1–2.5 into2.3and comparing the coefficients of each power ofFζin both sides, we will get an overdetermined system of nonlinear algebraic equations with respect toν, a0, a1, . . .. We will solve the over-determined system of nonlinear algebraic equations by use of Mathematica.
The relations between values ofA,B,C, and corresponding JEF solution Fζ of2.4are given inTable 1. Substituting the values ofA,B,C, and the corresponding JEF solutionFζ chosen fromTable 1into the general form of solution, then an ideal periodic wave solution expressed by JEF can be obtained.
snζ, cnζ, and dnζare the JE sine function, JE cosine function, and the JEF of the third kind, respectively. And
cn2ζ 1−sn2ζ, dn2ζ 1−m2sn2ζ, 2.8
with the modulusm0< m <1.
Whenm → 1. the Jacobi functions degenerate to the hyperbolic functions, that is,
snζ → tanhζ, cnζ → sechζ, dnζ → sechζ, 2.9
whenm → 0, the Jacobi functions degenerate to the triangular functions, that is,
snζ → sinζ, cnζ → cosζ, dn → 1. 2.10
3. Generalized (1+1)-Dimensional Ito Equation
We first consider the generalized11-dimensional Ito equation1.1as follows:
uttuxxxt3uxutuuxt 3uxx
x
∞utdx0, 3.1 if we use the transformationuvx, it carries3.1into
vxttvxxxxt3vxxvxtvxvxxt 3vxxxvt0, 3.2
if we useζx−νttransforms3.2into the ODE, we have
−νVVv−3ν
VVVV
−3νVV0, 3.3
where by integrating once we obtain, upon setting the constant of integration to zero,
V3 V2
−νV0, 3.4
if we use the transformationWV, then3.4can be written as follows:
W3W2−νW 0. 3.5
Balancing the termWwith the termW2we obtainN2 then
Wζ a0a1ψa−1ψ−1a2ψ2a−2ψ−2, ψ ABψ2Cψ4. 3.6 Substituting3.6into3.5and comparing the coefficients of each power ofψin both sides, we will get an over-determined system of nonlinear algebraic equations with respect to ν, ai,i 0,1,−1,−2,2. Solving the over-determined system of nonlinear algebraic equations by use of Mathematica, we obtain three groups of constants
1
a−1a−10, a0 −2B
3 , a2−2C, a−2−2A, ν−2
B212AC
B ,
3.7
2
a−1 a2a−1 0, a0−2B
3 , a−2 −2A, ν−2
B2−3AC
B , 3.8
3
a−1a−2a−10, a0−2B
3 , a2−2C, ν−2
B2−3AC
B . 3.9
The solutions of3.1are
u1 − 2 1m2
3 −2m2sn2
⎛
⎜⎝x−2
1m2212m2
1m2 t
⎞
⎟⎠
−2ns2
⎛
⎜⎝x−2
1m2212m2
1m2 t
⎞
⎟⎠,
3.10
u2 − 2 1m2
