New exact double periodic wave and complex ... - Semantic Scholar
Applied Mathematics and Computation 229 (2014) 159–172
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New exact double periodic wave and complex wave solutions for a generalized sinh–Gordon equation Bin He ⇑, Weiguo Rui, Yao Long College of Mathematics, Honghe University, Mengzi, Yunnan 661100, PR China
a r t i c l e
i n f o
Keywords: A generalized sinh–Gordon equation Binary F-expansion method Elliptic equation Constraint condition Exact solution
a b s t r a c t In this paper, dependent and independent variable transformations are introduced to solve a generalized sinh–Gordon equation by using the binary F-expansion method and the knowledge of elliptic equation and Jacobian elliptic functions. Many different new exact solutions such as double periodic wave and complex wave solutions are obtained. Some previous results are extended. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that the exact solutions of the sinh–Gordon equations have been extensively studied in the field of theoretical physics (see Refs. [1–7] and references cited therein). In 2006, Wazwaz [8] studied the following generalized Sinh–Gordon equation:
utt auxx þ b sinhðnuÞ ¼ 0;
ð1Þ
where n is a positive integer and a; b are two constants. And he derived families of exact solutions using the reliable tanh method. Tang et al. [9] studied the bifurcation behaviors and exact solutions of the Eq. (1) under three different functions transformations by using the bifurcation theory of dynamical system. In this paper, we aim to extend the previous works in Refs. [8,9], we shall obtain many new exact solutions of Eq. (1), including double periodic wave and complex wave solutions. This paper is organized as follows. In Section 2, we introduce the binary F-expansion method briefly. In Section 3, we give many exact solutions of Eq. (1). In Section 4, a short conclusion will be given. 2. The binary F-expansion method For a given nonlinear partial differential equation
Uðf ðuÞ; ux ; ut ; uxx ; utt ; uxt ; . . .Þ ¼ 0;
ð2Þ
where f ðuÞ is a composite function which is similar to sinðnuÞ or sinhðnuÞ ðn ¼ 1; 2; . . .Þ etc. As in Ref. [15], the binary F-expansion method is simply represented as follows: Step 1: We make a transformation
UðnÞ ; u¼/ VðgÞ
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (B. He). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.040
ð3Þ
160
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
where n ¼ k1 ðx þ c1 tÞ; g ¼ k2 ðx þ c2 tÞ; k1 ; k2 ; c1 ; c2 are unknown parameters which to be further determined. The transfor UðnÞ mation u ¼ / VUðnÞ was first given by Lamb and used it to solve the sine–Gordon equation [12], u ¼ 4n tan1 Vð ðgÞ gÞ and 1 UðnÞ 4 are its two special cases. Substituting (3) into (2), yields u ¼ n tanh VðgÞ
UðU; U 0 ; U 00 ; . . . ; V; V 0 ; V 00 ; . . .Þ ¼ 0:
ð4Þ
Step 2: On some constraint conditions, if Eq. (4) can be differentiated as follows
U 02 ¼ P1 þ Q 1 U 2 þ R1 U 4 ;
ð5Þ
V 02 ¼ P2 þ Q 2 V 2 þ R2 V 4 ;
ð6Þ
where P1 ; Q 1 ; R1 ; P 2 ; Q 2 ; R2 are some parameters, then, with the aid of Table 1 (see Ref. [14]), we can get the solutions UðnÞ; VðgÞ of Eqs. (5) and (6). Step 3: Substituting the UðnÞ; VðgÞ into (3), many exact solutions of Eq. (2) can be obtained. 3. Exact solutions of Eq. (1) First, let us recall some properties of Jacobian elliptic functions. We know that there exist twelve kinds of Jacobian elliptic functions [10,11]
snðs; mÞ; cnðs; mÞ; dnðs; mÞ; scðs; mÞ; sdðs; mÞ; cdðs; mÞ; nsðs; mÞ; ncðs; mÞ; ndðs; mÞ; csðs; mÞ; dsðs; mÞ; dcðs; mÞ; where m ð0 < m < 1Þ is a modulus of Jacobian elliptic functions. When m ! 1, the Jacobian functions degenerate to the hyperbolic functions, that is
snðs; mÞ ! tanhðsÞ; cnðs; mÞ ! sechðsÞ; dnðs; mÞ ! sechðsÞ; scðs; mÞ ! sinhðsÞ; sdðs; mÞ ! sinhðsÞ; cdðs; mÞ ! 1; nsðs; mÞ ! cothðsÞ; ncðs; mÞ ! coshðsÞ; ndðs; mÞ ! coshðsÞ; csðs; mÞ ! cschðsÞ; dsðs; mÞ ! cschðsÞ; dcðs; mÞ ! 1: When m ! 0, the Jacobian functions degenerate to the trigonometric functions, i.e.
snðs; mÞ ! sinðsÞ; cnðs; mÞ ! cosðsÞ; dnðs; mÞ ! 1; scðs; mÞ ! tanðsÞ; sdðs; mÞ ! sinðsÞ; cdðs; mÞ ! cosðsÞ; nsðs; mÞ ! cscðsÞ; ncðs; mÞ ! secðsÞ; ndðs; mÞ ! 1; csðs; mÞ ! cotðsÞ; dsðs; mÞ ! cscðsÞ; dcðs; mÞ ! secðsÞ: Next, we study Eq. (1). Considering the following transformation:
n ¼ kðx þ ctÞ;
a c
g ¼ k x þ t ; a – c2 ;
ð7Þ
where k; c are two parameters to be determined later, Eq. (1) can be rewritten as 2
k2 c2 ðc2 aÞunn þ k2 aða c2 Þugg þ bc sinhðnuÞ ¼ 0:
ð8Þ
By means of a similar ansatz as given in Refs. [12,13], letting Table 1 Relations between values of ðP; Q ; RÞ and corresponding FðsÞ in ODE F 0 2 ¼ P þ QF 2 þ RF 4 P
Q
R
FðsÞ
1
ð1 þ m2 Þ 2m2 1 2 m2 ð1 þ m2 Þ 2m2 1 2 m2 2 m2 2m2 1 2 m2 2m2 1
m2 m2 1 1
snðs; mÞ; cdðs; mÞ cnðs; mÞ
12m2 2 1þm2 2 m2 2 2 m2 2 2
1 4
1 m2 m2 1 m2 m2 1 1 1 1 m2 m2 ð1 m2 Þ 1 4 1m2 4 m2 4 m2 4
1 m2 m2 1 1 m2 m2 ð1 m2 Þ 1 1
dnðs; mÞ nsðs; mÞ; dcðs; mÞ ncðs; mÞ ndðs; mÞ scðs; mÞ sdðs; mÞ csðs; mÞ dsðs; mÞ nsðs; mÞ csðs; mÞ
1m2 4 1 4
ncðs; mÞ scðs; mÞ
m2 4
snðs; mÞ icsðs; mÞ; i ¼ 1
nsðs; mÞ dsðs; mÞ 2
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 1 UðnÞ tanh n VðgÞ
161
ð9Þ
for Eq. (8), yields
V gg U nn 2 k2 c2 ðc2 aÞ ðU 2 V 2 Þ 2U 2n þ k2 aða c2 Þ ðV 2 U 2 Þ 2V 2g ¼ nbc ðU 2 þ V 2 Þ: U V
ð10Þ
Successive differentiation of this result with respect to both n and g results in
c2 U nn a V gg ¼ 0; UU n U n VV g V g
ð11Þ
and from (11), one has
c2 U nn a V gg ¼ ¼ x; UU n U n VV g V g
ð12Þ
where x is a parameter to be determined later, i.e.
