Propagation of nonlinear waves with a weak dispersion
via coupled ( 2 + 1 ) -dimensional Konopelchenko–Dubrovsky dynamical equation
ALY R SEADAWY1,∗, DAVID YARO2and DIANCHEN LU2
1Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2Faculty of Science, Jiangsu University, Zhenjiang 212013, Jiangsu, People’s Republic of China
∗Corresponding author. E-mail: Aly742001@yahoo.com
MS received 15 April 2019; revised 4 September 2019; accepted 24 September 2019
Abstract. This work applies the modified extended direct algebraic method to construct some novel exact travelling wave solutions for the coupled(2+1)-dimensional Konopelchenko–Dubrovsky (KD) equation. Soliton, periodic, solitary wave, Jacobi elliptic function, new elliptic, Weierstrass elliptic function solutions and so on are obtained, which have several implementations in the field of applied sciences and engineering. In addition, we discuss the dynamics of some solutions like periodic, soliton and dark-singular combo soliton by their evolutionary shapes.
Keywords. (2+1)-Dimensional Konopelchenko–Dubrovsky equation; modified extended direct algebraic method; travelling wave solution; Jacobi elliptic function.
PACS Nos 02.30.Jr; 05.45.Yv; 47.10.A; 47.35.Fg
1. Introduction
In recent years, several researchers have devoted their energy to study the propagation of wave solutions for coupled nonlinear evolution equations (NEEs), which perform a crucial role in explaining the features of nonlinear problems in science and engineering. Sev- eral nonlinear coupled wave models are often used to illustrate many of the problems in physics, such as heat flow occurrence, plasma physics, biology, elec- tricity, quantum mechanics, fibre optics etc. [1–6]. In addition, several problems in other areas like epidemi- ology, ecology, chemical material reactivity, etc. are also described by NEEs. Obtaining exact and numer- ical solutions for coupled nonlinear wave is crucial and important in the study of physical occurrences. On the other hand, the exact solution of coupled nonlinear problems can help people understand these occurrences better than numerical solutions. Therefore, the study of exact solutions of NEEs is very crucial, and many powerful methods have been developed to construct the exact solutions. Backlund transformation, Darboux¨ transformation, Cole–Hopf transformation, various tanh methods, various Jacobi elliptic function methods, vari- able separation approach, Painlevé method, homoge- neous balance method, similarity reduction method, etc.
[7–28] are very effective algorithms for constructing exact solutions of many NEEs [29–36].
More periodic wave solutions expressed by Jacobi elliptic functions for the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) equations were obtained by using the extended F-expansion method.
The solitary wave solutions and trigonometric func- tion solutions for the KD equations were found in [37].
By using the further improved F-expansion method, (2+1)-dimensional KD equation were given the Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions [38]. The similarity transformation method generated infinite-dimensional Lie algebra and commutation relations of the KD equation [39]. The extended tanh method, the sech–
csch ansatz, the Hirota’s bilinear formalism combined with the simplified Hereman form and the Darboux transformation method were applied to determine the travelling wave solutions and other kinds of exact solutions for the two-dimensional KD equation and abundant new soliton solutions, kink solutions, periodic wave solutions and complexiton solutions were formally derived [40]. The tanh–sech method, the cosh–sinh method and exponential function method were effi- ciently employed to handle the(2+1)-dimensional KD equation [41].
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In this paper we consider the coupled (2+1)- dimensional nonlinear KD equation. By applying the modified extended direct algebraic method, we con- structed some novel exact travelling wave solutions for the coupled(2+1)-dimensional KD equation. Soliton, periodic, solitary wave, Jacobi elliptic function, new elliptic, Weierstrass elliptic function solutions and so on are obtained, which have several implementations in the field of applied sciences and engineering. The pur- pose is to find new exact solutions of the equation, and hence we consider the following KD equation:
hξ −hζζζ −6γhhζ +3
2β2h2hζ −3gι+3βhζg =0,
gζ =hι, (1)
where h=h(ζ, ι, ξ) and g=g(ζ, ι, ξ), the subscripts represent the partial differentiation, γ and β are the real parameters. For hι = 0, eq. (1) is reduced to the Gardner equation and it is also converted to the Kadomtsev–Petviashvili equation, where β = 0. Fur- thermore, it becomes modified Kadomtsev–Petviashvili equation withγ =0, which is valuable in soliton theory.