3 −2m2cd2
⎛
⎜⎝x− 2
1m22
12m2
1m2 t
⎞
⎟⎠
−2dc2
⎛
⎜⎝x−2
1m2212m2
1m2 t
⎞
⎟⎠,
3.11
u3 − 2
2m2−1
3 2m2cn2
⎛
⎜⎝x 2 12m2
m2−1
2m2−12
2m2−1 t
⎞
⎟⎠
−2
1−m2 nc2
⎛
⎜⎝x2 12m2
m2−1
2m2−12
2m2−1 t
⎞
⎟⎠,
3.12
u4 − 2 2−m2
3 2dn2
⎛
⎜⎝x−2
2−m22−12
−1m2
2−m2 t
⎞
⎟⎠
−2
m2−1 nd2
⎛
⎜⎝x− 2
2−m22
−12
−1m2
2−m2 t
⎞
⎟⎠,
3.13
u5 − 2 1m2
3 −2ns2
⎛
⎜⎝x−2
1m22−3m2
1m2 t
⎞
⎟⎠, 3.14
u6 − 2 2−m2
3 −2
1−m2 sc2
⎛
⎜⎝x 2 12
1−m2
2−m22
2−m2 t
⎞
⎟⎠
−2cs2
⎛
⎜⎝x2 12
1−m2
2−m22
2−m2 t
⎞
⎟⎠,
3.15
u7 2
2m2−1
3 2m2
1m2 sd2
⎛
⎜⎝x2
−12m2 1−m2
−12m22
2m2−1 t
⎞
⎟⎠
−2ds2
⎛
⎜⎝x2
−12m2 1−m2
−12m22
2m2−1 t
⎞
⎟⎠,
3.16
u8 2
1−2m2 6
−1 2
⎛
⎜⎝ns
⎛
⎜⎝x2 0.75
0.5−m22 0.5−m2 t
⎞
⎟⎠cs
⎛
⎜⎝x2 0.75
0.5−m22 0.5−m2 t
⎞
⎟⎠
⎞
⎟⎠
2
−1 2
⎛
⎜⎝ns
⎛
⎜⎝x2 0.75
0.5−m22 0.5−m2 t
⎞
⎟⎠cs
⎛
⎜⎝x2 0.75
0.5−m22 0.5−m2 t
⎞
⎟⎠
⎞
⎟⎠
−2
, 3.17 u9 − 1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝x2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22
0.5m20.5 t
⎞
⎟⎠
sc
⎛
⎜⎝x2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22
0.5m20.5 t
⎞
⎟⎠
⎞
⎟⎠
2
1−m2 2
⎛
⎜⎝nc
⎛
⎜⎝x2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22
0.5m20.5 t
⎞
⎟⎠
sc
⎛
⎜⎝x 2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22
0.5m20.5 t
⎞
⎟⎠
⎞
⎟⎠
−2
, 3.18
u10 − m2−2 3
−m2 2
⎛
⎜⎝ns
⎛
⎜⎝x2
0.75m2
−10.5m22 0.5m2−1 t
⎞
⎟⎠
ds
⎛
⎜⎝x2
0.75m2
−10.5m22 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
2
−1 2
⎛
⎜⎝ns
⎛
⎜⎝x2
0.75m2
−10.5m22 0.5m2−1 t
⎞
⎟⎠
ds
⎛
⎜⎝x2
0.75m2
−10.5m22 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.19 u11 m2−2
3
−m2 2
⎛
⎜⎝sn
⎛
⎜⎝x2
0.75m4
−10.5m22 0.5m2−1 t
⎞
⎟⎠
ics
⎛
⎜⎝x2
0.75m4
−10.5m22 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
2
m2 2
⎛
⎜⎝sn
⎛
⎜⎝x2
0.75m4
−10.5m22 0.5m2−1 t
⎞
⎟⎠
ics
⎛
⎜⎝x2
0.75m4
−10.5m22 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.20
u12 − 2 1m2
3 −2dc2
⎛
⎜⎝x−2
1m22
−3m2
1m2 t
⎞
⎟⎠, 3.21
u13 − 2
2m2−1
3 −2
1−m2 nc2
⎛
⎜⎝x2
−3m2 m2−1
2m2−12
2m2−1 t
⎞
⎟⎠, 3.22
u14 − 2 2−m2
3 −2
m2−1 nd2
⎛
⎜⎝x−2
2−m223
−1m2
2−m2 t
⎞
⎟⎠, 3.23
u15 − 2 2−m2
3 −2cs2
⎛
⎜⎝x2
2−m22−3
1−m2
2−m2 t
⎞
⎟⎠, 3.24
u16 2
2m2−1 3 −2ds2
⎛
⎜⎝x2 3m2
1−m2
−12m22
2m2−1 t
⎞
⎟⎠, 3.25
u17 2
1−2m2
6 −1
2
⎛
⎜⎝ns
⎛
⎜⎝x2
0.5−m22−3/16 0.5−m2 t
⎞
⎟⎠
cs
⎛
⎜⎝x2
0.5−m22
−3/16 0.5−m2 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.26