U 2n ¼
x 4c2
U 4 þ l1 U 2 þ m1 ;
V 2g ¼
x 4a
V 4 þ l2 V 2 þ m2 ;
ð13Þ
where l1 ; m1 ; l2 ; m2 are integral constants. Considering (10) and (13), we have the corresponding constraint conditions 2
k2 c2 ðc2 aÞl1 þ k2 aða c2 Þl2 þ nbc ¼ 0;
c2 m1 am2 ¼ 0:
ð14Þ
Obviously, not all Jacobi elliptic functions satisfying (13) can satisfy the constraint conditions (14). Only some combinations of these Jacobi elliptic functions are the solutions that the Eq. (1) can admit. With the aid of Table 1, we will show the details. There are 31 cases which need to be addressed. Case 1. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ ndðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼ m2 1;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð15Þ
Substituting (15) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
nb ; 4að1 m2 Þ
x ¼ 4aðm2 1Þ; k2 ¼
c2 ¼ a;
ð16Þ
then the double periodic wave solutions to the Eq. (1) are
u¼
4 1 tanh ðsnðn; kÞdnðg; mÞÞ; n
ð17Þ
u¼
4 1 tanh ðcdðn; kÞdnðg; mÞÞ: n
ð18Þ
and
The profile of (17) is shown in Fig. 1(1–1). Case 2. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð19Þ
Substituting (19) into the constraint conditions (14), the parameters can be determined as
k ¼ 1; x ¼ a;
k2 ¼
2nb ; 3að2m2 1Þ
c2 ¼
a ; 4
ð20Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð21Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
6 3
3
4
2
2
2
1
1
0
0
1
1
2
2
0 2 4
3
3 100
50
0
50
100
3
2
1
0
1
2
3
4
2
1
0
1
2
3
4
3 0.6
6
0.4
2
0.2
1
2
0
0
0.0
4
0.2
2
1 0.4
4 2
0.6
6 3 5
0
20
5
10
0
10
20
10
5
15
10
5
0
5
10
15
4 1.5 1.0
2
0.5 0
0.0 0.5
2 1.0 1.5
4 10
5
0
5
10
0
5
10
pffiffi 3 Fig. 1. Various travelling waves described by solutions (17),pffiffi(21), (49), (69), (76) and (107). (1–1) n ¼ 3; a ¼ 25; b ¼ 16 ; c ¼ 5; m ¼ 13 ; k ¼ 2 3 2 ; k ¼ 40 . pffiffi pffiffi 2 5 3 2 5 (1–2) and (1–3) n¼ 3; a ¼ 16; b ¼ 1; c ¼ 2; m p ¼ffiffi 3 ; k ¼ 3 ; k ¼ 4 . (1–4) n ¼ 3; a ¼ 36; b ¼ 1; c ¼ 3, k ¼ 15 . (1–5) q n ffiffiffiffi ¼ 3; a ¼ 36; b ¼ 1; c ¼ 3; pffiffiffiffi pffiffi 57 3 3 7 10 k ¼ 57 . (1–6) n ¼ 3; a ¼ 25; b ¼ 1; c ¼ 5; k ¼ 10 . (1–7) and (1–8) n ¼ 3; a ¼ 16; b ¼ 10; c ¼ 2; m ¼ 4 ; k ¼ 4 ; k ¼ 2 71.
and
u¼
4 1 1 ; tanh n nsðg; mÞ csðg; mÞ
ð22Þ
The profiles of (21) are shown in Fig. 1(1–2) and (1–3). When m ! 0, the solutions (21), (22) turn to be another complex wave and periodic wave solutions respectively
u¼
4 tanhðnÞ 1 ; tanh n cscðgÞ cotðgÞ
ð23Þ
u¼
4 1 1 : tanh n cscðgÞ cotðgÞ
ð24Þ
and
Case 3. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 2
m2 ¼
1 m2 : 4
ð25Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
Substituting (25) into the constraint conditions (14), the parameters can be determined as
x ¼ að1 m2 Þ; k2 ¼
k ¼ 1;
2nbðm2 1Þ ; aðm2 þ 1Þðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð26Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð27Þ
u¼
4 1 1 ; tanh n ncðg; mÞ scðg; mÞ
ð28Þ
and
when m ! 0, the solutions (27), (28) turn to be another complex wave and periodic wave solutions respectively
u¼
4 tanhðnÞ 1 ; tanh n secðgÞ tanðgÞ
ð29Þ
u¼
4 1 1 : tanh n secðgÞ tanðgÞ
ð30Þ
and
Case 4. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð31Þ
Substituting (31) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 1;
x ¼ a; k2 ¼
2nb ; 3a
c2 ¼
a ; 4
ð32Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n cothðgÞ cschðgÞ
ð33Þ
u¼
4 1 1 : tanh n cothðgÞ cschðgÞ
ð34Þ
and
2
Case 5. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð35Þ
Substituting (35) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ am2 ; k2 ¼
aðm2
2m2 nb ; 2Þðm2 4Þ
c2 ¼
m2 a ; 4
ð36Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð37Þ
u¼
4 1 1 ; tanh n snðg; mÞ icsðg; mÞ
ð38Þ
and
when m ! 1, the solutions (37), (38) turn to be another complex wave solutions respectively
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 tanhðnÞ 1 ; tanh n tanhðgÞ icschðgÞ
ð39Þ
u¼
4 1 1 : tanh n tanhðgÞ icschðgÞ
ð40Þ
and
Case 6. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ sdðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼ m2 ð1 m2 Þ;
l2 ¼ 2m2 1; m2 ¼ 1:
ð41Þ
Substituting (41) into the constraint conditions (14), the parameters can be determined as 2
k ¼
m2 ð1 m2 Þ ; 1 þ m2 m4
x ¼ 4am2 ðm2 1Þ; k2 ¼
nbð1 þ m2 m4 Þ ; am4 ð1 m4 Þ
c2 ¼ að1 þ m2 m4 Þ;
ð42Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðcnðn; kÞdsðg; mÞÞ: n
ð43Þ
Case 7. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼ 1;
l2 ¼ 2m2 1; m2 ¼ m2 ð1 m2 Þ:
ð44Þ
Substituting (44) into the constraint conditions (14), the parameters can be determined as 2
k ¼
1 ; 1 þ m2 m4
x ¼ 4a; c2 ¼ að1 þ m2 m4 Þ; k2 ¼
nbð1 þ m2 m4 Þ ; am2 ð2 þ 3m2 m6 Þ
ð45Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðcnðn; kÞsdðg; mÞÞ: n
ð46Þ
Case 8. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ nsðg; mÞ þ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð47Þ
Substituting (47) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 0;
x ¼ a; k2 ¼
4nb ; 15a
a c2 ¼ ; 4
ð48Þ
then the complex wave solution to the Eq. (1) is
u¼
4 1 1 tanh sechðnÞ sinðgÞ : n 2
ð49Þ
The profile of (49) is shown in Fig. 1(1–4). Case 9. From Table 1, choosing U ¼ dnðn; kÞ and V ¼ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ k2 1;
x 4a
¼ 1;
l2 ¼ 2 m2 ; m2 ¼ 1 m2 :
ð50Þ
Substituting (50) into the constraint conditions (14), the parameters can be determined as
k ¼ m;
x ¼ 4a; k2 ¼
nb ; 4að2 m2 Þ
c2 ¼ a;
ð51Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðdnðn; mÞscðg; mÞÞ; n
ð52Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
165
when m ! 0, the solution (52) turn to be another periodic wave solution
u¼
4 1 tanh ðtanðgÞÞ: n
ð53Þ
Case 10. From Table 1, choosing U ¼ dnðn; kÞ and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
x
l1 ¼ 2 k2 ; m1 ¼ k2 1;
4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð54Þ
Substituting (54) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
x ¼ a; k2 ¼
4nb ; 15aðm2 1Þ
a c2 ¼ ; 4
ð55Þ
then the complex wave solution to the Eq. (1) is
u¼
4 dnðn; kÞ 1 : tanh n nsðg; mÞ dsðg; mÞ
ð56Þ
Case 11. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð57Þ
Substituting (57) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ a; k2 ¼