Several researchers have studied in detail the KD sys- tem and has proposed some efficient methods for its exact solutions. For instance, Wuet al[42] successfully obtained exact solutions of the equation by using the Hirota direct method and linear superposition principle, whereas Kumar and Tiwari [39] presented various new group-invariant solutions of the KD equation by using Lie symmetry approach. In addition, Wang and Wei [40] combined the extended tanh method, the sech–csch ansatz, the Hirota bilinear formalism with the simplified Hereman form and the Darboux transformation method to obtain the travelling wave solutions of the KD equa- tions. Wazwaz [41] obtained exact solutions of the KD equation with periodic, soliton and kink wave proper- ties.
Under the impetus of the above literature, the ansatz equation is extended in a new general formula in the modified extended direct algebraic method to find travel- ling wave solutions of the coupled KD equation [43–45].
Subsequently, more or less novel and further versatile exact travelling wave solutions have been constructed.
The following subsection provides a brief overview of the modified extended direct algebraic method.
1.1 A summary of the method
We assume that the given NEE of h(ζ, ι, ξ) is in the form
Q(h,hξ,hι,hζ,hξξ,hιι,hζζ, . . . )=0, (2)
whereQis the polynomial in its parameters. The nature of the method can be given in the subsequent stages [46–48]:
Stage1: To obtain the solutions of eq. (2), takeh(ζ, ι, ξ)
= H(φ), φ =κ1ζ +κ2ι−αξ and transform eq. (2) to the ordinary differential equation (ODE)
R(H,H,H,H, . . . )=0, (3) where (prime) represents the derivative with respect toφ.
Stage2: By introducing the solutionH(φ)of eq. (3) in a finite series form given by [50]
H(φ)= M
j=−M
djθj(φ), (4)
where dj (real constants withdM = 0) and M (posi- tive integer) are to be determined.θ(φ)represents the solution of the equation below:
θ(φ)
=
f0+ f1θ + f2θ2+ f3θ3+ f4θ4+ f5θ5+ f6θ6, (5) where fj are constants.
Stage3: CalculateMby using the homogeneous balance principle on eq. (3).
Stage 4: By [49,50], using Mathematica to solve the system and based on the value of the parameters fj we can get the exact solutions of eq. (1).
The structure of this paper is as follows: The intro- duction is given in §1. In §2, we applied the method to construct more or less novel exact soliton, periodic and Jacobi elliptic solutions of the coupled KD equation.
Analysis and discussion of the solution is given in §3.
Finally, the conclusion is given §4.
2. The (2+1)-dimensional nonlinear KD equation Here we consider the coupled (2 + 1)-dimensional nonlinear KD equation discussed in §1as follows:
hξ −hζζζ −6γhhζ + 3
2β2h2hζ −3gι+3βhζg=0,
gζ =hι, (6)
whereγ andβ are nonzero real parameters. By consid- ering the travelling wave solution as
h(ζ, ι, ξ)=h(φ)= M
j=−M
djθj(φ),
g(ζ, ι, ξ)=g(φ)= N k=−N
ekθk(φ), (7) θ(φ)
=
f0+f1θ+f2θ2+ f3θ3+ f4θ4+ f5θ5+ f6θ6,
φ =κ1ζ+κ2ι−αξ, (8)
wheredj,ek, f0, f1, f2, f3, f4, f5, f6, γ, β, κ1, κ2and α are constants,M,N are positive integers to be deter- mined later. The values of M and N are normally obtained by applying the homogeneous balance princi- ple (i.e. balancing the highest-order linear term together with the highest order of nonlinear terms). Substituting eq. (7) on eq. (6), results
−αh−κ13h−6γhh+ 3
2β2κ1h2h
−3κ2g+3βκ1hg=0, (9)
κ1g=κ2h. (10)
By integrating eq. (10) with respect toφ, we get g= κ2h
κ1 +A, (11)
for which Ais the integration constant. By putting eq.