u18 − 1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝x2
0.50.5m22
−3
0.5−0.5m2
0.25−0.25m2
0.5m20.5 t
⎞
⎟⎠
sc
⎛
⎜⎝x2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2
0.5m20.5 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.27
u19 − m2−2 3 −1
2
⎛
⎜⎝ns
⎛
⎜⎝x 2
0.5m2−12
−3/16 0.5m2−1 t
⎞
⎟⎠
ds
⎛
⎜⎝x2
0.5m2−12−3/16 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.28
u20 m2−2 3 m2
2
⎛
⎜⎝sn
⎛
⎜⎝x2
0.5m2−12
−m4/16 0.5m2−1 t
⎞
⎟⎠
ics
⎛
⎜⎝x2
0.5m2−12−m4/16 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
−2
,
3.29
u21 − 2 1m2
3 −2m2sn2
⎛
⎜⎝x−2
1m22−3m2
1m2 t
⎞
⎟⎠, 3.30
u22 − 2 1m2
3 −2m2cd2
⎛
⎜⎝x− 2
1m22−3m2
1m2 t
⎞
⎟⎠, 3.31
u23 − 2
2m2−1
3 2m2cn2
⎛
⎜⎝x 2
2m2−12−3m2
m2−1
2m2−1 t
⎞
⎟⎠, 3.32
u24 − 2 2−m2
3 2dn2
⎛
⎜⎝x−2
2−m22 3
−1m2
2−m2 t
⎞
⎟⎠, 3.33
u25 −2 2−m2
3 −2
1−m2 sc2
⎛
⎜⎝x2
2−m22
−3
1−m2
2−m2 t
⎞
⎟⎠, 3.34
u26 2
2m2−1
3 2m2
1m2 sd2
⎛
⎜⎝x2 3m2
1−m2
−12m22
2m2−1 t
⎞
⎟⎠, 3.35
u27 2
1−2m2
6 −1
2
⎛
⎜⎝ns
⎛
⎜⎝x2
0.5−m22−3/16 0.5−m2 t
⎞
⎟⎠
cs
⎛
⎜⎝x2
0.5−m22−3/16 0.5−m2 t
⎞
⎟⎠
⎞
⎟⎠
2
,
3.36
u28 − 1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝x2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2
0.5m20.5 t
⎞
⎟⎠
sc
⎛
⎜⎝x2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2
0.5m20.5 t
⎞
⎟⎠
⎞
⎟⎠
2
,
3.37
u29 − m2−2 3 −m2
2
⎛
⎜⎝ns
⎛
⎜⎝x2
0.5m2−12−m2/16 0.5m2−1 t
⎞
⎟⎠
ds
⎛
⎜⎝x2
0.5m2−12−m2/16 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
2
,
3.38
u30 m2−2 3 − m2
2
⎛
⎜⎝sn
⎛
⎜⎝x2
0.5m2−12
−m4/16 0.5m2−1 t
⎞
⎟⎠
ics
⎛
⎜⎝x2
0.5m2−12−
m4/16 0.5m2−1 t
⎞
⎟⎠
⎞
⎟⎠
2
.
3.39
3.1. Soliton Solutions
Some solitary wave solutions can be obtained, if the modulusmapproaches to 1 in3.10–
3.39as follows:
u31 −4
3 −2tanh2x−16t−2coth2x−16t, u32 −2
3 2sech2x2t, u33 −2
3 2sech2x−2t, u34 −4
3 −2coth2x−t, u35 −2
3 −2csch2x2t, u36 2
3 4sinh2x2t−2csch2x2t, u37 −1
3 −1
2cothx−4t cschx−4t2−1
2cothx−4t cschx−4t−2, u38 −1
3 −1
2tanhx−4t icschx−4t21
2tanhx−4t icschx−4t−2, u39 −1
3 −1 2
coth
x− 1
4t
csch
x−1 4t
−2 , u40 −1
3 1 2
tanh
x−1
4t
icsch
x−1 4t
−2 , u41 −4
3 −2tanh2x−t, u42 2
3 4sinh2x2t, u43 −1
3 −1 2
coth
x− 1
4t
csch
x−1 4t
2 , u44 1
3 −1
2cothx2t cschx2t2, u45 −1
3 −1 2
tanh
x−3
4t
icsch
x−3 4t
2 .