nb ; 3aðm2 1Þ
c2 ¼
a ; 4
ð58Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð59Þ
when m ! 0, the solution (59) turn to be another complex wave solution
u¼
4 cothðnÞ 1 : tanh n cscðgÞ cotðgÞ
ð60Þ
Case 12. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð61Þ
Substituting (61) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; að3 þ m2 Þ
c2 ¼
að1 m2 Þ ; 4
ð62Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð63Þ
when m ! 0, the solution (63) turn to be another complex wave solution
u¼
4 cothðnÞ 1 : tanh n secðgÞ tanðgÞ
ð64Þ
Case 13. From Table 1, choosing U ¼ nsðn; kÞ (or U ¼ dcðn; kÞ) and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
Substituting (65) into the constraint conditions (14), the parameters can be determined as
ð65Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
k ¼ m;
x ¼ a; k2 ¼
4nb ; 9að1 m2 Þ
c2 ¼
a ; 4
ð66Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 nsðn; kÞ 1 tanh n nsðg; mÞ dsðg; mÞ
ð67Þ
u¼
4 dcðn; kÞ 1 ; tanh n nsðg; mÞ dsðg; mÞ
ð68Þ
and
when m ! 0, the solutions (67) and (68) turn to be another double periodic wave solutions respectively
u¼
4 1 1 cscðnÞ sinðgÞ ; tanh n 2
ð69Þ
u¼
4 1 1 secðnÞ sinðgÞ : tanh n 2
ð70Þ
and
The profile of (69) is shown in Fig. 1(1–5). Case 14. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð71Þ
Substituting (71) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ am2 ; k2 ¼
m2 nb ; að1 m2 Þð4 m2 Þ
c2 ¼
am2 ; 4
ð72Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð73Þ
Case 15. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2k2 1; m1 ¼ k2 ;
x 4a
¼ 1 m2 ;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð74Þ
Substituting (74) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 1;
x ¼ 0; k2 ¼
nb ; 4a
c2 ¼ a;
ð75Þ
then the complex wave solution to the Eq. (1) is
u¼
4 1 coshðnÞ : tanh n sinhðgÞ
ð76Þ
The profile of (76) is shown in Fig. 1(1–6). Case 16. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ sdðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2k2 1; m1 ¼ k2 ;
x 4a
¼ m2 ð1 m2 Þ;
l2 ¼ 2m2 1; m2 ¼ 1:
ð77Þ
Substituting (77) into the constraint conditions (14), the parameters can be determined as
1 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ m2 m4
x ¼ 4am2 ðm2 1Þ; k2 ¼
nbð1 þ m2 m4 Þ ; m2 að2 þ 3m2 m6 Þ
c2 ¼ aðm4 m2 1Þ;
ð78Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðncðn; kÞdsðg; mÞÞ: n
ð79Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
167
Case 17. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
x
l1 ¼ 2k2 1; m1 ¼ k2 ;
4a
l2 ¼ 2m2 1; m2 ¼ m2 ð1 m2 Þ:
¼ 1;
ð80Þ
Substituting (80) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ð1 m2 Þð1 þ m2 m4 Þ ; 1 þ m2 m4
x ¼ 4a; k2 ¼
nbð1 þ m2 m4 Þ ; am4 ð1 m4 Þ
c2 ¼ að1 þ m2 m4 Þ;
ð81Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðncðn; kÞsdðg; mÞÞ: n
ð82Þ
Case 18. From Table 1, choosing U ¼ ndðn; kÞ and V ¼ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k 1;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼ 1 m2 ;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð83Þ
Substituting (83) into the constraint conditions (14), the parameters can be determined as
k ¼ m;
x ¼ 4að1 m2 Þ; k2 ¼
nb ; 4að2 m2 Þ
c2 ¼ a;
ð84Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðndðn; kÞcsðg; mÞÞ; n
ð85Þ
when m ! 0, the solution (85) turn to be another periodic wave solution
4 1 u ¼ tanh ðcotðgÞÞ: n
ð86Þ
Case 19. From Table 1, choosing U ¼ scðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð87Þ
Substituting (87) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ a; k2 ¼
nb ; 3am2
c2 ¼
a ; 4
ð88Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð89Þ
when m ! 1, the solution (89) turn to be another complex wave solution
u¼
4 tanðnÞ 1 : tanh n cothðgÞ cschðgÞ
ð90Þ
Case 20. From Table 1, choosing U ¼ scðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð91Þ
Substituting (91) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; am2 ðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð92Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð93Þ
168
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172 2
Case 21. From Table 1, choosing U ¼ scðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
m2 ; 4
¼
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð94Þ
Substituting (94) into the constraint conditions (14), the parameters can be determined as 2
k ¼ 0;
x ¼ am2 ; k2 ¼
nbm ; að4 m2 Þ
c2 ¼
am2 ; 4
ð95Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð96Þ
when m ! 1, the solution (96) turn to be another complex wave solution
u¼
4 tanðnÞ 1 : tanh n tanhðgÞ icschðgÞ
ð97Þ
Case 22. From Table 1, choosing U ¼ csðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
1 ; 4
¼
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð98Þ
Substituting (98) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ a; k2 ¼
nb ; 3am2
c2 ¼
a ; 4
ð99Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð100Þ
when m ! 1, the solution (100) turn to be another complex wave solution
u¼
4 cotðnÞ 1 : tanh n cothðgÞ cschðgÞ
ð101Þ
Case 23. From Table 1, choosing U ¼ csðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
1 m2 ; 4
¼
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð102Þ
Substituting (102) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; am2 ðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð103Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð104Þ
Case 24. From Table 1, choosing U ¼ csðn; kÞ and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð105Þ
Substituting (105) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
x ¼ a; k2 ¼
4nb ; 3að5 m2 Þ
then the complex wave solution to the Eq. (1) is
c2 ¼
a ; 4
ð106Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 csðn; kÞ 1 ; tanh n nsðg; mÞ dsðg; mÞ
169
ð107Þ
The profiles of (107) are shown in Fig. 1(1–7) and (1–8). When m ! 0, the solution (107) turn to be another complex wave solution
4 1 1 u ¼ tanh cschðnÞ sinðgÞ : n 2
ð108Þ
2
Case 25. From Table 1, choosing U ¼ csðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
¼ 1;
x
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð109Þ
