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All meromorphic solutions for two forms of odd order algebraic differential equations and its applications
Applied Mathematics and Computation 240 (2014) 240–251
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
All meromorphic solutions for two forms of odd order algebraic differential equations and its applications q Wenjun Yuan a,b,⇑, Yonghong Wu c, Qiuhui Chen d,*, Yong Huang e a
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China c Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia d Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China e School of Computer Science and Education Software, Guangzhou University, Guangzhou 510006, China b
a r t i c l e
i n f o
Keywords: Differential equation Exact solution Meromorphic function Elliptic function
a b s t r a c t In this article, we employ the Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for two classes of odd order algebraic differential equations with the weak hp; qi and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of some generalized Bretherton equations by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics, and using the traveling wave nobody can find other new exact solutions for many nonlinear partial differential equations by any method. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and main results In this paper, a meromorphic function wðzÞ means that wðzÞ is holomorphic in the complex plane C except for poles. }ðz; g 2 ; g 3 Þ is the Weierstrass elliptic function with invariants g 2 and g 3 . It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory, such as
Tðr; f Þ; mðr; f Þ; Nðr; f Þ Nðr; f Þ; . . . : For detail of Nevanlinna’s value-distribution theory, please see [1–3]. We denote by Sðr; f Þ any function satisfying Sðr; f Þ ¼ ofTðr; f Þg, as r ! 1, possibly outside of a set of finite measure. Recently, some authors find the exact solutions of certain PDEs combining with the complex ODEs. In 2013, Yuan et al. [4,5] researched the existence of meromorphic solutions of some algebraic differential equations with constant coefficients by using the Nevanlinna’s value distribution theory, and gave the representations of all meromorphic solutions. At the same time, the complex method to find all traveling wave exact solutions of corresponding partial differential equations was given. q
This work was completed with the support with the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121).
⇑ Corresponding authors. Address: Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China (Q. Chen). E-mail addresses: [email protected] (W. Yuan), [email protected] (Y. Wu), [email protected] (Q. Chen), [email protected] (Y. Huang). http://dx.doi.org/10.1016/j.amc.2014.04.099 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
241
In order to state these results, we need some concepts and notations: Set m 2 N :¼ f1; 2; 3; . . .g; rj 2 N0 ¼ N [ f0g; r ¼ ðr 0 ; r 1 ; . . . ; r m Þ; j ¼ 0; 1; . . . ; m. r
r
rm
Mr ½wðzÞ :¼ ½wðzÞr0 ½w0 ðzÞ 1 ½w00 ðzÞ 2 ½wðmÞ ðzÞ : dðrÞ :¼ r 0 þ r 1 þ þ rm is called the degree of M r ½w. Definition 1.1. A differential polynomial with constant coefficients is defined by
X ar M r ½w;
P½w :¼
r2I
where ar are constants, and I is a finite index set. The total degree deg P½w of P½w is defined by deg P½w :¼ maxr2I fdðrÞg. Consider the differential equations
Eðz; wÞ :¼ P½w þ w0 wðmÞ
1 h ðmþ1Þ i2 w 2 awn ¼ 0; 2
ð1:1Þ
where a – 0 is a constant, n 2 N. Definition 1.2. Let p; q 2 N. Suppose that the meromorphic solution w of the Eq. (1.1) has at least one pole. We say that the Eq. (1.1) satisfies hp; qi condition if there exactly exist p distinct meromorphic solutions of the Eq. (1.1) with pole of multiplicity q at z ¼ 0. We say that the Eq. (1.1) satisfies weak hp; qi condition if substituting Laurent series 1 X
wðzÞ ¼
c k zk ;
q > 0; cq – 0
ð1:2Þ
k¼q
into the Eq. (1.1), we can determine p distinct principle 1 X
c k zk
k¼q
with pole of multiplicity q at z ¼ 0. Definition 1.3. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of eaz ; a 2 C, or a rational function of z. Theorem 1.4 ([4,5]). Let p; l; m; n 2 N; deg P½w < n, and the Eq. (1.1) satisfy hp; qi condition. Then all meromorphic solutions w of the Eq. (1.1) must be one of the following four cases: (I) w is a constant. (II) w is a rational function with lð6 pÞ distinct poles of multiplicity q, and w :¼ RðzÞ has the form of
RðzÞ ¼
q l X X
cij
i¼1 j¼1
ðz zi Þj
þ c0 :
ð1:3Þ
(III) w is a rational function RðnÞ of n ¼ eaz ða 2 CÞ. RðnÞ has lð6 pÞ distinct poles of multiplicity q, and has the form (1.3). (IV) w is an elliptic function with double periods 2x1 ; 2x2 , which has lð6 pÞ distinct poles of multiplicity q per parallelogram of periods. And w has one of the following three forms.
(1.1) If w is even, and the pole of }ðzÞ is the pole of w, then w is a rational Q ðnÞ in n :¼ }ðzÞ and has the form of
Q ðnÞ ¼
q l X X i¼1 j¼1
cij ðn ni Þ
j
þ
q0 X ci ni ; i¼0
where q0 6 2q. If the pole of }ðzÞ is not the pole of w, then q0 ¼ 0. (1.2) If w is odd, then }0wðzÞ is a rational Q ðnÞ of n :¼ }ðzÞ. (1.3) If w is non-odd and non-even, then
wðzÞ ¼ Q 1 ð}ðzÞÞ þ }0 ðzÞQ 2 ð}ðzÞÞ; where Q 1 ðnÞ and Q 2 ðnÞ are rational functions with the form (1.4).
ð1:4Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
Each elliptic function solution with pole z ¼ 0 can be expressed as the form
wðzÞ ¼
! 2 j2 j2 q q j l1 X l1 X X ð1Þj cij d 1 }0 ðzÞ þ Bi ci1 }0 ðzÞ þ Bi X ð1Þ clj d þ }ðzÞ þ }ðzÞ þ c0 ; j2 4 }ðzÞ Ai ðj 1Þ! dz 2 }ðzÞ Ai ðj 1Þ! dzj2 i¼1 j¼2 i¼1 j¼2
where cij can be determined by (1.2), }ðzÞ is the Weierstrass elliptic function, B2i ¼ 4A3i g 2 Ai g 3 and The authors proposed a conjecture below in [4]:
Pl
i¼1 ci1
ð1:5Þ ¼ 0.