3.40
3.2. Triangular Periodic Solutions
Some trigonometric function solutions can be obtained, if the modulusmapproaches to zero in3.10–3.39as follwos:
u46 −2
3 −2csc2x−2t, u47 −2
3 −2sec2x−2t, u48 2
3−2sec2x−2t,
2
0 1
t=0
−2 −1
x 0
0.1 0.2
0.3 0.40.5
t 0
20 40 u
a
200 400 600 800 1000 1200 1400
−2 −1
x
1 2
t=0
b
Figure 1: The modulus of solitary wave solutionu13.10wherem0.5.
u49 −4
3 −2tan2x16t−2cot2x16t, u50 1
3− 1
2cscx4t cotx4t2−1
2cscx4t cotx4t−2, u51 −1
3 1
2secx7t tanx7t2 1
2secx7t tanx7t−2, u52 2
3−sin2x−2t, u53 −4
3 −2cot2xt, u54 1
3− 1 2
csc
x1
4t
cot
x1 4t
−2 ,
u55 −1 3 1
2
sec
x−1 2t
tan
x−1
2t −2
, u56 2
3− 1 2sin2
x3
8t
, u57 −4
3 −2tan2xt, u58 1
3− 1 2
csc
x1
4t
cot
x1 4t
2 ,
u59 −1 3 1
2
sec
x−1 2t
tan
x−1
2t 2
.
3.41 The modulus of solitary wave solutionsu1,u2,u21, andu23 is displayed in Figures1, 2,3, and4, respectively, with values of parameters listed in their captions.
0
0.10.20.30.40.5 t 2
0 1
−2 −1 x
t=0
0 20 40 u
a
100 200 300 400 500 600 700
2 1
−1
−2
x t=0
b
Figure 2: The modulus of solitary wave solutionu23.11wherem0.5.
0 0.5
11.522.5
1 2
−2 −1 t
0 x 0.4
0.80.6 u
a
2 4
0.5 0.6 0.7 0.8
−4 −2 x
t=0
b
Figure 3: The modulus of solitary wave solutionu213.30wherem0.5.
0
0.511.522.5 y 2
0 1
−2 −1 x 0.60.4
0.8 u
a
1 2 3
0.4 0.6 0.8 1.2
−3 −2 −1 t=0
x
b
Figure 4: The modulus of solitary wave solutionu233.32wherem0.5.
4. Generalized (2+1)-Dimensional Ito Equation
In this section we consider the generalized21-dimensional Ito equation1.2as follows:
uttuxxxt32uxutuuxt 3uxx
x
∞utdxαuytβuxt0, 4.1 if we use the transformationuvx, it carries4.1into
vxttvxxxxt32vxxvxtvxvxxt 3vxxxvtαvxytβvxxt0, 4.2
if we useζxy−νtcarries4.2into the ODE, we have ν−α−β
V−Vv−3 V2
0, 4.3
where by integrating twice we obtain, upon setting the constant of integration to zero, ν−α−β
V−V−3 V2
0, 4.4
if we use the transformationWV, it carries4.4into ν−α−β
W−W−3W20. 4.5
Balancing the termWwith the termW2, we obtainN2, then
Wζ a0a1ψa−1ψ−1a2ψ2a−2ψ−2, ψ ABψ2Cψ4. 4.6
Proceeding as in the previous case, we obtain 1
a−1a−10, a0 −2B
3 , a2−2C, a−2−2A, ναβ−2
B212AC
B ,
4.7
2
a−1 a2a−10, a0−2B
3 , a−2−2A, ναβ−2
B2−3AC
B , 4.8
3
a−1 a−2a−10, a0−2B
3 , a2 −2C, ναβ−2
B2−3AC
B . 4.9
The solutions of4.1are
u1 −2 1m2
3 −2m2sn2
⎛
⎜⎝x
⎛
⎜⎝αβ−2
1m2212m2 1m2
⎞
⎟⎠t
⎞
⎟⎠
−2ns2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
1m2212m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u2 −2 1m2
3 −2m2cd2
⎛
⎜⎝xy
⎛
⎜⎝αβ− 2
1m2212m2 1m2
⎞
⎟⎠t
⎞
⎟⎠
−2dc2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
1m2212m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u3 −2
2m2−1
3 2m2cn2
⎛
⎜⎝xy
⎛
⎜⎝αβ 2 12m2
m2−1
2m2−12 2m2−1
⎞
⎟⎠t
⎞
⎟⎠
−2
1−m2 nc2
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12m2
m2−1
2m2−12 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u4 −2 2−m2
3 2dn2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
2−m22
−12
−1m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠
−2
m2−1 nd2
⎛
⎜⎝xy
⎛
⎜⎝αβ− 2
2−m22
−12
−1m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u5 −2 1m2
3 −2ns2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
1m22
−3m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u6 −2 2−m2
3 −2
1−m2 sc2
⎛
⎜⎝xy
⎛
⎜⎝αβ 2 12
1−m2
2−m22 2−m2
⎞
⎟⎠t
⎞
⎟⎠
−2cs2
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12
1−m2
2−m22 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u7 2
2m2−1 3 2m2
1m2 sd2
⎛
⎜⎝xy
⎛
⎜⎝αβ2
−12m2 1−m2
−12m22 2m2−1
⎞
⎟⎠t
⎞
⎟⎠
−2ds2
⎛
⎜⎝xy
⎛
⎜⎝αβ2
−12m2 1−m2
−12m22 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u82
1−2m2 6
−1 2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2 0.75
0.5−m22 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
cs
⎛
⎜⎝xy
⎛
⎜⎝αβ2 0.75
0.5−m22 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
−1 2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2 0.75
0.5−m22 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
cs
⎛
⎜⎝xy
⎛
⎜⎝αβ2 0.75
0.5−m22 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u9 −1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
sc
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
sc
⎛
⎜⎝xy
⎛
⎜⎝αβ2 12
0.5−0.5m2
0.25−0.25m2