Substituting (109) into the constraint conditions (14), the parameters can be determined as
x ¼ am2 ; k2 ¼
k ¼ 0;
m2 nb ; að4 m2 Þ
c2 ¼
am2 ; 4
ð110Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð111Þ
when m ! 1, the solution (111) turn to be another complex wave solution
u¼
4 cotðnÞ 1 : tanh n tanhðgÞ icschðgÞ
ð112Þ
Case 26. From Table 1, choosing U ¼ nsðn; kÞ csðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼
1 ; 4
2
l1 ¼
1 2k ; 2
1 4
m1 ¼ ;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð113Þ
Substituting (113) into the constraint conditions (14), the parameters can be determined as 2
k ¼
nb þ ak2 m4 m2 nb ak2 m2 ðm2 1Þ
x ¼ að1 m2 Þ; c2 ¼ að1 m2 Þ;
;
ð114Þ
then the complex wave solution to the Eq. (1) is
u¼
4 nsðn; kÞ csðn; kÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð115Þ
2
Case 27. From Table 1, choosing U ¼ nsðn; kÞ csðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1 respectively, and then from (13), we have
x 4c2
¼
1 ; 4
2
l1 ¼
1 2k ; 2
1 4
m1 ¼ ;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð116Þ
Substituting (116) into the constraint conditions (14), the parameters can be determined as 2
k ¼
am2 k2 þ m2 nb ak2 ak2 m2 ðm2 1Þ
x ¼ am2 ; c2 ¼ am2 ;
;
ð117Þ
then the complex wave solution to the Eq. (1) is
u¼
4 nsðn; kÞ csðn; kÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð118Þ
Case 28. From Table 1, choosing U ¼ ncðn; kÞ scðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ respectively, and then from (13), we have
x 4c2
2
¼
1k ; 4
2
l1 ¼
1þk ; 2
2
m1 ¼
1k ; 4
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
Substituting (119) into the constraint conditions (14), the parameters can be determined as
ð119Þ
170
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172 2
x ¼ að1 m2 Þ; k2 ¼
nbðk 1Þð1 m2 Þ 2
aðk m2 Þ
;
c2 ¼
að1 m2 Þ 1k
2
ð120Þ
;
and
k ¼ 1;
x ¼ 0; k2 ¼
m ¼ 1;
nbc
2
ðc2 aÞ
2
ð121Þ
;
then the complex wave solutions to the Eq. (1) are
u¼
4 ncðn; kÞ scðn; kÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð122Þ
u¼
4 1 coshðnÞ sinhðnÞ ; tanh n coshðgÞ sinhðgÞ
ð123Þ
and
when k ! 0, the solution (122) turn to be another complex wave solution
u¼
4 secðnÞ tanðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð124Þ
when m ! 0, the solution (122) turn to be another complex wave solution
u¼
4 1 ncðn; kÞ scðn; kÞ : tanh n secðgÞ tanðgÞ
ð125Þ
2
Case 29. From Table 1, choosing U ¼ ncðn; kÞ scðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼
1k ; 4
2
l1 ¼
1þk ; 2
2
m1 ¼
1k ; 4
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð126Þ
Substituting (126) into the constraint conditions (14), the parameters can be determined as 2
x ¼ am2 ; k2 ¼
m2 nbðk 1Þ 2
2 2
aðk m4 ð1 k Þ ð1 þ m2 ÞÞ
;
c2 ¼
am2 2
1k
ð127Þ
;
and 2
k ¼ 1;
m ¼ 0;
x ¼ 0; k2 ¼
nbc ; c2
ð128Þ
a2
then the complex wave solutions to the Eq. (1) are
u¼
4 ncðn; kÞ scðn; kÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð129Þ
u¼
4 1 coshðnÞ sinhðnÞ ; tanh n sinðgÞ i cotðgÞ
ð130Þ
and
when k ! 0, the solution (129) turn to be another complex wave solution
u¼
4 secðnÞ tanðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð131Þ
when m ! 1, the solution (129) turn to be another complex wave solution
u¼
4 ncðn; kÞ scðn; kÞ 1 : tanh n tanhðgÞ icschðgÞ
ð132Þ
2
Case 30. From Table 1, choosing U ¼ nsðn; kÞ dsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
x 4c2
¼
2
1 ; 4
l1 ¼
k 2 ; 2
2
m1 ¼
x
k ; 4
¼
4a
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
171
ð133Þ
Substituting (133) into the constraint conditions (14), the parameters can be determined as
x ¼ am2 ; k2 ¼
k ¼ 1;
m2 nb að1 m2 Þ
2
c2 ¼ am2 ;
;
ð134Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ cschðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð135Þ
2
Case 31. From Table 1, choosing U ¼ snðn; kÞ icsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼
k ; 4
2
l1 ¼
k 2 ; 2
2
m1 ¼
x
k ; 4
4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð136Þ
Substituting (136) into the constraint conditions (14), the parameters can be determined as 2
x ¼ am2 ; k2 ¼
k m2 nb 2
aðk m2 Þ
2
;
c2 ¼
am2 2
k
;
ð137Þ
;
ð138Þ
and
m ¼ 0;
k ¼ 0;
x ¼ 0; k2 ¼
nbc ðc2
2
aÞ
2
then the complex wave solutions to the Eq. (1) are
u¼
4 snðn; kÞ icsðn; kÞ 1 tanh ; n snðg; mÞ icsðg; mÞ
ð139Þ
u¼
4 1 sinðnÞ i cotðnÞ ; tanh n sinðgÞ i cotðgÞ
ð140Þ
and
when k ! 1, the solution (139) turn to be another complex wave solution
u¼
4 tanhðnÞ icschðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð141Þ
4. Conclusion In this paper, we obtained many exact solutions of a generalized sinh–Gordon equation, these solutions involve combinations of the arctanh function, Jacobi elliptic functions, hyperbolic functions and trigonometric functions. Compare with Refs. [8,9], most solutions of our results are entirely new. Acknowledgements This research was supported by Natural Science Foundation of Yunnan Province, China (2013FZ117). We thank the referees’ valuable suggestions. References [1] A.M. Wazwaz, The tanh method: exact solutions of the sine–Gordon and the sinh–Gordon equations, Appl. Math. Comput. 167 (2005) 1196–1210. [2] K.W. Chow, A class of doubly periodic waves for nonlinear evolution equations, Wave Motion 35 (2002) 71–90. [3] H.T. Chen, H.C. Yin, Double elliptic equation method and new exact solutions of the (n+1)-dimensional sinh–Gordon equation, J. Math. Phys. 48 (2007) 013504-1–013504-7. [4] J.B. Li, M. Li, Bounded travelling wave solutions for the (n+1)-dimensional sine- and sinh–Gordon equations, Chaos Soliton Fract. 25 (2005) 1037–1047. [5] K. Narita, Deformed sine- and sinh–Gordon equations, deformed Liouville equation, and their discrete models, J. Phys. Soc. Jpn. 72 (2003) 1339–1349. [6] H. Zhang, New exact solutions for the sinh–Gordon equation, Chaos Soliton Fract. 28 (2006) 489–496. [7] Z.T. Fu, S.K. Liu, S.D. Liu, Exact Jacobian elliptic function solutions to sinh–Gordon equation, Commun. Theor. Phys. (Beijing, China) 45 (2006) 55–60. [8] A.M. Wazwaz, Exact solutions for the generalized sine–Gordon and the generalized sinh–Gordon equations, Chaos Soliton Fract. 28 (2006) 127–135.