Conjecture. Let p; l; m; n 2 N; deg P½w < n, and the Eq. (1.1) satisfy weak hp; qi condition. Then the conclusion of Theorem 1.4 holds. In order to state and prove our result, we need some notations and concepts [6]: Definition 1.5. Let Eðz; wÞ ¼ 0 be a m-th order algebraic differential equation, w is a meromorphic solution. The involved term of Eðz; wÞ determining the multiplicity q of w is called the dominant term. The dominant part of Eðz; wÞ is consisted by ^ wÞ. The multiplicity of pole of each term in Eðz; ^ wÞ is the same integer denoted by all dominant terms, and is denoted by Eðz; ^ wÞ is denoted by Dr ðqÞ. DðqÞ. The multiplicity of pole of each monomial M r ½w in Eðz; wÞ Eðz; Obviously
Dr ðqÞ ¼ qdðrÞ þ r 1 þ 2r 2 þ þ mr m < DðqÞ: ^ ^ wÞ with respect to w is denoted by E ^0 ðz; wÞ :¼ @ Eðz;wÞ The derivative of dominant part Eðz; , which can be defined in the fol@w lowing form:
for any
v : E^0 ðz; wÞv ¼ lim k!0
^ w þ kv Þ Eðz; ^ wÞ Eðz; : k
ð1:6Þ
The root of the equation
^0 ðx; cq xq Þxiq ¼ 0 PðiÞ ¼ limxiþDðqÞ E
ð1:7Þ
x!0
is called the Fuchs index of the equation Eðz; wÞ ¼ 0. In this paper, we obtain two results for some odd order Eq. (1.1) to show that the conjecture holds. By our results, then we give the corresponding complex method of finding all travel wave exact solutions of some partial differential equations and an example of finding all meromorphic solutions of the generalized Bretherton equation and three modified Kawahara equations by using our method and results. The generalized Bretherton equation [7]
1 b m 2 2 w0 w000 ðw00 Þ ðw0 Þ awn þ bw þ c ¼ 0; 2 2
ð1:8Þ
where a – 0; b; c and b are constants. Three modified Kawahara equations [8]
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ w3 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:9Þ
and
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ 30w4 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:10Þ
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ 4w6 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:11Þ
and
where b; C 0 ; C 1 and C 2 are constants. Recently, Kudryashov et al. [7,8] obtained the following result by using their method for the Eq. (1.8). Theorem 1.6. All meromorphic solutions w of the Eq. (1.8), (1.9), (1.10) and (1.11) belong to the class W for the pair of ðm; nÞ ¼ ð3; 6Þ; ð4; 6Þ; ð5; 6Þ, and ð3; 4Þ, respectively. Furthermore, without loss of generality, the parameter a in the Eq. (1.12) reduces 4; 30 or 1 corresponding to n ¼ 6; 4 or 3, respectively. At the same time, the elliptic solutions of the Eq. (1.8) corresponding to these cases are of the forms:
we;3;6 ðzÞ ¼
b }0 ðz z0 ; 0; g 3 Þ 20 ; 2}ðz z0 ; 0; g 3 Þ 2
3
b b where n ¼ 6; m ¼ 3; a ¼ 4; c ¼ 200 ; b ¼ 0, and g 3 ¼ 400 and z0 2 C.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
we;4;6 ðzÞ ¼
}0 ðz z0 ; g 2 ; g 3 Þ 2}ðz z0 ; g 2 ; g 3 Þ 2bb 60
243
;
where n ¼ 6; m ¼ 4; a ¼ 4; c ¼ ð2bbÞð2bþbÞð9bþ2bÞ ; g 2 ¼ bð2bbÞ and g 3 ¼ ð13bþ4bÞð2bbÞ and z0 2 C. 6000 120 216000 3
we;5;6 ðzÞ ¼
5b }0 ðz z0 ; g 2 ; g 3 Þ þ 864
2}ðz z0 ; g 2 ; g 3 Þ
7b2 864
þ
b ; 24
6
4
6
2
b b b 5b where n ¼ 6; m ¼ 5; a ¼ 4; c ¼ 11943936 ; g 2 ¼ 497664 and g 3 ¼ 806215680 ; b ¼ 144 and z0 2 C.
" #2 1 }0 ðz z0 ; g 2 ; g 3 Þ b we;3;4 ðzÞ ¼ 2}ðz z0 ; g 2 ; g 3 Þ ; b 4 }ðz z0 ; g 2 ; g 3 Þ þ 60 60 4
2
3
11b b b where n ¼ 4; m ¼ 3; a ¼ 30; c ¼ 303750 ; g 2 ¼ 540 and g 3 ¼ 81000 and z0 2 C. All elliptic solutions of the Eq. (1.9) are of forms
we;9 ðzÞ ¼ 280}2 ðz z0 ; g 2 ; g 3 Þ
140b 31b2 C 0 14g 2 }ðz z0 ; g 2 ; g 3 Þ þ ; 39 3042 6 3
3
31b 2 where g 3 ¼ 7bg 4745520 ; g 2 is a solution of quadratic equation 780
g 22
b2 1457b4 C 20 C1 g2 þ ¼ 0; 1014 662158224 23184 1932
and
! 287 2 1148 41328b4 372b6 5C 20 b2 5b2 C 1 C3 C0C1 g2 þ C0 þ C1 þ þ 0 ; 207 69 656903 4826809 18252 1521 108 6
C2 ¼
and z0 2 C. All elliptic solutions of the Eq. (1.10) are of two forms
we;10;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ 2
b ; 60
3
0 b 1 b where g 2 ¼ 10C180 ; g 3 ¼ 5C 0 bþ450C ; C2 ¼ 5400
we;10;2 ðzÞ ¼
32b4 220C 0 b2 þ125C 20 8100C 1 b 324000
10800}2 ðz z0 ; g 2 ; g 3 Þ 360b}ðz z0 ; g 2 ; g 3 Þ þ 25C 0 b2 ; 180ð60}ðz z0 ; g 2 ; g 3 Þ b
2
2
3
2
0 Þð4b 0 þb 0 bþ2b where g 2 ¼ 5C540 ; g 3 ¼ 25C ; C 1 ¼ 0; C 2 ¼ ð11b 100C 1215000 162000 All elliptic solutions of the Eq. (1.11) are of two forms
we;11;1 ðzÞ ¼ 2
and z0 2 C.
60}0 ðz z0 ; g 2 ; g 3 Þ ; 2ð60}ðz z0 ; g 2 ; g 3 Þ þ bÞ
we;11;2 ðzÞ ¼
3
3
150C 0 b 0 where g 2 ¼ b 10C ; g 3 ¼ 13b108000 ; C 1 ¼ 0; C 2 ¼ 9b 120 Our main results follows.
100C 0 b 3000
25C 0 Þ
and z0 2 C.
60i}0 ðz z0 ; g 2 ; g 3 Þ 2ð60}ðz z0 ; g 2 ; g 3 Þ þ bÞ and z0 2 C.