(11) into eq. (9) we get
−2κ14h+3β2κ12h2h+6κ1(βκ2−2γ κ1)hh
+2(3βκ12A−ακ1−3κ22)h=0. (12) Now by balancing the highest-order derivativeh and the nonlinear termh2h, we attainM =1 and consider the solution of eq. (12) as
h(φ)= d−1
θ +d0+d1θ. (13)
Now by substituting eqs (8) and (13) into eq. (12) and applying [49,50], the solutions of eq. (6) can be expressed as follows:
Case1: f0= f1= f5 = f6 =0 (i) d−1 =0, d0= −βf3κ12−2βκ2
√f4+4γ κ1
√f4 2β2√
f4κ1
, d1= ±2κ1√ f4
β ,
A= 8αβ2f4κ1−3β2f32κ14+8β2f2f4κ14+36β2f4κ22−48βγf4κ2κ1+48γ2f4κ12
24β3f4κ12 . (14)
(ii) d−1=0, d0 = βf3κ12−2βκ2√
f4+4γ κ1√ f4
2β2√
f4κ1 , d1= ±2κ1
√f4
β ,
A= 8αβ2f4κ1−3β2f32κ14+8β2f2f4κ14+36β2f4κ22−48βγf4κ2κ1+48γ2f4κ12
24β3f4κ12 . (15)
In this case, the solutions of eq. (6) are in the following forms:
h1(φ)=d0± 4f2κ1√
f4sech(√ f2φ) β(√
− f4sech(√ f2φ)), g1(φ)= κ2h1
κ1 + A, f2 >0, >0 (16) h2(φ)=d0∓ 4f2κ1√
f4sech(√ f2φ) β(√
+ f4sech(√ f2φ)), g2(φ)= κ2h2
κ1 +A, f2 >0, >0 (17) h3(φ)=d0± 4f2κ1
√f4csch(√ f2φ) β(√
−− f4csch(√ f2φ)), g3(φ)= κ2h3
κ1 + A, f2 >0, <0 (18) h4(φ)=d0∓ 4f2κ1√
f4csch(√ f2φ) β(√
−+ f4csch(√ f2φ)), g4(φ)= κ2h4
κ1 +A, f2 >0, <0 (19) h5(φ)=d0∓2f2κ1
β√ f4
1±tanh √
f2φ 2
g5(φ)= κ2h5
κ1 + A, f2 >0, =0; (20) h6(φ)=d0∓2f2κ1
β√ f4
1±coth √
f2φ 2
g6(φ)= κ2h6
κ1
+ A, f2 >0, =0, (21) where
φ =κ1ζ +κ2ι−αξ, = f32−4f2f4, and
A= 8αβ2f4κ1−3β2f32κ14+8β2f2f4κ14+36β2f4κ22−48βγf4κ2κ1+48γ2f4κ12
24β3f4κ12 .
Case2: f0 = f1= f3= f5 = f6 =0 d−1 =0, d0= 2γ κ1−βκ2
β2κ1 , d1 = ±2κ1√ f4
β ,
A
= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 .
(22)
In this case, we obtain the following solutions of eq.
(6) by putting eq. (22) in eq. (13) together with the solu- tions of eq. (8):
h1(φ)= 2γ κ1−βκ2
β2κ1 ± 4f2κ1
√f4exp(√ f2φ) β(1± f2f4exp(2√
f2φ)), g1(φ)= κ2h1
κ1 +A, f2>0, f4<0; (23) h2(φ)= 2γ κ1−βκ2
β2κ1 ±2κ1
√−f2
β csch(
−f2φ), g2(φ)= κ2h2
κ1 +A, f2<0, f4>0; (24) h3(φ)= 2γ κ1−βκ2
β2κ1
∓2κ1√
−f2
β csc(
−f2φ), g3(φ)= κ2h3
κ1 +A, f2<0, f4>0; (25) h4(φ)= 2γ κ1−βκ2
β2κ1 ∓ 4f2κ1√
f4 exp(√ f2φ) β(f2f4 exp(2√
f2φ)−1), g4(φ)= κ2h4
κ1 +A, f2>0, f4>0, (26) where
φ =κ1ζ +κ2ι−αξ and
A=2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 .