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[9] Y.N. Tang, W. Xu, J.W. Shen, L. Gao, Bifurcations of traveling wave solutions for a generalized Sinh–Gordon equation, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1048–1055. [10] D.V. Patrick, Elliptic Function and Elliptic Curves, Cambridge University Press, New York, 1973. [11] K. Chamdrasekharan, Elliptic Functions, Springer-Verlag, Berlin, 1985. [12] G.L. Lamb Jr., Elements of Soliton Theory, Wiley, New York, 1980. [13] P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge University Press, New York, 1989. [14] E. Yomba, A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations, Phys. Lett. A 372 (2008) 1048–1060. [15] W.G. Rui, B. He, Y. Long, The binary F-expansion method and its application for solving the ðn þ 1Þ-dimensional sine–Gordon equation, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1245–1258.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
New exact double periodic wave and complex wave solutions for a generalized sinh–Gordon equation Bin He ⇑, Weiguo Rui, Yao Long College of Mathematics, Honghe University, Mengzi, Yunnan 661100, PR China
a r t i c l e
i n f o
Keywords: A generalized sinh–Gordon equation Binary F-expansion method Elliptic equation Constraint condition Exact solution
a b s t r a c t In this paper, dependent and independent variable transformations are introduced to solve a generalized sinh–Gordon equation by using the binary F-expansion method and the knowledge of elliptic equation and Jacobian elliptic functions. Many different new exact solutions such as double periodic wave and complex wave solutions are obtained. Some previous results are extended. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that the exact solutions of the sinh–Gordon equations have been extensively studied in the field of theoretical physics (see Refs. [1–7] and references cited therein). In 2006, Wazwaz [8] studied the following generalized Sinh–Gordon equation:
utt auxx þ b sinhðnuÞ ¼ 0;
ð1Þ
where n is a positive integer and a; b are two constants. And he derived families of exact solutions using the reliable tanh method. Tang et al. [9] studied the bifurcation behaviors and exact solutions of the Eq. (1) under three different functions transformations by using the bifurcation theory of dynamical system. In this paper, we aim to extend the previous works in Refs. [8,9], we shall obtain many new exact solutions of Eq. (1), including double periodic wave and complex wave solutions. This paper is organized as follows. In Section 2, we introduce the binary F-expansion method briefly. In Section 3, we give many exact solutions of Eq. (1). In Section 4, a short conclusion will be given. 2. The binary F-expansion method For a given nonlinear partial differential equation
Uðf ðuÞ; ux ; ut ; uxx ; utt ; uxt ; . . .Þ ¼ 0;
ð2Þ
where f ðuÞ is a composite function which is similar to sinðnuÞ or sinhðnuÞ ðn ¼ 1; 2; . . .Þ etc. As in Ref. [15], the binary F-expansion method is simply represented as follows: Step 1: We make a transformation
UðnÞ ; u¼/ VðgÞ
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (B. He). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.040
ð3Þ
160
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
where n ¼ k1 ðx þ c1 tÞ; g ¼ k2 ðx þ c2 tÞ; k1 ; k2 ; c1 ; c2 are unknown parameters which to be further determined. The transfor UðnÞ mation u ¼ / VUðnÞ was first given by Lamb and used it to solve the sine–Gordon equation [12], u ¼ 4n tan1 Vð ðgÞ gÞ and 1 UðnÞ 4 are its two special cases. Substituting (3) into (2), yields u ¼ n tanh VðgÞ
UðU; U 0 ; U 00 ; . . . ; V; V 0 ; V 00 ; . . .Þ ¼ 0:
ð4Þ
Step 2: On some constraint conditions, if Eq. (4) can be differentiated as follows
U 02 ¼ P1 þ Q 1 U 2 þ R1 U 4 ;
ð5Þ
V 02 ¼ P2 þ Q 2 V 2 þ R2 V 4 ;
ð6Þ
where P1 ; Q 1 ; R1 ; P 2 ; Q 2 ; R2 are some parameters, then, with the aid of Table 1 (see Ref. [14]), we can get the solutions UðnÞ; VðgÞ of Eqs. (5) and (6). Step 3: Substituting the UðnÞ; VðgÞ into (3), many exact solutions of Eq. (2) can be obtained. 3. Exact solutions of Eq. (1) First, let us recall some properties of Jacobian elliptic functions. We know that there exist twelve kinds of Jacobian elliptic functions [10,11]
snðs; mÞ; cnðs; mÞ; dnðs; mÞ; scðs; mÞ; sdðs; mÞ; cdðs; mÞ; nsðs; mÞ; ncðs; mÞ; ndðs; mÞ; csðs; mÞ; dsðs; mÞ; dcðs; mÞ; where m ð0 < m < 1Þ is a modulus of Jacobian elliptic functions. When m ! 1, the Jacobian functions degenerate to the hyperbolic functions, that is
snðs; mÞ ! tanhðsÞ; cnðs; mÞ ! sechðsÞ; dnðs; mÞ ! sechðsÞ; scðs; mÞ ! sinhðsÞ; sdðs; mÞ ! sinhðsÞ; cdðs; mÞ ! 1; nsðs; mÞ ! cothðsÞ; ncðs; mÞ ! coshðsÞ; ndðs; mÞ ! coshðsÞ; csðs; mÞ ! cschðsÞ; dsðs; mÞ ! cschðsÞ; dcðs; mÞ ! 1: When m ! 0, the Jacobian functions degenerate to the trigonometric functions, i.e.
snðs; mÞ ! sinðsÞ; cnðs; mÞ ! cosðsÞ; dnðs; mÞ ! 1; scðs; mÞ ! tanðsÞ; sdðs; mÞ ! sinðsÞ; cdðs; mÞ ! cosðsÞ; nsðs; mÞ ! cscðsÞ; ncðs; mÞ ! secðsÞ; ndðs; mÞ ! 1; csðs; mÞ ! cotðsÞ; dsðs; mÞ ! cscðsÞ; dcðs; mÞ ! secðsÞ: Next, we study Eq. (1). Considering the following transformation:
n ¼ kðx þ ctÞ;
a c
g ¼ k x þ t ; a – c2 ;
ð7Þ
where k; c are two parameters to be determined later, Eq. (1) can be rewritten as 2
k2 c2 ðc2 aÞunn þ k2 aða c2 Þugg þ bc sinhðnuÞ ¼ 0:
ð8Þ
By means of a similar ansatz as given in Refs. [12,13], letting Table 1 Relations between values of ðP; Q ; RÞ and corresponding FðsÞ in ODE F 0 2 ¼ P þ QF 2 þ RF 4 P
Q
R
FðsÞ
1
ð1 þ m2 Þ 2m2 1 2 m2 ð1 þ m2 Þ 2m2 1 2 m2 2 m2 2m2 1 2 m2 2m2 1
m2 m2 1 1
snðs; mÞ; cdðs; mÞ cnðs; mÞ
12m2 2 1þm2 2 m2 2 2 m2 2 2
1 4
1 m2 m2 1 m2 m2 1 1 1 1 m2 m2 ð1 m2 Þ 1 4 1m2 4 m2 4 m2 4
1 m2 m2 1 1 m2 m2 ð1 m2 Þ 1 1
dnðs; mÞ nsðs; mÞ; dcðs; mÞ ncðs; mÞ ndðs; mÞ scðs; mÞ sdðs; mÞ csðs; mÞ dsðs; mÞ nsðs; mÞ csðs; mÞ
1m2 4 1 4
ncðs; mÞ scðs; mÞ
m2 4
snðs; mÞ icsðs; mÞ; i ¼ 1
nsðs; mÞ dsðs; mÞ 2
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 1 UðnÞ tanh n VðgÞ
161
ð9Þ
for Eq. (8), yields
V gg U nn 2 k2 c2 ðc2 aÞ ðU 2 V 2 Þ 2U 2n þ k2 aða c2 Þ ðV 2 U 2 Þ 2V 2g ¼ nbc ðU 2 þ V 2 Þ: U V
ð10Þ
Successive differentiation of this result with respect to both n and g results in
c2 U nn a V gg ¼ 0; UU n U n VV g V g
ð11Þ
and from (11), one has
c2 U nn a V gg ¼ ¼ x; UU n U n VV g V g
ð12Þ
where x is a parameter to be determined later, i.e.