Theorem 1.7. Let p; l; m; n 2 N; deg P½w < n; m be an odd integer and mþ1 be an even positive integer. If the Eq. (1.1) satisfies h mþ1 i22 ^ wÞ ¼ w0 wðmÞ 1 wð 2 Þ awn , then all meromorphic solutions w of the Eq. weak hp; qi condition and the dominant part Eðz; 2 (1.1) belong to the class W. Furthermore, the conclusions of Theorem 1.4 hold. Theorem 1.8. Let p; l; m; n 2 N; deg P½w < n; m and mþ1 be two odd positive integers. If the Eq. (1.1) satisfies weak hp; qi conh mþ1 i22 0 ðmÞ 1 ð Þ ^ dition and the dominant part Eðz; wÞ ¼ w w þ 2 w 2 awn , then all meromorphic solutions w of the Eq. (1.1) belong to the class W. Furthermore, the conclusions of Theorem 1.4 hold. Remark 1.9. By Theorem 1.6 and taking m ¼ 3 in Theorem 1.7, we obtain the following corollary. Corollary 1.10. Consider the equation
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ awn þ PðzÞ ¼ 0; 2 2
ð1:12Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where b and a – 0 are constants, PðzÞ is a polynomial such that deg P < n and n is a positive integer. Then all meromorphic solutions w of the Eq. (1.12) belong to the class W iff n ¼ 6; 4 and 3. Furthermore, without loss of generality, the Eq. (1.12) reduces (1.10),
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ 30w4 þ P1 ðwÞ; 2 2
ð1:13Þ
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ 4w6 þ P 2 ðwÞ; 2 2
ð1:14Þ
or
where P 1 ðwÞ ¼ C 3 w3 þ C20 w2 þ C 1 w þ C 2 and P2 ðwÞ ¼ C 5 w5 þ C 4 w4 þ C 3 w3 þ C20 w2 þ C 1 w þ C 2 ; b; C 0 ; C 1 ; C 2 ; C 3 ; C 4 and C 5 are constants. All elliptic solutions of the Eq. (1.13) are of
we;13;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ þ C2
C3 b ; 120 60
7C 3
2
7C 2 b
3
2
C2
C0 7C 0 b b 7b b 1 0 C3 3 3 3 where g 2 ¼ 18 þ 720 180 , g 3 ¼ 43200 þ 43200 7C720 þ 1080 7C61 þ 10800 , C 0 ¼ 10 403 ; C 1 ¼ 19200 C 2 and C 3 are three constants, z0 2 C. And
we;13;2 ðzÞ ¼
2304000C 2 þC 43 þ80b4 48C 3 b3 8C 23 b2 þ4C 33 b , C 3 þ3b
b3 216000}3 þ 60ð240} þ C 3 þ 2bÞ 3600}2 108000} þ b2 ; 7200 3600}2 ðz z0 ; g 2 ; g 3 Þ 108000}ðz z0 ; g 2 ; g 3 Þ b2
C2
C3
2
2
4
C4
C 2 b2
b 3b 3 3 3 where C 0 ¼ 403 b5 ; C 1 ¼ 14400 þ C600 ; C 2 ¼ 15000 6912000 144000 ; } ¼ }ðz z0 ; g 2 ; g 3 Þ; b and C 3 are two constants, z0 2 C. All elliptic solutions of the Eq. (1.14) are of
we;14;1 ðzÞ ¼
1 }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24
where g 2 ¼ 0; g 3 ¼ 4A3 B2 , and C 0 ¼ 120 c430 þ 6 c230 b þ 360 c230 A þ 120 Bc30 6 b A; C 1 ¼ 2 b c330 þ 2 b B þ 24 c530 120 c330 A 6 c30 b A; C 2 ¼ 4 c630 10 A3 2 g 3 3b A2 þ 2b c430 2 c30 b B þ 30 c430 A þ 20 Bc330 12 Bc30 A 3 c230 b A; C 3 ¼ 2
60 Bc230 þ 12 BAþ
80 c330 2 c30 b 120 c30 A 20 B; C 4 ¼ 60 c230 þ 2b þ 30 A, and c30 ¼ C245 ; b and C 5 are two given constants, A; B and z0 2 C. Or, some other elliptic solutions of the Eq. (1.14) are of
we;14;2 ðzÞ ¼
i }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24 2
where g 2 ¼ 0; g 3 ¼ 4A3 B2 , and C 0 ¼ 120c440 6bc240 360c240 A 6bA 120ic40 B; C 1 ¼ 24 c540 þ 2 b c340 þ 120 c340 A þ 60 ic40 B þ2ibB þ 12iBA þ 6c40 bA; C 2 ¼ 2 c40 b þ 120 c40 A þ 20 iB; C 4 ¼
3
10A þ 2g 3 þ 3b A2 4c640 2b 2 2 b 2 60 c40 2 30 A; i ¼ 1;
3 30c440 A 20ic40 B 2ic40 bB 12ic40 AB 3c240 bA; C5 c30 ¼ 24 ; b and C 5 are two given constants, A; B and z0
c440
C3 ¼
80c340 þ
2 C.
2. Preliminary lemmas and the complex method In order to prove our results, we need the following three lemmas. Lemma 2.1 ((Clunie Lemma) [2,3]). Let w be a meromorphic solution of an equation wn Pðz; w; w0 ; . . . ; wðmÞ Þ ¼ Q ðz; w; w0 ; . . . ; wðmÞ Þ, where P and Q are two differential polynomials in w and its derivatives w0 ; . . . ; wðmÞ with meromorphic coefficients fak jk 2 Ig, and mðr; ak Þ ¼ Sðr; wÞ. If the total degree of Q ðz; w; w0 ; . . . ; wðmÞ Þ in w and its derivatives w0 ; . . . ; wðmÞ satisfies deg Q ðz; w; w0 ; . . . ; wðmÞ Þ 6 n, then
mðr; Pðz; w; w0 ; . . . ; wðmÞ ÞÞ ¼ Sðr; wÞ: Lemma 2.2 ((Mohon’ko Theorem) [2,3]). Let w be a meromorphic function,
Rðz; wÞ :¼
Pp ai ðzÞwi Pðz; wÞ ¼ Pqi¼0 j Qðz; wÞ j¼0 bj ðzÞw
be an irreducible rational function in w with meromorphic coefficients ai ðzÞ; bj ðzÞ , and Tðr; ai Þ ¼ Sðr; wÞ; Tðr; bj Þ ¼ Sðr; wÞ; i ¼ 0; . . . ; p; j ¼ 0; . . . ; q. Then
Tðr; Rðz; wÞÞ ¼ dTðr; wÞ þ Sðr; wÞ; where d ¼ maxðp; qÞ.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
245
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [9]. 1 Let x1 ; x2 be two given complex numbers such that Im x x2 > 0; L ¼ L½2x1 ; 2x2 be discrete subset L½2x1 ; 2x2 ¼ fxjx ¼ 2nx1 þ 2mx2 ; n; m 2 Zg, which is isomorphic to Z Z. The discriminant D ¼ Dðc1 ; c2 Þ :¼ c31 27c22 and
sn ¼ sn ðLÞ :¼
X
1
xn x2Lnf0g
:
Weierstrass elliptic function }ðzÞ :¼ }ðz; g 2 ; g 3 Þ is a meromorphic function with double periods 2x1 ; 2x2 and satisfying the equation 2
ð}0 ðzÞÞ ¼ 4}ðzÞ3 g 2 }ðzÞ g 3 ;
ð2:1Þ
where g 2 ¼ 60s4 ; g 3 ¼ 140s6 and Dðg 2 ; g 3 Þ – 0. If changing (2.1) to the form as 2