Case3: f3= f4 = f5 = f6=0 (i) d−1 = ±2κ1√
f0
β , d0= 4γ κ1
√f0−βf1κ12−2βκ2
√f0
2β2κ1√
f0 , d1 =0,
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (27)
(ii) d−1= ±2κ1
√f0
β , d0 = 4γ κ1√
f0+βf1κ12−2βκ2√ f0
2β2κ1
√f0 , d1=0,
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (28)
In this case, by substituting eq. (27) into eq. (13), eq. (28) into eq. (13) together with the solutions of eq.
(8), we respectively obtain the following solitary solu- tions of eq. (6):
h1(φ)= 4γ κ1
√f0−βf1κ12−2βκ2
√f0 2β2κ1
√f0
± 8f2κ1√ f0
β(exp(√
f2φ)−2f1+(f12−4f0f2)exp(−√ f2φ)), g1(φ)= κ2h1
κ1 + A, f0 =0, f1 =0, f2 >0; (29) h2(φ)= 4γ κ1
√f0+βf1κ12−2βκ2
√f0 2β2κ1
√f0
± 8f2κ1
√f0
β(exp(√
f2φ)−2f1+(f12−4f0f2)exp(−√ f2φ)), g2(φ)= κ2h2
κ1 +A, f0=0, f1 =0, f2 >0, (30) where
φ =κ1ζ+κ2ι−αξ
and
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0, κ12 .
Case4: f1= f3= f4 = f5 = f6 =0 d−1 = ±2κ1
√f0
β , d0 = 2γ κ1−βκ2
β2κ1 , d1=0, A
= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 .
(31) Now by putting eq. (31) into eq. (13) together with the solutions of eq. (8), the following solutions of eq. (6) are obtained:
h1(φ)= 2γ κ1−βκ2
β2κ1
± 4f2κ1√ f0
β(exp(√
f2φ)− f0f2exp(−√ f2φ)), g1(φ)= κ2h1
κ1 +A, f0=0, f2 >0; (32)
h2(φ)= 2γ κ1−βκ2
β2κ1 ± 2κ1√
−f2
βsin(√
−f2φ), g2(φ)= κ2h2
κ1 +A, f0 >0, f2 <0; (33)
h3(φ)= 2γ κ1−βκ2
β2κ1 ∓ 2κ1
√f0f2
β√
−f0sin(√
−f2φ), g3(φ)= κ2h3
κ1 + A, f0 <0, f2 <0, (34) where
φ =κ1ζ +κ2ι−αξ
and A
= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 .
Case5: f1= f3 = f5 = f6=0 (i) d−1 = ±2κ1
√f0
β , d0= 2γ κ1−βκ2
β2κ1 , d1 =0, A= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 . (35)
(ii) d−1= ±2κ1
√f0
β , d0 = 2γ κ1−βκ2
β2κ1 , d1 = ±2κ1
√f4
β ,
A= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14−12β2κ14√ f0f4
6β3κ12 . (36)
(iii) d−1 = ±2κ1√ f0
β , d0= 2γ κ1−βκ2
β2κ1 , d1= ±2κ1√ f4
β ,
A= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14+12β2κ14√ f0f4
6β3κ12 . (37)
(iv) d−1=0, d0 = 2γ κ1−βκ2
β2κ1 , d1 = ±2κ1
√f4
β , A= 2αβ2κ1+9β2κ22−12βγ κ2κ1+12γ2κ12+2β2f2κ14
6β3κ12 .