0.50.5m22 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u10 −m2−2 3 − m2
2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m2
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ds
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m2
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
−1 2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m2
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ds
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m2
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u11 m2−2 3 −m2
2
⎛
⎜⎝sn
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m4
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ics
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m4
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
m2 2
⎛
⎜⎝sn
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m4
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ics
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.75m4
−10.5m22 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u12 −2 1m2
3 −2dc2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
1m22−3m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u13 −2
2m2−1
3 −2
1−m2 nc2
⎛
⎜⎝xy
⎛
⎜⎝αβ2
−3m2 m2−1
2m2−12 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u14 −2 2−m2
3 −2
m2−1 nd2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
2−m22
3
−1m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u15 −2 2−m2
3 −2cs2
⎛
⎜⎝xy
⎛
⎜⎝αβ2
2−m22−3
1−m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u16 2
2m2−1 3 −2ds2
⎛
⎜⎝xy
⎛
⎜⎝αβ2 3m2
1−m2
−12m22 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u17 2
1−2m2
6 −1
2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5−m22−3/16 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
cs
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5−m22
−3/16 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u18 −1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.50.5m22
−3
0.5−0.5m2
0.25−0.25m2 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
sc
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u19 −m2−2 3
−1 2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12−3/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ds
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12
−3/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u20 m2−2 3 m2
2
×
⎛
⎜⎝sn
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12
−m4/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ics
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12−m4/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
−2
,
u21 −2 1m2
3 −2m2sn2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
1m22
−3m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u22 −2 1m2
3 −2m2cd2
⎛
⎜⎝xy
⎛
⎜⎝αβ− 2
1m22
−3m2 1m2
⎞
⎟⎠t
⎞
⎟⎠,
u23 −2
2m2−1
3 2m2cn2
⎛
⎜⎝xy
⎛
⎜⎝αβ 2
2m2−12−3m2
m2−1 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u24 −2 2−m2
3 2dn2
⎛
⎜⎝xy
⎛
⎜⎝αβ−2
2−m22
3
−1m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u25 −2 2−m2
3 −2
1−m2 sc2
⎛
⎜⎝xy
⎛
⎜⎝αβ 2
2−m22−3
1−m2 2−m2
⎞
⎟⎠t
⎞
⎟⎠,
u26 2
2m2−1
3 2m2
1m2 sd2
⎛
⎜⎝xy
⎛
⎜⎝αβ 2 3m2
1−m2
−12m22 2m2−1
⎞
⎟⎠t
⎞
⎟⎠,
u27 2
1−2m2 6
−1 2
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5−m22
−3/16 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
cs
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5−m22−3/16 0.5−m2
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
,
u28 −1m2
3 1−m2 2
×
⎛
⎜⎝nc
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
sc
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.50.5m22−3
0.5−0.5m2
0.25−0.25m2 0.5m20.5
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
,
u29 −m2−2 3 − m2
2
×
⎛
⎜⎝ns
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12−m2/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ds
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12
−m2/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
,
u30m2−2 3 −m2
2
×
⎛
⎜⎝sn
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12−m4/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
ics
⎛
⎜⎝xy
⎛
⎜⎝αβ2
0.5m2−12
−m4/16 0.5m2−1
⎞
⎟⎠t
⎞
⎟⎠
⎞
⎟⎠
2
.