ð}0 ðzÞÞ ¼ 4ð}ðzÞ e1 Þð}ðzÞ e2 Þð}ðzÞ e2 Þ; we have e1 ¼ }ðx1 Þ; e2 ¼ }ðx2 Þ; e3 ¼ }ðx1 þ x2 Þ. Inversely, given two complex numbers g 2 and g 3 such that Dðg 2 ; g 3 Þ – 0, then there exists double periods 2x1 ; 2x2 Weierstrass elliptic function }ðzÞ such that above relations hold. It is easy to see that the set of poles of Weierstrass elliptic function }ðzÞ is L; }ðzÞ has 4 distinct complete multiple values e1 ; e2 ; e3 and infinite, and thus whose any other value must be simple. Lemma 2.3 ([9,10]). Weierstrass elliptic functions }ðzÞ :¼ }ðz; g 2 ; g 3 Þ have two successive degeneracies and addition formula below: (I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential eaz Þ according to
3d 2 coth }ðz; 3d ; d Þ ¼ 2d 2 2
3
rffiffiffiffiffiffi 3d z; 2
ð2:2Þ
if one root ej is double (Dðg 2 ; g 3 Þ ¼ 0Þ. (II) Degeneracy to rational functions of z according to
}ðz; 0; 0Þ ¼
1 z2
if one root ej is triple (g 2 ¼ g 3 ¼ 0). (III) Addition formula
}ðz z0 Þ ¼ }ðzÞ }ðz0 Þ þ
2 1 }0 ðzÞ þ }0 ðz0 Þ : 4 }ðzÞ }ðz0 Þ
ð2:3Þ
By above lemmas, we can give the complex method to find exact solutions of some PDEs. Step 1 Substitute the transform T : uðx; tÞ ! wðzÞ; ðx; tÞ ! z into a given PDE gives an odd order non-linear ordinary differential Eq. (1.1). Step 2 Substitute (1.2) into the Eq. (1.1) to verify that the weak hp; qi condition holds. Step 3 By indeterminant relations (1.3), (1.4) and (1.5) we find the elliptic, rational and simply periodic solutions wðzÞ of the Eq. (1.1) with pole at z ¼ 0, respectively. Step 4 By Theorem 1.4 we obtain all meromorphic solutions wðz z0 Þ. Step 5 Substitute the inverse transform T 1 into these meromorphic solutions wðz z0 Þ, then we get all exact solutions uðx; tÞ of the original given PDE. 3. Proofs of Theorem 1.7 and 1.8
Proof. By Theorem 1.4, we only need to prove that w 2 W. We consider two cases: Case 1 w is an entire function. We will first prove that w is not transcendental. Otherwise, rewrite (1.1) as follows
wn1 aw ¼ w0 wðmÞ
1 h ðmþ1Þ i2 w 2 þ P½w: 2
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
Since deg P½w < n and the Eq. (1.1) satisfies weak hp; qi condition, we see that n P 3, and then, making use of Lemma 2.1, we get
mðr; wÞ ¼ Sðr; wÞ: Combining with Nðr; wÞ ¼ 0, we have
Tðr; wÞ ¼ mðr; wÞ ¼ Sðr; wÞ ¼ oðTðr; wÞÞ: This is impossible. We know that w is a polynomial. Let w be a polynomial of degree kðk P 1Þ, that is
w ¼ ak zk þ ak1 zk1 þ þ a0 ðak – 0Þ: Substituting w into (1.1), we know that the coefficient aank of the highest degree term in z is not zero. Hence w is constant. (I) occurs. w 2 W. Case 2 w is a meromorphic function with poles. We consider two subcases. Subcase 2.1 w has finite poles. using of the logarithmic derivative lemma and properties of Nevanlinna’s characteristic function we have ByðiÞ T r; ww ¼ Oðlog rÞ þ Sðr; wÞ. Change P½w to the form of
P½w ¼
0
n X w wðmÞ wi ; bi ;...; w w i¼0
where each bi ði ¼ 1; . . . ; n 1Þ is its variables polynomial, Lemma 2.2 gives
Tðr; P½wÞ 6 ðn 1ÞTðr; wÞ þ Oðlog rÞ þ Sðr; wÞ: By Lemma 2.2 once more, we have Tðr; awn Þ ¼ nTðr; wÞ þ Oð1:1Þ, and then Tðr; wÞ ¼ Sðr; wÞ þ Oðlog rÞ. Hence w is rational. w 2 W. Subcase 2.2 w has infinite poles. Claim: The Eq. (1.1): Eðz; wÞ ¼ 0 satisfies hp; qi condition. In fact, since the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies weak hp; qi condition and the dominant part h mþ1 i2 ^ Eðz; wÞ ¼ w0 wðmÞ 12 wð 2 Þ awn , we have DðqÞ ¼ m þ 1 þ 2q ¼ nq. Thus
m ¼ ðn 2Þq 1; Noting that limz!0 z
jþq
Dr ðqÞ < m þ 2q þ 1: ðjÞ
ð3:1Þ
j
w ðzÞ ¼ ð1Þ qðq þ 1Þ ðq þ j 1Þcq ,
r r m r limzmþ2qþ1 M r ½w ¼ limzmþ2qþ1Dr ðqÞ zDr ðqÞ M r ½w ¼ limzmþ2qþ1Dr ðqÞ ðzq wÞ 0 zqþ1 w0 1 zqþm wðmÞ ¼ 0; z!0
z!0
z!0
by the Eq. (1.1): Eðz; wÞ ¼ 0, we get 2 acn2 q ¼ q ðq þ 1Þ ðq þ m 1Þ
2 1 mþ1 qðq þ 1Þ q þ 1 : 2 2
ð3:2Þ
h i2 ^ wÞ ¼ w0 wðmÞ 1 wðmþ1 2 Þ Note that Eðz; awn , (1.6) gives 2
! mþ1 m mþ1 d 2 0 q iq q 0 d q ðmÞ d q n1 ð Þ ^ 2 xiq E ðx; cq x Þx ¼ ðcq x Þ m þ ðcq x Þ w naðcq x Þ mþ1 dx dx dx 2 ¼ qði qÞði q 1Þ ði q m þ 1Þ þ ðqÞðq 1Þ ðq m 1Þði qÞ
mþ1 mþ1 cq xðn1Þqþiq : þ 1 ði qÞði q 1Þ i q þ 1 nacn2 qÞðq 1Þ q q 2 2 ð3:3Þ
Hence, by using of (3.1), (3.2), (1.7) and (3.3) and noting that m is an odd positive integer and ger, we deduce that the Fuchs index of the Eq. (1.1): Eðz; wÞ ¼ 0 are zeros of the function
0mþ1 1 mþ1
mþ1 2 2 1 2 2 2 Y Y mþ1 1 Y @ PðiÞ ¼ q ðq i þ jÞ qiþjþ ðq þ 1 þ jÞA þ ðq þ jÞ 2 2 j¼0 j¼0 j¼0 j¼0 0 1 mþ1 mþ1
2 2 2 2 Y Y m þ 1 @ð2q þ m þ 1Þ ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þ j A: 2 j¼0 j¼0 mþ1 1 2
Y
mþ1 2
is an even positive inte-
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
Clearly, Pð1Þ ¼ 0 and for any i 2 N
0mþ1 1 mþ1
2 2 2 2 Y Y m þ 1 ðq þ jÞ @ ðq þ 1 þ jÞ qþ þ j A < 0: 2 j¼0 j¼0
mþ1 1 2
Y
P1 ðiÞ :¼
j¼0
ð3:4Þ
If i 2 N; i < q, then
0mþ1 1
mþ1 2 2 2 2 Y Y m þ 1 PðiÞ < q ðq þ jÞ @ qþjþ ðq þ 1 þ jÞA 2 j¼0 j¼0 j¼0 0mþ1 1 mþ1 mþ1
1 2 2 2 2 2 Y Y 2q þ m þ 1 Y m þ 1 mþ1 þ ðq þ jÞ @ ðq þ 1 þ jÞ qþ þj A¼ P 1 ðiÞ: 2 2 2 j¼0 j¼0 j¼0 mþ1 1 2
Y
Thus (3.4) gives that PðiÞ < 0. If i 2 N; q 6 i 6 q þ mþ1 2 1, then
0 1 mþ1 mþ1 mþ1
1 2 2 2 2 2 Y Y 1 Y m þ 1 mþ1 P1 ðiÞ PðiÞ ¼ ðq þ jÞ @ð2q þ m þ 1Þ ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A< qþ 2 j¼0 2 2 j¼0 j¼0 < 0:
If i 2 N; q þ mþ1 6 i 6 q þ m 1, noting that 2 mþ1 1 2
PðiÞ ¼ q
Y
mþ1 2
mþ1 2 2
ði q jÞ
Y
j¼0
0
@ð2q þ m þ 1Þ
j¼0
j¼0
1 Y ðq þ jÞ 2 j¼0
1
Y mþ1 ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A 2 j¼0