U 2n ¼
x 4c2
U 4 þ l1 U 2 þ m1 ;
V 2g ¼
x 4a
V 4 þ l2 V 2 þ m2 ;
ð13Þ
where l1 ; m1 ; l2 ; m2 are integral constants. Considering (10) and (13), we have the corresponding constraint conditions 2
k2 c2 ðc2 aÞl1 þ k2 aða c2 Þl2 þ nbc ¼ 0;
c2 m1 am2 ¼ 0:
ð14Þ
Obviously, not all Jacobi elliptic functions satisfying (13) can satisfy the constraint conditions (14). Only some combinations of these Jacobi elliptic functions are the solutions that the Eq. (1) can admit. With the aid of Table 1, we will show the details. There are 31 cases which need to be addressed. Case 1. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ ndðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼ m2 1;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð15Þ
Substituting (15) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
nb ; 4að1 m2 Þ
x ¼ 4aðm2 1Þ; k2 ¼
c2 ¼ a;
ð16Þ
then the double periodic wave solutions to the Eq. (1) are
u¼
4 1 tanh ðsnðn; kÞdnðg; mÞÞ; n
ð17Þ
u¼
4 1 tanh ðcdðn; kÞdnðg; mÞÞ: n
ð18Þ
and
The profile of (17) is shown in Fig. 1(1–1). Case 2. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð19Þ
Substituting (19) into the constraint conditions (14), the parameters can be determined as
k ¼ 1; x ¼ a;
k2 ¼
2nb ; 3að2m2 1Þ
c2 ¼
a ; 4
ð20Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð21Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
6 3
3
4
2
2
2
1
1
0
0
1
1
2
2
0 2 4
3
3 100
50
0
50
100
3
2
1
0
1
2
3
4
2
1
0
1
2
3
4
3 0.6
6
0.4
2
0.2
1
2
0
0
0.0
4
0.2
2
1 0.4
4 2
0.6
6 3 5
0
20
5
10
0
10
20
10
5
15
10
5
0
5
10
15
4 1.5 1.0
2
0.5 0
0.0 0.5
2 1.0 1.5
4 10
5
0
5
10
0
5
10
pffiffi 3 Fig. 1. Various travelling waves described by solutions (17),pffiffi(21), (49), (69), (76) and (107). (1–1) n ¼ 3; a ¼ 25; b ¼ 16 ; c ¼ 5; m ¼ 13 ; k ¼ 2 3 2 ; k ¼ 40 . pffiffi pffiffi 2 5 3 2 5 (1–2) and (1–3) n¼ 3; a ¼ 16; b ¼ 1; c ¼ 2; m p ¼ffiffi 3 ; k ¼ 3 ; k ¼ 4 . (1–4) n ¼ 3; a ¼ 36; b ¼ 1; c ¼ 3, k ¼ 15 . (1–5) q n ffiffiffiffi ¼ 3; a ¼ 36; b ¼ 1; c ¼ 3; pffiffiffiffi pffiffi 57 3 3 7 10 k ¼ 57 . (1–6) n ¼ 3; a ¼ 25; b ¼ 1; c ¼ 5; k ¼ 10 . (1–7) and (1–8) n ¼ 3; a ¼ 16; b ¼ 10; c ¼ 2; m ¼ 4 ; k ¼ 4 ; k ¼ 2 71.
and
u¼
4 1 1 ; tanh n nsðg; mÞ csðg; mÞ
ð22Þ
The profiles of (21) are shown in Fig. 1(1–2) and (1–3). When m ! 0, the solutions (21), (22) turn to be another complex wave and periodic wave solutions respectively
u¼
4 tanhðnÞ 1 ; tanh n cscðgÞ cotðgÞ
ð23Þ
u¼
4 1 1 : tanh n cscðgÞ cotðgÞ
ð24Þ
and
Case 3. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 2
m2 ¼
1 m2 : 4
ð25Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
Substituting (25) into the constraint conditions (14), the parameters can be determined as
x ¼ að1 m2 Þ; k2 ¼
k ¼ 1;
2nbðm2 1Þ ; aðm2 þ 1Þðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð26Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð27Þ
u¼
4 1 1 ; tanh n ncðg; mÞ scðg; mÞ
ð28Þ
and
when m ! 0, the solutions (27), (28) turn to be another complex wave and periodic wave solutions respectively
u¼
4 tanhðnÞ 1 ; tanh n secðgÞ tanðgÞ
ð29Þ
u¼
4 1 1 : tanh n secðgÞ tanðgÞ
ð30Þ
and
Case 4. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð31Þ
Substituting (31) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 1;
x ¼ a; k2 ¼
2nb ; 3a
c2 ¼
a ; 4
ð32Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n cothðgÞ cschðgÞ
ð33Þ
u¼
4 1 1 : tanh n cothðgÞ cschðgÞ
ð34Þ
and
2
Case 5. From Table 1, choosing U ¼ snðn; kÞ (or U ¼ cdðn; kÞ) and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼k ;
l1 ¼ ð1 k2 Þ; m1 ¼ 1;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð35Þ
Substituting (35) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ am2 ; k2 ¼
aðm2
2m2 nb ; 2Þðm2 4Þ
c2 ¼
m2 a ; 4
ð36Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 tanhðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð37Þ
u¼
4 1 1 ; tanh n snðg; mÞ icsðg; mÞ
ð38Þ
and
when m ! 1, the solutions (37), (38) turn to be another complex wave solutions respectively
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 tanhðnÞ 1 ; tanh n tanhðgÞ icschðgÞ
ð39Þ
u¼
4 1 1 : tanh n tanhðgÞ icschðgÞ
ð40Þ
and
Case 6. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ sdðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼ m2 ð1 m2 Þ;
l2 ¼ 2m2 1; m2 ¼ 1:
ð41Þ
Substituting (41) into the constraint conditions (14), the parameters can be determined as 2
k ¼
m2 ð1 m2 Þ ; 1 þ m2 m4
x ¼ 4am2 ðm2 1Þ; k2 ¼
nbð1 þ m2 m4 Þ ; am4 ð1 m4 Þ
c2 ¼ að1 þ m2 m4 Þ;
ð42Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðcnðn; kÞdsðg; mÞÞ: n
ð43Þ
Case 7. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼ 1;
l2 ¼ 2m2 1; m2 ¼ m2 ð1 m2 Þ:
ð44Þ
Substituting (44) into the constraint conditions (14), the parameters can be determined as 2
k ¼
1 ; 1 þ m2 m4
x ¼ 4a; c2 ¼ að1 þ m2 m4 Þ; k2 ¼
nbð1 þ m2 m4 Þ ; am2 ð2 þ 3m2 m6 Þ
ð45Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðcnðn; kÞsdðg; mÞÞ: n
ð46Þ
Case 8. From Table 1, choosing U ¼ cnðn; kÞ and V ¼ nsðg; mÞ þ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k ;
x
l1 ¼ 2k2 1; m1 ¼ 1 k2 ;
4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð47Þ
Substituting (47) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 0;
x ¼ a; k2 ¼
4nb ; 15a
a c2 ¼ ; 4
ð48Þ
then the complex wave solution to the Eq. (1) is
u¼
4 1 1 tanh sechðnÞ sinðgÞ : n 2
ð49Þ
The profile of (49) is shown in Fig. 1(1–4). Case 9. From Table 1, choosing U ¼ dnðn; kÞ and V ¼ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ k2 1;
x 4a
¼ 1;
l2 ¼ 2 m2 ; m2 ¼ 1 m2 :
ð50Þ
Substituting (50) into the constraint conditions (14), the parameters can be determined as
k ¼ m;
x ¼ 4a; k2 ¼
nb ; 4að2 m2 Þ
c2 ¼ a;
ð51Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðdnðn; mÞscðg; mÞÞ; n
ð52Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
165
when m ! 0, the solution (52) turn to be another periodic wave solution
u¼
4 1 tanh ðtanðgÞÞ: n
ð53Þ
Case 10. From Table 1, choosing U ¼ dnðn; kÞ and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
x
l1 ¼ 2 k2 ; m1 ¼ k2 1;
4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð54Þ
Substituting (54) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
x ¼ a; k2 ¼
4nb ; 15aðm2 1Þ
a c2 ¼ ; 4
ð55Þ
then the complex wave solution to the Eq. (1) is
u¼
4 dnðn; kÞ 1 : tanh n nsðg; mÞ dsðg; mÞ
ð56Þ
Case 11. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð57Þ
Substituting (57) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ a; k2 ¼
nb ; 3aðm2 1Þ
c2 ¼
a ; 4
ð58Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð59Þ