4.10
4.1. Soliton Solutions
Some solitary wave solutions can be obtained, if the modulusmapproaches to 1 in4.10as follows:
u31 −4
3 −2tanh2
xy
αβ−16 t
−2coth2
xy
αβ−16 t
, u32 −2
3 2sech2
xy
αβ2 t
, u33 −2
3 2sech2
xy
αβ−2 t
, u34 −4
3 −2coth2
xy
αβ−1 t
, u35 −2
3 −2csch2
xy
αβ2 t
, u36 2
3 4sinh2
xy
αβ2 t
−2csch2
xy
αβ2 t
, u37 −1
3 −1 2
coth
xy
αβ−4 t
csch
xy
αβ−4 t2
−1 2
coth
xy
αβ−4 t
csch
xy
αβ−4 t−2
, u38 −1
3 −1 2
tanh
xy
αβ−4 t
icsch
xy
αβ−4 t2 1
2 tanh
xy
αβ−4 t
icsch
xy
αβ−4 t−2
, u39−1
3 −1 2
coth
xy
αβ−1 4
t
csch
xy
αβ− 1 4
t
−2 ,
u40−1 3 1
2
tanh
xy
αβ−1 4
t
icsch
xy
αβ−1 4
t
−2 , u41 −4
3 −2tanh2
xy
αβ−1 t
,
u42 2
3 4sinh2
xy
αβ2 t
, u43−1
3 −1 2
coth
xy
αβ−1 4
t
csch
xy
αβ− 1 4
t
2 , u44 1
3 −1 2
coth
xy
αβ2 t
csch
xy
αβ2 t2
, u45−1
3 −1 2
tanh
xy
αβ−3 4
t
icsch
xy
αβ−3 4
t
2 .
4.11
4.2. Triangular Periodic Solutions
Some trigonometric function solutions can be obtained, if the modulusmapproaches to zero in4.10as follows:
u46 − 2
3−2csc2
xy
αβ−2 t
, u47 − 2
3−2sec2
xy
αβ−2 t
, u48 2
3 −2sec2
xy
αβ−2 t
, u49 − 4
3−2tan2
xy
αβ16 t
−2cot2
xy
αβ16 t
, u50 1
3 −1 2
csc
xy
αβ4 t
cot
xy
αβ4 t2
−1 2
csc
xy
αβ4 t
cot
xy
αβ4 t−2
, u51 − 1
31 2
sec
xy
αβ7 t
tan
xy
αβ7 t2 1
2 sec
xy
αβ7 t
tan
xy
αβ7 t−2
, u52 2
3 −sin2
xy
αβ−2 t
, u53 − 4
3−2cot2
xy
αβ1 t
, u54 1
3 −1 2
csc
xy
αβ 1 4
t
cot
xy
αβ1 4
t
−2 ,
u55 − 1 31
2
sec
xy
αβ−1 2
t
tan
xy
αβ−1 2
t
−2 , u56 2
3 −1 2sin2
xy
αβ3 8
t
,