mþ1 2 2
Y
ðq þ 1 þ jÞ þ
is an even positive integer, then mþ1 1 2
mþ1 2 2
mþ1 P1 ðiÞ < 0: < qþ 2 If i 2 N; i P q þ m, noting that m is an odd positive integer and
mþ1 2
is an even positive integer, then
0mþ1 1 mþ1 mþ1
mþ1 1 2 2 1 2 2 2 2 Y Y Y m þ 1 1 Y þ PðiÞ ¼ q ðq i þ jÞ @ iqj ðq þ 1 þ jÞA þ ðq þ jÞ 2 2 j¼0 j¼0 j¼0 j¼0 0 1 mþ1 mþ1
2 2 2 2 Y Y mþ1 mþ1 @ P1 ðiÞ < 0: ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A< qþ ð2q þ m þ 1Þ 2 2 j¼0 j¼0 Therefore, PðiÞ < 0 if i 2 N and m is an odd positive integer and mþ1 is an even positive integer. In other word, the equation 2 is an even positive integer i.e., the Eq. (1.1): PðiÞ ¼ 0 has no positive integer solution if m is an odd positive integer and mþ1 2 Eðz; wÞ ¼ 0 has no positive integer Fuchs index. This means that (Ref. [6], p. 90): The Laurent series of w can be determined by the principle part. Thus, weak hp; qi condition implies that hp; qi condition holds. Claim holds. Assume that z1 , z2 ; . . . ; zpþ1 are of p þ 1 distinct poles of w on the whole C, then wðz þ z1 Þ; . . . ; wðz þ zpþ1 Þ are of solutions of the Eq. (1.1), which have pole at z ¼ 0. By using of our claim, we see that there exist at least two of them are equal, say wðz þ z1 Þ wðz þ zpþ1 Þ, i.e., wðzÞ wðz z1 þ zpþ1 Þ, Hence w is periodic. Hence, there exist l 6 p distinct poles z1 ; . . . ; zl such that the set of poles of w can be expressed as z1 þ C; . . . ; zl þ C, where C is a nontrivial discrete subgroup on ðC; þÞ. Then C isomorphic to Z or Z Z. If C isomorphic to Z , then C=C ¼ C ¼ C n 0, and w is a simple periodic meromorphic function. So w can be expressed by RðexpðazÞÞ, where R is a meromorphic function and has and only has l 6 p distinct poles n1 ; . . . ; nl with multiplicity q on C . Now we will prove that R is a rational function. Set n ¼ expðazÞ, then substituting w ¼ RðnÞ into the Eq. (1.1), we have 2 mþ1 mþ1 1 mþ1 P½RðnÞ þ amþ1 nR0 ½nR0 þ þ nm RðmÞ a 2 fnR0 þ þ n 2 Rð 2 Þ g ¼ aRn ðnÞ: 2
ð3:5Þ
By the similarly arguments in Case 1, if R is transcendental, by using Lemma 2.1 to Eq. (3.5), we get mðr; RÞ ¼ Sðr; RÞ. Noting that R has and only has l 6 p distinct poles, we see that Tðr; RÞ ¼ Sðr; RÞ, a contradiction. Therefore R is a rational function w 2 W. If C isomorphic to Z Z, then w is an elliptic function and has at most lð6 pÞ distinct poles n1 ; ; nl with multiplicity q per parallelogram of periods.
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
The proof of Theorem 1.7 is completed.
h
Proof. The proof of Theorem 1.8 is similar to the proof of Theorem 1.7, we only need to prove the claim below. Claim: The Eq. (1.1): Eðz; wÞ ¼ 0 satisfies hp; qi condition. ^ wÞ ¼ w0 wðmÞ In fact, since the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies weak hp; qi condition and the dominant part Eðz; 2
mþ1
þ 12 ½wð 2 Þ awn , we have that DðqÞ ¼ m þ 1 þ 2q ¼ nq and (3.1) holds. Noting that limz!0 zjþq wðjÞ ðzÞ ¼ ð1Þj qðq þ 1Þ ðq þ j 1Þcq ,
r r m r limzmþ2qþ1 M r ½w ¼ limzmþ2qþ1Dr ðqÞ zDr ðqÞ M r ½w ¼ limzmþ2qþ1Dr ðqÞ ðzq wÞ 0 zqþ1 w0 1 zqþm wðmÞ ¼ 0; z!0
z!0
z!0
by the Eq. (1.1): Eðz; wÞ ¼ 0, we get 2 acn2 q ¼ q ðq þ 1Þ ðq þ m 1Þ þ
2 1 mþ1 qðq þ 1Þ q þ 1 : 2 2
ð3:6Þ
h i2 ^ wÞ ¼ w0 wðmÞ þ 1 wðmþ1 2 Þ Note that Eðz; awn , (1.6) gives 2
! mþ1 m 2 0 q iq q 0 d q ðmÞ d ðmþ1 Þ d q n1 ^ 2 xiq E ðx; cq x Þx ¼ ðcq x Þ m þ ðcq x Þ þw naðcq x Þ mþ1 dx dx dx 2 ¼ qði qÞði q 1Þ ði q m þ 1Þ þ ðqÞðq 1Þ ðq m 1Þði qÞ
mþ1 mþ1 cq xðn1Þqþiq : þ 1 ði qÞði q 1Þ i q þ 1 nacn2 þ qÞðq 1Þ q q 2 2 ð3:7Þ
Hence, by using of (3.1), (3.6), (1.7) and (3.7) and noting that m and Fuchs index of the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies the equation
mþ1 2
are odd positive integers, we deduce that the
0 mþ1 1 mþ1 mþ1 1 1 1 2 2 2 Y Y 1 Y @ PðiÞ ¼ q ðq i þ jÞ ðq þ i þ m þ 1Þ ðq þ jÞ þ ðq þ jÞ 2 ðq i þ jÞ n ðq þ jÞA ¼ 0: 2 j¼0 j¼0 j¼0 j¼0 j¼0 m1 Y
m1 Y
Clearly, Pð1Þ ¼ 0. If i 2 N; i 6 q, then m 1 Y
q
ðq i þ jÞ < ðq þ i þ m þ 1Þ
j¼0
m 1 Y
ðq þ jÞ
j¼0
and mþ1 1 2
2
Y
mþ1 1 2
ðq i þ jÞ < n
j¼0
Y
ðq þ jÞ:
j¼0
Thus pðiÞ < 0. If i 2 N; q < i 6 q þ mþ1 2 1, then
2mþ1 32 1 2 1 4Y PðiÞ ¼ ðq þ i þ m þ 1Þ ðq þ jÞ n ðq þ jÞ5 < 0: 2 j¼0 j¼0 m1 Y
If i 2 N; q þ mþ1 6 i 6 q þ m 1, noting that 2 mþ1 1 2
PðiÞ ¼
Y
ðq þ jÞ
j¼0
8mþ1 1
j¼0
PðiÞ ¼
Y j¼0
ðq þ jÞ
8mþ1 1
j¼0
is an odd positive integer, then
9 2mþ1 32
1 2 Y mþ1 = 1 4 Y ði q jÞ þ ðq þ i þ m þ 1Þ qþjþ n ðq þ jÞ5 < 0: ; 2 2 j¼0 j¼0 mþ1 2 2
If i 2 N; i P q þ m, noting that m and mþ1 1 2
mþ1 2
mþ1 2
are two odd positive integers, then
9 2mþ1 32
1 m1 2 Y Y mþ1 = 1 4Y ði q jÞ þ ðq þ i þ m þ 1Þ qþjþ q ði q jÞ n ðq þ jÞ5 < 0: ; 2 2 j¼0 j¼0 j¼0 mþ1 2 2
Therefore, PðiÞ < 0 if i 2 N and m and mþ1 are two odd positive integers. In other word, the equation PðiÞ ¼ 0 has no posi2 tive integer solution if m and mþ1 are two odd positive integers, i.e., the Eq. (1.1): Eðz; wÞ ¼ 0 has no positive integer Fuchs 2 index. This means that (Ref. [6], p. 90): The Laurent series of w can be determined by the principle part. Thus, weak hp; qi condition implies that hp; qi condition holds. Claim holds.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
The proof of Theorem 1.8 is completed.