(38)
For Case 5, eq. (6) has the following solutions in terms of the Jacobi elliptic function [50] with different solutions of eq. (8):
h1(φ)=d−1dcφ+d0+d1cdφ, g1(φ)= κ2h1
κ1 +A, f0=1,
f2 = −(m2+1), f4 =m2; (39) h2(φ)=d−1nsφ+d0+d1snφ,
g2(φ)= κ2h2
κ1 +A, f0=1,
f2 = −(m2+1), f4 =m2; (40) h3(φ)=d−1snφ+d0+d1nsφ,
g3(φ)= κ2h3
κ1 +A, f0=m2,
f2 = −m2−1, f4=1; (41) h4(φ)=d−1cdφ+d0+d1dcφ,
g4(φ)= κ2h4
κ1 +A, f0=m2,
f2 = −m2−1, f4=1; (42)
h5(φ)=d−1ndφ+d0+d1dnφ, g5(φ)= κ2h5
κ1 +A, f0=m2−1
f2 = −m2, f4= −1; (43) h6(φ)=d−1ncφ+d0+d1cnφ,
g6(φ)= κ2h6
κ1 +A, f0=1−m2,
f2 =2m2−1, f4 = −m2; (44) h7(φ)=d−1cnφ+d0+d1ncφ,
g7(φ)= κ2h7
κ1 +A, f0= −m2,
f2 = −1+2m2, f4 =1−m2; (45) h8(φ)=d−1dnφ+d0+d1ndφ,
g8(φ)= κ2h8
κ1 +A, f0= −1,
f2 =2−m2, f4 =m2−1; (46) h9(φ)=d−1scφ+d0+d1csφ,
g9(φ)= κ2h9
κ1 +A, f0=1−m2,
f2 =2−m2, f4 =1; (47) h10(φ)=d−1csφ+d0+d1scφ,
g10(φ)= κ2h10
κ1 +A, f0=1,
f2=2−m2, f4=1−m2; (48) h11(φ)=d−1dsφ+d0+d1sdφ,
g11(φ)= κ2h11
κ1 +A, f0=1,
f2=2m2−1, f4 =m2(−1+m2); (49) h12(φ)=d−1sdφ+d0+d1dsφ,
g12(φ)= κ2h12
κ1 +A, f0=m2(−1+m2),
f2=2m2−1, f4 =1; (50) h13(φ)= d−1
(ns(φ)±cs(φ))+d0+d1(ns(φ)±cs(φ)), g13(φ)= κ2h13
κ1 +A, f0= 1
4, f2 = (1−m2)
2 , f4 = 1
4; (51)
h14(φ)= d−1
(nc(φ)±sc(φ))+d0+d1(nc(φ)±sc(φ)), g14(φ)= κ2h14
κ1 +A, f0= (1−m2)
4 , f2= (1+m2)
2 , f4 = 1
4; (52) h15(φ)= d−1
(ns(φ)±ds(φ))
+d0+d1(ns(φ)±ds(φ)), g15(φ)= κ2h15
κ1 +A, f0= m2
4 , f2 = (m2−2)
2 , f4 = 1
4; (53)
h16(φ)= d−1
(sn(φ)±i cn(φ))
+d0+d1(sn(φ)±i cn(φ)), g16(φ)= κ2h16
κ1 +A, f0= m2 4 , f2= (m2−2)
2 , f4= m2
4 ; (54)
h17(φ)=d−1
i
1−m2sn(φ)±cn(φ)
nd(φ) +d0+d1
dn(φ) (i√
1−m2sn(φ)±cn(φ)),
g17(φ)= κ2h17
κ1 +A, f0 = m2 4 , f2 = (m2−2)
2 , f4 = m2
4 ; (55)
h18(φ)=d−1cn(φ)ns(φ)nd(φ)+d0 +d1sn(φ)dn(φ)nc(φ), g18(φ)= κ2h18
κ1 +A,
f0 =1, f2 =2−4m2, f4 =1; (56) h19(φ)=d−1B1(1+sn(φ))(1+msn(φ))nd(φ)nc(φ)
+d0+d1 dn(φ)cn(φ)
B1(1+sn(φ))(1+msn(φ)), g19(φ)= κ2h19
κ1 +A, f0 = (m−1)2
4B12 , f2 = (1+m2+6m)
2 ,
f4 = B12(−1+m)2
4 ; (57)