when m ! 0, the solution (59) turn to be another complex wave solution
u¼
4 cothðnÞ 1 : tanh n cscðgÞ cotðgÞ
ð60Þ
Case 12. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð61Þ
Substituting (61) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; að3 þ m2 Þ
c2 ¼
að1 m2 Þ ; 4
ð62Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð63Þ
when m ! 0, the solution (63) turn to be another complex wave solution
u¼
4 cothðnÞ 1 : tanh n secðgÞ tanðgÞ
ð64Þ
Case 13. From Table 1, choosing U ¼ nsðn; kÞ (or U ¼ dcðn; kÞ) and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
Substituting (65) into the constraint conditions (14), the parameters can be determined as
ð65Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
k ¼ m;
x ¼ a; k2 ¼
4nb ; 9að1 m2 Þ
c2 ¼
a ; 4
ð66Þ
then the complex wave solutions to the Eq. (1) are
u¼
4 nsðn; kÞ 1 tanh n nsðg; mÞ dsðg; mÞ
ð67Þ
u¼
4 dcðn; kÞ 1 ; tanh n nsðg; mÞ dsðg; mÞ
ð68Þ
and
when m ! 0, the solutions (67) and (68) turn to be another double periodic wave solutions respectively
u¼
4 1 1 cscðnÞ sinðgÞ ; tanh n 2
ð69Þ
u¼
4 1 1 secðnÞ sinðgÞ : tanh n 2
ð70Þ
and
The profile of (69) is shown in Fig. 1(1–5). Case 14. From Table 1, choosing U ¼ nsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ ð1 þ k2 Þ; m1 ¼ k2 ;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð71Þ
Substituting (71) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
x ¼ am2 ; k2 ¼
m2 nb ; að1 m2 Þð4 m2 Þ
c2 ¼
am2 ; 4
ð72Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð73Þ
Case 15. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2k2 1; m1 ¼ k2 ;
x 4a
¼ 1 m2 ;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð74Þ
Substituting (74) into the constraint conditions (14), the parameters can be determined as
k ¼ 1;
m ¼ 1;
x ¼ 0; k2 ¼
nb ; 4a
c2 ¼ a;
ð75Þ
then the complex wave solution to the Eq. (1) is
u¼
4 1 coshðnÞ : tanh n sinhðgÞ
ð76Þ
The profile of (76) is shown in Fig. 1(1–6). Case 16. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ sdðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2k2 1; m1 ¼ k2 ;
x 4a
¼ m2 ð1 m2 Þ;
l2 ¼ 2m2 1; m2 ¼ 1:
ð77Þ
Substituting (77) into the constraint conditions (14), the parameters can be determined as
1 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ m2 m4
x ¼ 4am2 ðm2 1Þ; k2 ¼
nbð1 þ m2 m4 Þ ; m2 að2 þ 3m2 m6 Þ
c2 ¼ aðm4 m2 1Þ;
ð78Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðncðn; kÞdsðg; mÞÞ: n
ð79Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
167
Case 17. From Table 1, choosing U ¼ ncðn; kÞ and V ¼ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
x
l1 ¼ 2k2 1; m1 ¼ k2 ;
4a
l2 ¼ 2m2 1; m2 ¼ m2 ð1 m2 Þ:
¼ 1;
ð80Þ
Substituting (80) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ð1 m2 Þð1 þ m2 m4 Þ ; 1 þ m2 m4
x ¼ 4a; k2 ¼
nbð1 þ m2 m4 Þ ; am4 ð1 m4 Þ
c2 ¼ að1 þ m2 m4 Þ;
ð81Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðncðn; kÞsdðg; mÞÞ: n
ð82Þ
Case 18. From Table 1, choosing U ¼ ndðn; kÞ and V ¼ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼ k 1;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼ 1 m2 ;
l2 ¼ 2 m2 ; m2 ¼ 1:
ð83Þ
Substituting (83) into the constraint conditions (14), the parameters can be determined as
k ¼ m;
x ¼ 4að1 m2 Þ; k2 ¼
nb ; 4að2 m2 Þ
c2 ¼ a;
ð84Þ
then the double periodic wave solution to the Eq. (1) is
u¼
4 1 tanh ðndðn; kÞcsðg; mÞÞ; n
ð85Þ
when m ! 0, the solution (85) turn to be another periodic wave solution
4 1 u ¼ tanh ðcotðgÞÞ: n
ð86Þ
Case 19. From Table 1, choosing U ¼ scðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼
1 ; 4
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð87Þ
Substituting (87) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ a; k2 ¼
nb ; 3am2
c2 ¼
a ; 4
ð88Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð89Þ
when m ! 1, the solution (89) turn to be another complex wave solution
u¼
4 tanðnÞ 1 : tanh n cothðgÞ cschðgÞ
ð90Þ
Case 20. From Table 1, choosing U ¼ scðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð91Þ
Substituting (91) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; am2 ðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð92Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð93Þ
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B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172 2
Case 21. From Table 1, choosing U ¼ scðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼1k ;
l1 ¼ 2 k2 ; m1 ¼ 1;
x 4a
m2 ; 4
¼
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð94Þ
Substituting (94) into the constraint conditions (14), the parameters can be determined as 2
k ¼ 0;
x ¼ am2 ; k2 ¼
nbm ; að4 m2 Þ
c2 ¼
am2 ; 4
ð95Þ
then the complex wave solution to the Eq. (1) is
u¼
4 tanðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð96Þ
when m ! 1, the solution (96) turn to be another complex wave solution
u¼
4 tanðnÞ 1 : tanh n tanhðgÞ icschðgÞ
ð97Þ
Case 22. From Table 1, choosing U ¼ csðn; kÞ and V ¼ nsðg; mÞ csðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
1 ; 4
¼
l2 ¼
1 2m2 ; 2
1 4
m2 ¼ :
ð98Þ
Substituting (98) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ a; k2 ¼
nb ; 3am2
c2 ¼
a ; 4
ð99Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 ; tanh n nsðg; mÞ csðg; mÞ
ð100Þ
when m ! 1, the solution (100) turn to be another complex wave solution
u¼
4 cotðnÞ 1 : tanh n cothðgÞ cschðgÞ
ð101Þ
Case 23. From Table 1, choosing U ¼ csðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
1 m2 ; 4
¼
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð102Þ
Substituting (102) into the constraint conditions (14), the parameters can be determined as
k ¼ 0;
x ¼ að1 m2 Þ; k2 ¼
nbðm2 1Þ ; am2 ðm2 þ 3Þ
c2 ¼
að1 m2 Þ ; 4
ð103Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð104Þ
Case 24. From Table 1, choosing U ¼ csðn; kÞ and V ¼ nsðg; mÞ dsðg; mÞ, respectively, and then from (13), we have
x 4c2
¼ 1;
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
x 4a
¼
1 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð105Þ
Substituting (105) into the constraint conditions (14), the parameters can be determined as
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 ;
x ¼ a; k2 ¼
4nb ; 3að5 m2 Þ
then the complex wave solution to the Eq. (1) is
c2 ¼
a ; 4
ð106Þ
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
u¼
4 csðn; kÞ 1 ; tanh n nsðg; mÞ dsðg; mÞ
169
ð107Þ
The profiles of (107) are shown in Fig. 1(1–7) and (1–8). When m ! 0, the solution (107) turn to be another complex wave solution