249
h
4. Proof of Corollary 1.10 Proof. By Theorem 1.7, relation 3.1 and taking m ¼ 3, noting that 4 ¼ ðn 2Þq has only integer solution pairs of ðn; qÞ ¼ ð6; 1Þ; ð4; 2Þ and ð3; 4Þ, we know that all meromorphic solutions w of the Eq. (1.12) belong to the class W iff n ¼ 6; 4 and 3. Making use of relation (3.2) once more and for the sake of convenient and simplicity, cn2 q ¼ 1 implies that a ¼ 4; 30 and 1 for n ¼ 6; 4 and 3, respectively. Hence the Eq. (1.12) reduces the Eqs. (1.9), (1.13) and (1.14). By Theorem 1.6, we only need to determine all elliptic function solutions of the Eqs. (1.13) and (1.14). Consider the Eq. (1.13). Thus n ¼ 4; q ¼ 2 and c2 ¼ 1. Taking c2 ¼ 1 and substituting the Laurent series (1.2) into the Eq. (1.13), we obtain
wðzÞ ¼
! ! 1 C3 b C0 C 23 b2 C 33 C 23 b C0C3 C0b C1 b3 2 z z4 þ : þ þ þ þ þ þ þ z2 120 60 360 14400 3600 172800 172800 2880 4320 24 43200
ð4:1Þ
By the (1.5) of Theorem 1.4 with l ¼ 1, we should take an elliptic solution in the form
wðzÞ ¼ }ðz; g 2 ; g 3 Þ þ c10 :
ð4:2Þ
Put (4.2) into the Eq. (1.13), and the Laurent series for this function at the point z = 0 is the following
1 g g þ c10 þ 2 z2 þ 3 z4 þ : z2 20 28
wðzÞ ¼
ð4:3Þ
Comparing coefficients of the series (4.3) with coefficients of the series (4.1), we find the parameters of the elliptic solution (4.2)
C3 b ; 120 60
c10 ¼
g2 ¼
C0 C2 b2 þ 3 ; 18 720 180
g3 ¼
7C 33 7C 23 b 7C 0 C 3 7C 0 b 7C 1 7b3 þ þ þ ; 1080 43200 43200 720 6 10800
ð4:4Þ
and the correlation on the parameters of the Eq. (1.13)
C0 ¼
b2 C 23 ; 10 40
C1 ¼
1 2304000C 2 þ C 43 þ 80b4 48C 3 b3 8C 23 b2 þ 4C 33 b : 19200 C 3 þ 3b
ð4:5Þ
Therefore, some elliptic solutions of the Eq. (1.13) are of
we;13;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ þ
C3 b ; 120 60
where both (4.4) and (4.5) hold, C 0 and C 1 are given by (4.5), b; C 2 and C 3 are three constants, z0 2 C. By the (1.5) of Theorem 1.4 with l ¼ 2, we should take an elliptic solution in the form
wðzÞ ¼
2 1 }0 ðz; g 2 ; g 3 Þ þ B þ 2}ðz; g 2 ; g 3 Þ þ c20 ; 4 }ðz; g 2 ; g 3 Þ A
ð4:6Þ
where B2 ¼ 4A3 g 2 A g 3 . Put (4.6) into the Eq. (1.13), and the Laurent series for this function at the point z = 0 is the following
1 3g 2 2g 3 g2 A 4 2 3 2 z z þ : 2A þ c þ 3A þ 4A þ 20 z2 2 10 7
wðzÞ ¼
ð4:7Þ
Comparing coefficients of the series (4.7) with coefficients of the series (4.1), we find the parameters of the elliptic solution (4.6)
c20 ¼
C3 b þ ; 120 60
A¼
b ; 60
B ¼ 0; ð4:8Þ
3
g 2 ¼ 0;
g3 ¼
b ; 54000
and the correlation on the parameters of the Eq. (1.13)
C0 ¼
C 23 b2 ; 40 5
C1 ¼
C 33 C 3 b2 þ ; 14400 600
C2 ¼
b4 C 43 C 23 b2 : 15000 6912000 144000
Therefore, some other elliptic solutions of the Eq. (1.13) are of
we;13;2 ðzÞ ¼
b3 216000}3 þ 60ð240} þ C 3 þ 2bÞ 3600}2 108000} þ b2 ; 7200 3600}2 ðz z0 ; g 2 ; g 3 Þ 108000}ðz z0 ; g 2 ; g 3 Þ b2
ð4:9Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where both (4.8) and (4.9) hold, } ¼ }ðz z0 ; g 2 ; g 3 Þ; b and C 3 are two constants, z0 2 C. Consider the Eq. (1.14). Thus n ¼ 6; q ¼ 1 and c1 ¼ 1; i. Taking c1 ¼ 1 or i and substituting the Laurent series (1.2) into the Eq. (1.14), we obtain
! ! 1 C5 C52 b C4 C3 C4 C5 C 35 wðzÞ ¼ þ þ þ þ þ zþ z2 þ ; z 24 288 60 30 40 240 3456
ð4:10Þ
or
! ! i C5 C52 b C4 C3 C4C5 C 35 wðzÞ ¼ þ þ þ þ þ iz z2 þ : z 24 288 60 30 40 240 3456
ð4:11Þ
By the (1.5) of Theorem 1.4 with l ¼ 2, we should take an elliptic solution in the form
we;14;1 ðzÞ ¼
1 }0 ðz; g 2 ; g 3 Þ þ B þ c30 ; 2 }ðz; g 2 ; g 3 Þ A
ð4:12Þ
we;14;2 ðzÞ ¼
i }0 ðz; g 2 ; g 3 Þ þ B þ c40 ; 2 }ðz; g 2 ; g 3 Þ A
ð4:13Þ
or
2
where i ¼ 1; B2 ¼ 4A3 g 2 A g 3 . Put (4.12) or (4.13) into the Eq. (1.14), and the Laurent series for this function at the point z = 0 is the following
wðzÞ ¼
1 B AB 4 3 A2 B 6 Ag þ c30 þ Az z2 þ A2 z3 z þ A3 g 3 z5 z þ A4 3 z 7 þ ; z 2 2 28 2 7
ð4:14Þ
or
i Bi ABi 4 33 A2 Bi 6 3960A 3 5 7 wðzÞ ¼ þ c40 þ Aiz z2 þ A2 iz z þ A3 g 3 iz z þ A4 g 3 iz þ ; z 2 2 308 2 27720
ð4:15Þ
Comparing coefficients of the series (4.14) or (4.15) with coefficients of the series (4.10) or (4.11), we find the parameters of the elliptic solution (4.12)
c30 ¼
C5 ; 24
g 2 ¼ 0;
g 3 ¼ 4A3 B2 ;
ð4:16Þ
and the correlation on the parameters of the Eq. (1.14)
C 0 ¼ 120 c430 þ 6 c230 b þ 360 c230 A þ 120 Bc30 6 b A; C 1 ¼ 2 b c330 þ 2 b B þ 24 c530 120 c330 A 60 Bc230 þ 12 BA þ 6 c30 b A; 3b 2 b 4 A þ c30 2 c30 b B þ 30 c430 A þ 20 Bc330 12 Bc30 A 3 c230 b A; C 2 ¼ 4 c630 10 A3 2 g 3 2 2 C 3 ¼ 80 c330 2 c30 b 120 c30 A 20 B; b C 4 ¼ 60 c230 þ þ 30 A: 2
ð4:17Þ
Or we find the parameters of the elliptic solution (4.13)
c40 ¼
C5 ; 24
g 2 ¼ 0;
g 3 ¼ 4A3 B2 ;
ð4:18Þ
and the correlation on the parameters of the Eq. (1.14)
C 0 ¼ 120 c440 6 b c240 360 c240 A 6 b A 120 ic40 B; 2