h20(φ)=d−1B1(1+sn(φ))(1−msn(φ))nd(φ)nc(φ) +d0+d1
dn(φ)cn(φ)
B1(1+sn(φ))(1−msn(φ)), g20(φ)= κ2h20
κ1 + A, f0 = (m+1)2 4B12 , f2 = (1+m2+6m)
2 ,
f4 = B12(1+m)2
4 ; (58)
h21(φ)=d−1(1+msn2(φ))nd(φ)nc(φ) m
+d0+d1
mdn(φ)cn(φ) (1+msn2(φ)), g21(φ)= κ2h21
κ1 +A,
f0 = −2m3+m4+m2, f2= −4 m ,
f4 =6m−m2−1; (59)
h22(φ)=d−1(msn2(φ)−1)nd(φ)nc(φ) m
+d0+d1
mdn(φ)cn(φ) msn2(φ)−1,
g22(φ)= κ2h22
κ1 +A,
f0=2m3+m4+m2, f2= −6m−m2−1, f4= −4
m ; (60)
h23(φ)=d−1(√
1−m2−dn2(φ))ns(φ)nc(φ) m2
+d0+d1
m2sn(φ)cn(φ) (√
1−m2−dn2(φ)), g23(φ)= κ2h23
κ1 +A, f0=2+2
1−m2−m2, f2=6
1−m2−m2+2, f4=4
1−m2; (61)
h24(φ)= − d−1(√
1−m2+dn2(φ))ns(φ)nc(φ)
m2 ;
+d0+d1
m2sn(φ)cn(φ) (√
1−m2+dn2(φ))
, g24(φ)= κ2h24
κ1 +A, f0=2−2
1−m2−m2, f2=6
1−m2−m2+2, f4 = −4
1−m2; (62) h25(φ)=d−1(B2cn(φ)+B3dn(φ))ns(φ)
(B22−B32)/(B22−B32m2) +d0
+d1
[(B22−B32)/(B22−B32m2)]sn(φ)
B2cn(φ)+B3dn(φ) , g25(φ)= κ2h25
κ1 +A, f0= m2−1
4(B32m2−B22), f2 = m2+1
2 ,
f4= (B32m2−B22)(m2−1)
4 ; (63)
h26(φ)=d−1(B2sn(φ)+B3cn(φ))nd(φ)
(B22+B32−B32m2)/(B22+B32)+d0
+d1
[(B22+B32−B32m2)/(B22+B32)]dn(φ) B2sn(φ)+B3cn(φ) , g26(φ)= κ2h26
κ1 +A, f0= m2 4(B32+B22), f2= m2−2
2 , f4 = (B32+B22)
4 ; (64)
h27(φ)=d−1B2(msn2(φ)+1) msn2(φ)−1 +d0+d1 (msn2(φ)−1)
B2
msn2(φ)+1, g27(φ)= κ2h27
κ1 +A, f0 = 2m−m2−1
B22 , f2 =2m2+2,
f4 = −B22m2−B22−2B22m; (65) h28(φ)=d−1B2(msn2(φ)−1)
msn2(φ)+1 +d0+d1 (msn2(φ)+1)
B2
msn2(φ)−1, g28(φ)= κ2h28
κ1 +A, f0 = −2m+m2+1
B22 , f2=2m2+2,
f4 = −B22m2+B22+2B22m; (66) h29(φ)=d−1(mcn(φ)±i
1−m2)nd(φ) +d0+d1
dn(φ) (mcn(φ)±i√
1−m2), g29(φ)= κ2h29
κ1 +A, f0 = f4= 1
4, f2 = 1−2m2
2 ; (67)
h30(φ)=d−1
1
msn(φ)±i dn(φ)
+d0+d1(msn(φ)±i dn(φ)) , g30(φ)= κ2h30
κ1 + A, f0 = f4= 1
4, f2 = 1−2m2
2 ; (68)
h31(φ)=d−1
1
mns(φ)±cs(φ)
+d0+d1(mns(φ)±cs(φ)) , g31(φ)= κ2h31
κ1 +A, f0 = f4= 1
4, f2 = 1−2m2
2 ; (69)
h32(φ)=d−1(1±cn(φ))ns(φ) +d0+d1 sn(φ)
(1±cn(φ)), g32(φ)= κ2h32
κ1 +A, f0= f4= 1
4, f2 = 1−2m2
2 ; (70)
h33(φ)=d−1(1±msn(φ))nd(φ) +d0+d1
dn(φ) (1±msn(φ)), g33(φ)= κ2h33
κ1 +A, f0= f4= m2−1