4 1 1 u ¼ tanh cschðnÞ sinðgÞ : n 2
ð108Þ
2
Case 25. From Table 1, choosing U ¼ csðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
¼ 1;
x
l1 ¼ 2 k2 ; m1 ¼ 1 k2 ;
4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð109Þ
Substituting (109) into the constraint conditions (14), the parameters can be determined as
x ¼ am2 ; k2 ¼
k ¼ 0;
m2 nb ; að4 m2 Þ
c2 ¼
am2 ; 4
ð110Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cotðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð111Þ
when m ! 1, the solution (111) turn to be another complex wave solution
u¼
4 cotðnÞ 1 : tanh n tanhðgÞ icschðgÞ
ð112Þ
Case 26. From Table 1, choosing U ¼ nsðn; kÞ csðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ, respectively, and then from (13), we have
x 4c2
¼
1 ; 4
2
l1 ¼
1 2k ; 2
1 4
m1 ¼ ;
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
ð113Þ
Substituting (113) into the constraint conditions (14), the parameters can be determined as 2
k ¼
nb þ ak2 m4 m2 nb ak2 m2 ðm2 1Þ
x ¼ að1 m2 Þ; c2 ¼ að1 m2 Þ;
;
ð114Þ
then the complex wave solution to the Eq. (1) is
u¼
4 nsðn; kÞ csðn; kÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð115Þ
2
Case 27. From Table 1, choosing U ¼ nsðn; kÞ csðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1 respectively, and then from (13), we have
x 4c2
¼
1 ; 4
2
l1 ¼
1 2k ; 2
1 4
m1 ¼ ;
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð116Þ
Substituting (116) into the constraint conditions (14), the parameters can be determined as 2
k ¼
am2 k2 þ m2 nb ak2 ak2 m2 ðm2 1Þ
x ¼ am2 ; c2 ¼ am2 ;
;
ð117Þ
then the complex wave solution to the Eq. (1) is
u¼
4 nsðn; kÞ csðn; kÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð118Þ
Case 28. From Table 1, choosing U ¼ ncðn; kÞ scðn; kÞ and V ¼ ncðg; mÞ scðg; mÞ respectively, and then from (13), we have
x 4c2
2
¼
1k ; 4
2
l1 ¼
1þk ; 2
2
m1 ¼
1k ; 4
x 4a
¼
1 m2 ; 4
l2 ¼
1 þ m2 ; 2
m2 ¼
1 m2 : 4
Substituting (119) into the constraint conditions (14), the parameters can be determined as
ð119Þ
170
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172 2
x ¼ að1 m2 Þ; k2 ¼
nbðk 1Þð1 m2 Þ 2
aðk m2 Þ
;
c2 ¼
að1 m2 Þ 1k
2
ð120Þ
;
and
k ¼ 1;
x ¼ 0; k2 ¼
m ¼ 1;
nbc
2
ðc2 aÞ
2
ð121Þ
;
then the complex wave solutions to the Eq. (1) are
u¼
4 ncðn; kÞ scðn; kÞ 1 : tanh n ncðg; mÞ scðg; mÞ
ð122Þ
u¼
4 1 coshðnÞ sinhðnÞ ; tanh n coshðgÞ sinhðgÞ
ð123Þ
and
when k ! 0, the solution (122) turn to be another complex wave solution
u¼
4 secðnÞ tanðnÞ 1 ; tanh n ncðg; mÞ scðg; mÞ
ð124Þ
when m ! 0, the solution (122) turn to be another complex wave solution
u¼
4 1 ncðn; kÞ scðn; kÞ : tanh n secðgÞ tanðgÞ
ð125Þ
2
Case 29. From Table 1, choosing U ¼ ncðn; kÞ scðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼
1k ; 4
2
l1 ¼
1þk ; 2
2
m1 ¼
1k ; 4
x 4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð126Þ
Substituting (126) into the constraint conditions (14), the parameters can be determined as 2
x ¼ am2 ; k2 ¼
m2 nbðk 1Þ 2
2 2
aðk m4 ð1 k Þ ð1 þ m2 ÞÞ
;
c2 ¼
am2 2
1k
ð127Þ
;
and 2
k ¼ 1;
m ¼ 0;
x ¼ 0; k2 ¼
nbc ; c2
ð128Þ
a2
then the complex wave solutions to the Eq. (1) are
u¼
4 ncðn; kÞ scðn; kÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð129Þ
u¼
4 1 coshðnÞ sinhðnÞ ; tanh n sinðgÞ i cotðgÞ
ð130Þ
and
when k ! 0, the solution (129) turn to be another complex wave solution
u¼
4 secðnÞ tanðnÞ 1 ; tanh n snðg; mÞ icsðg; mÞ
ð131Þ
when m ! 1, the solution (129) turn to be another complex wave solution
u¼
4 ncðn; kÞ scðn; kÞ 1 : tanh n tanhðgÞ icschðgÞ
ð132Þ
2
Case 30. From Table 1, choosing U ¼ nsðn; kÞ dsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
B. He et al. / Applied Mathematics and Computation 229 (2014) 159–172
x 4c2
¼
2
1 ; 4
l1 ¼
k 2 ; 2
2
m1 ¼
x
k ; 4
¼
4a
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
171
ð133Þ
Substituting (133) into the constraint conditions (14), the parameters can be determined as
x ¼ am2 ; k2 ¼
k ¼ 1;
m2 nb að1 m2 Þ
2
c2 ¼ am2 ;
;
ð134Þ
then the complex wave solution to the Eq. (1) is
u¼
4 cothðnÞ cschðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð135Þ
2
Case 31. From Table 1, choosing U ¼ snðn; kÞ icsðn; kÞ and V ¼ snðg; mÞ icsðg; mÞ; i ¼ 1, respectively, and then from (13), we have
x 4c2
2
¼
k ; 4
2
l1 ¼
k 2 ; 2
2
m1 ¼
x
k ; 4
4a
¼
m2 ; 4
l2 ¼
m2 2 ; 2
m2 ¼
m2 : 4
ð136Þ
Substituting (136) into the constraint conditions (14), the parameters can be determined as 2
x ¼ am2 ; k2 ¼
k m2 nb 2
aðk m2 Þ
2
;
c2 ¼
am2 2
k
;
ð137Þ
;
ð138Þ
and
m ¼ 0;
k ¼ 0;
x ¼ 0; k2 ¼
nbc ðc2
2
aÞ
2
then the complex wave solutions to the Eq. (1) are
u¼
4 snðn; kÞ icsðn; kÞ 1 tanh ; n snðg; mÞ icsðg; mÞ
ð139Þ
u¼
4 1 sinðnÞ i cotðnÞ ; tanh n sinðgÞ i cotðgÞ
ð140Þ
and
when k ! 1, the solution (139) turn to be another complex wave solution
u¼
4 tanhðnÞ icschðnÞ 1 : tanh n snðg; mÞ icsðg; mÞ
ð141Þ
4. Conclusion In this paper, we obtained many exact solutions of a generalized sinh–Gordon equation, these solutions involve combinations of the arctanh function, Jacobi elliptic functions, hyperbolic functions and trigonometric functions. Compare with Refs. [8,9], most solutions of our results are entirely new. Acknowledgements This research was supported by Natural Science Foundation of Yunnan Province, China (2013FZ117). We thank the referees’ valuable suggestions. References [1] A.M. Wazwaz, The tanh method: exact solutions of the sine–Gordon and the sinh–Gordon equations, Appl. Math. Comput. 167 (2005) 1196–1210. [2] K.W. Chow, A class of doubly periodic waves for nonlinear evolution equations, Wave Motion 35 (2002) 71–90. [3] H.T. Chen, H.C. Yin, Double elliptic equation method and new exact solutions of the (n+1)-dimensional sinh–Gordon equation, J. Math. Phys. 48 (2007) 013504-1–013504-7. [4] J.B. Li, M. Li, Bounded travelling wave solutions for the (n+1)-dimensional sine- and sinh–Gordon equations, Chaos Soliton Fract. 25 (2005) 1037–1047. [5] K. Narita, Deformed sine- and sinh–Gordon equations, deformed Liouville equation, and their discrete models, J. Phys. Soc. Jpn. 72 (2003) 1339–1349. [6] H. Zhang, New exact solutions for the sinh–Gordon equation, Chaos Soliton Fract. 28 (2006) 489–496. [7] Z.T. Fu, S.K. Liu, S.D. Liu, Exact Jacobian elliptic function solutions to sinh–Gordon equation, Commun. Theor. Phys. (Beijing, China) 45 (2006) 55–60. [8] A.M. Wazwaz, Exact solutions for the generalized sine–Gordon and the generalized sinh–Gordon equations, Chaos Soliton Fract. 28 (2006) 127–135.
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