C 1 ¼ 24 c540 þ 2 b c340 þ 120 c340 A þ 60 ic40 B þ 2 ib B þ 12 iBA þ 6 c40 b A; 3b 2 b 3 A 4 c640 c440 30 c440 A 20 ic40 B 2 ic40 b B 12 ic40 AB 3 c240 b A; C 2 ¼ 10 A3 þ 2 g 3 þ 2 2 C 3 ¼ 80 c340 þ 2 c40 b þ 120 c40 A þ 20 iB; b C 4 ¼ 60 c240 30 A: 2 Therefore, some elliptic solutions of the Eq. (1.14) are of
we;14;1 ðzÞ ¼
1 }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24
ð4:19Þ
251
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where both (4.16) and (4.17) hold, b and C 5 are two given constants, A; B and z0 2 C. Or, some other elliptic solutions of the Eq. (1.14) are of
we;14;2 ðzÞ ¼
i }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24 2
where both (4.18) and (4.19) hold, i ¼ 1; b and C 5 are two given constants, A; B and z0 2 C.
h
Acknowledgment This work was supported by the Visiting Scholar Program of Department of Mathematics and Statistics at Curtin University of Technology - Australia when the first author worked as a visiting scholar (200001807894). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
L. Yang, Value-Distribution Theory its New Research, Science Press, Beijing, 1982. I. Laine, Nevanlinna Theroy and Complex Differential Equations, de Gruyter, Berlin, New York, 1993. Y.Z. He, X. Z Xiao, Algebriod Functions and Ordinary Differential Equations, Science Press, Beijing, 1988. W.J. Yuan, Y.D. Shang, Y. Huang, H. Wang, The representation of meromorphic solutions of certain ordinary differential equations and its applications, Sci. Sin. Math. 43 (6) (2013) 563–575, http://dx.doi.org/10.1360/012012-159. W.J. Yuan, Y. Huang, Y.D. Shang, All travelling wave exact solutions of two nonlinear physical models, Appl. Math. Comput. 219 (11) (2013) 6212–6223, http://dx.doi.org/10.1016/j.amc.2012.12.023. R. Conte, The Painlevé approach to nonlinear ordinary differential equations, in: The Painlev Property, One Century Later, in: R. Conte (Ed.), CRM Series in Mathematical Physics, Springer, New York, 1999, pp. 77–180. N.A. Kudryashov, D.I. Sinelshchikov, M.V. Demina, Exact solutions of the generalized Bretherton equation, Phys. Lett. A 375 (7) (2011) 1074–1079. M.V. Demina, N.A. Kudryashov, From Laurent series to exact meromorphic solutions: the Kawahara equation, Phys. Lett. A 374 (39) (2010) 4023–4029. S. Lang, Elliptic Functions, second ed., Springer Verlag, New York, 1987. R. Conte, M. Musette, Elliptic general analytic solutions, Stud. Appl. Math. 123 (1) (2009,) 63–81.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
All meromorphic solutions for two forms of odd order algebraic differential equations and its applications q Wenjun Yuan a,b,⇑, Yonghong Wu c, Qiuhui Chen d,*, Yong Huang e a
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China c Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia d Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China e School of Computer Science and Education Software, Guangzhou University, Guangzhou 510006, China b
a r t i c l e
i n f o
Keywords: Differential equation Exact solution Meromorphic function Elliptic function
a b s t r a c t In this article, we employ the Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for two classes of odd order algebraic differential equations with the weak hp; qi and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of some generalized Bretherton equations by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics, and using the traveling wave nobody can find other new exact solutions for many nonlinear partial differential equations by any method. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and main results In this paper, a meromorphic function wðzÞ means that wðzÞ is holomorphic in the complex plane C except for poles. }ðz; g 2 ; g 3 Þ is the Weierstrass elliptic function with invariants g 2 and g 3 . It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory, such as
Tðr; f Þ; mðr; f Þ; Nðr; f Þ Nðr; f Þ; . . . : For detail of Nevanlinna’s value-distribution theory, please see [1–3]. We denote by Sðr; f Þ any function satisfying Sðr; f Þ ¼ ofTðr; f Þg, as r ! 1, possibly outside of a set of finite measure. Recently, some authors find the exact solutions of certain PDEs combining with the complex ODEs. In 2013, Yuan et al. [4,5] researched the existence of meromorphic solutions of some algebraic differential equations with constant coefficients by using the Nevanlinna’s value distribution theory, and gave the representations of all meromorphic solutions. At the same time, the complex method to find all traveling wave exact solutions of corresponding partial differential equations was given. q
This work was completed with the support with the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121).