4 , f2 = 1+m2
2 ; (71)
h34(φ)=d−1 1
msd(φ)±nd(φ)
+d0+d1(msd(φ)±nd(φ)) , g34(φ)= κ2h34
κ1 +A, f0= f4= m2−1
4 , f2 = 1+m2
2 ; (72)
h35(φ)=d−1(1±sn(φ))nc(φ) +d0+d1
cn(φ) (1±sn(φ)), g35(φ)= κ2h35
κ1 +A, f0= f4= 1−m2
4 , f2 = 1+m2
2 ; (73)
h36(φ)=d−1 1
(nc(φ)±sc(φ)) +d0+d1(nc(φ)±sc(φ)) , g36(φ)= κ2h36
κ1 +A, f0= f4= 1−m2
4 , f2 = 1+m2
2 ; (74)
h37(φ)=d−1
1
(mcn(φ)±dn(φ)) +d0+d1(mcn(φ)±dn(φ)) , g37(φ)= κ2h37
κ1 +A, f0= −(1−m2)2
4 , f2 = 1+m2
2 , f4 = 1
4; (75)
h38(φ)=d−1(dn(φ)±cn(φ))ns(φ) +d0+d1 sn(φ)
(dn(φ)±cn(φ)), g38(φ)= κ2h38
κ1 +A, f0 = 1 4, f2 = 1+m2
2 , f4= (1−m2)2
4 ; (76)
h39(φ)=d−1(
1−m2±dn(φ))nc(φ) +d0+d1 cn(φ)
(√
1−m2±dn(φ)), g39(φ)= κ2h39
κ1 +A, f0 = 1 4, f2 = m2−2
2 , f4= m4
4 ; (77)
h40(φ)=d−1(1±dn(φ))ns(φ) +d0+d1 sn(φ)
(1±dn(φ)),
g40(φ)= κ2h40
κ1 +A, f0= 1 4, f2= m2−2
2 , f4 = m4
4 , (78)
whereφ =κ1ζ+κ2ι−αξfulfills eqs (35)–(78),m(0<
m<1)is a modulus,B1,B2,B3(B1B2B3=0)andB4
are arbitrary constants. The Jacobi elliptic functions are doubly periodic and have the following characteristics of triangular functions:
sn2(φ)+cn2(φ)=1, dn2(φ)=1−m2sn2(φ), sn(φ)=cn(φ)dn(φ), cn(φ)= −sn(φ)dn(φ), dn(φ)= −m2sn(φ)cn(φ).
When m → 1, the Jacobi functions reprobate to the hyperbolic functions. Thus,
sn(φ)→tanh(φ), cn(φ)→sech(φ).
When m → 0, the Jacobi functions reprobate to the trigonometric functions. Thus,
sn(φ)→sin(φ), cos(φ)→sech(φ).
Case6: f5= f6 =0 (i) d−1 = −2κ1√
f0
β , d0= −βf1κ12−2βκ2
√f0+4γ κ1
√f0
2β2κ1√
f0 , d1 =0, A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (79)
(ii) d−1= 2κ1
√f0
β , d0 = βf1κ12−2βκ2√
f0+4γ κ1√ f0
2β2κ1
√f0 , d1=0,
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (80)
(iii) d−1 =0, d0= −βf1κ12−2βκ2
√f0+4γ κ1
√f0
2β2κ1
√f0
, d1= −2κ1√ f0
β ,
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (81)
(iv) d−1 =0, d0= βf1κ12−2βκ2√
f0+4γ κ1√ f0
2β2κ1√
f0 , d1 = 2κ1
√f0
β ,
A= 8αβ2f0κ1−3β2f12κ14+8β2f0f2κ14+36β2f0κ22−48βγf0κ2κ1+48γ2f0κ12
24β3f0κ12 . (82)