⇑ Corresponding authors. Address: Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China (Q. Chen). E-mail addresses: [email protected] (W. Yuan), [email protected] (Y. Wu), [email protected] (Q. Chen), [email protected] (Y. Huang). http://dx.doi.org/10.1016/j.amc.2014.04.099 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
241
In order to state these results, we need some concepts and notations: Set m 2 N :¼ f1; 2; 3; . . .g; rj 2 N0 ¼ N [ f0g; r ¼ ðr 0 ; r 1 ; . . . ; r m Þ; j ¼ 0; 1; . . . ; m. r
r
rm
Mr ½wðzÞ :¼ ½wðzÞr0 ½w0 ðzÞ 1 ½w00 ðzÞ 2 ½wðmÞ ðzÞ : dðrÞ :¼ r 0 þ r 1 þ þ rm is called the degree of M r ½w. Definition 1.1. A differential polynomial with constant coefficients is defined by
X ar M r ½w;
P½w :¼
r2I
where ar are constants, and I is a finite index set. The total degree deg P½w of P½w is defined by deg P½w :¼ maxr2I fdðrÞg. Consider the differential equations
Eðz; wÞ :¼ P½w þ w0 wðmÞ
1 h ðmþ1Þ i2 w 2 awn ¼ 0; 2
ð1:1Þ
where a – 0 is a constant, n 2 N. Definition 1.2. Let p; q 2 N. Suppose that the meromorphic solution w of the Eq. (1.1) has at least one pole. We say that the Eq. (1.1) satisfies hp; qi condition if there exactly exist p distinct meromorphic solutions of the Eq. (1.1) with pole of multiplicity q at z ¼ 0. We say that the Eq. (1.1) satisfies weak hp; qi condition if substituting Laurent series 1 X
wðzÞ ¼
c k zk ;
q > 0; cq – 0
ð1:2Þ
k¼q
into the Eq. (1.1), we can determine p distinct principle 1 X
c k zk
k¼q
with pole of multiplicity q at z ¼ 0. Definition 1.3. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of eaz ; a 2 C, or a rational function of z. Theorem 1.4 ([4,5]). Let p; l; m; n 2 N; deg P½w < n, and the Eq. (1.1) satisfy hp; qi condition. Then all meromorphic solutions w of the Eq. (1.1) must be one of the following four cases: (I) w is a constant. (II) w is a rational function with lð6 pÞ distinct poles of multiplicity q, and w :¼ RðzÞ has the form of
RðzÞ ¼
q l X X
cij
i¼1 j¼1
ðz zi Þj
þ c0 :
ð1:3Þ
(III) w is a rational function RðnÞ of n ¼ eaz ða 2 CÞ. RðnÞ has lð6 pÞ distinct poles of multiplicity q, and has the form (1.3). (IV) w is an elliptic function with double periods 2x1 ; 2x2 , which has lð6 pÞ distinct poles of multiplicity q per parallelogram of periods. And w has one of the following three forms.
(1.1) If w is even, and the pole of }ðzÞ is the pole of w, then w is a rational Q ðnÞ in n :¼ }ðzÞ and has the form of
Q ðnÞ ¼
q l X X i¼1 j¼1
cij ðn ni Þ
j
þ
q0 X ci ni ; i¼0
where q0 6 2q. If the pole of }ðzÞ is not the pole of w, then q0 ¼ 0. (1.2) If w is odd, then }0wðzÞ is a rational Q ðnÞ of n :¼ }ðzÞ. (1.3) If w is non-odd and non-even, then
wðzÞ ¼ Q 1 ð}ðzÞÞ þ }0 ðzÞQ 2 ð}ðzÞÞ; where Q 1 ðnÞ and Q 2 ðnÞ are rational functions with the form (1.4).
ð1:4Þ
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Each elliptic function solution with pole z ¼ 0 can be expressed as the form
wðzÞ ¼
! 2 j2 j2 q q j l1 X l1 X X ð1Þj cij d 1 }0 ðzÞ þ Bi ci1 }0 ðzÞ þ Bi X ð1Þ clj d þ }ðzÞ þ }ðzÞ þ c0 ; j2 4 }ðzÞ Ai ðj 1Þ! dz 2 }ðzÞ Ai ðj 1Þ! dzj2 i¼1 j¼2 i¼1 j¼2
where cij can be determined by (1.2), }ðzÞ is the Weierstrass elliptic function, B2i ¼ 4A3i g 2 Ai g 3 and The authors proposed a conjecture below in [4]:
Pl
i¼1 ci1
ð1:5Þ ¼ 0.
Conjecture. Let p; l; m; n 2 N; deg P½w < n, and the Eq. (1.1) satisfy weak hp; qi condition. Then the conclusion of Theorem 1.4 holds. In order to state and prove our result, we need some notations and concepts [6]: Definition 1.5. Let Eðz; wÞ ¼ 0 be a m-th order algebraic differential equation, w is a meromorphic solution. The involved term of Eðz; wÞ determining the multiplicity q of w is called the dominant term. The dominant part of Eðz; wÞ is consisted by ^ wÞ. The multiplicity of pole of each term in Eðz; ^ wÞ is the same integer denoted by all dominant terms, and is denoted by Eðz; ^ wÞ is denoted by Dr ðqÞ. DðqÞ. The multiplicity of pole of each monomial M r ½w in Eðz; wÞ Eðz; Obviously
Dr ðqÞ ¼ qdðrÞ þ r 1 þ 2r 2 þ þ mr m < DðqÞ: ^ ^ wÞ with respect to w is denoted by E ^0 ðz; wÞ :¼ @ Eðz;wÞ The derivative of dominant part Eðz; , which can be defined in the fol@w lowing form:
for any
v : E^0 ðz; wÞv ¼ lim k!0
^ w þ kv Þ Eðz; ^ wÞ Eðz; : k
ð1:6Þ
The root of the equation
^0 ðx; cq xq Þxiq ¼ 0 PðiÞ ¼ limxiþDðqÞ E
ð1:7Þ
x!0
is called the Fuchs index of the equation Eðz; wÞ ¼ 0. In this paper, we obtain two results for some odd order Eq. (1.1) to show that the conjecture holds. By our results, then we give the corresponding complex method of finding all travel wave exact solutions of some partial differential equations and an example of finding all meromorphic solutions of the generalized Bretherton equation and three modified Kawahara equations by using our method and results. The generalized Bretherton equation [7]
1 b m 2 2 w0 w000 ðw00 Þ ðw0 Þ awn þ bw þ c ¼ 0; 2 2
ð1:8Þ
where a – 0; b; c and b are constants. Three modified Kawahara equations [8]
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ w3 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:9Þ
and
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ 30w4 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:10Þ
1 b C0 2 2 w0 w000 ðw00 Þ ðw0 Þ 4w6 þ w2 þ C 1 w þ C 2 ¼ 0; 2 2 2
ð1:11Þ
and
where b; C 0 ; C 1 and C 2 are constants. Recently, Kudryashov et al. [7,8] obtained the following result by using their method for the Eq. (1.8). Theorem 1.6. All meromorphic solutions w of the Eq. (1.8), (1.9), (1.10) and (1.11) belong to the class W for the pair of ðm; nÞ ¼ ð3; 6Þ; ð4; 6Þ; ð5; 6Þ, and ð3; 4Þ, respectively. Furthermore, without loss of generality, the parameter a in the Eq. (1.12) reduces 4; 30 or 1 corresponding to n ¼ 6; 4 or 3, respectively. At the same time, the elliptic solutions of the Eq. (1.8) corresponding to these cases are of the forms:
we;3;6 ðzÞ ¼
b }0 ðz z0 ; 0; g 3 Þ 20 ; 2}ðz z0 ; 0; g 3 Þ 2
3
b b where n ¼ 6; m ¼ 3; a ¼ 4; c ¼ 200 ; b ¼ 0, and g 3 ¼ 400 and z0 2 C.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
we;4;6 ðzÞ ¼
}0 ðz z0 ; g 2 ; g 3 Þ 2}ðz z0 ; g 2 ; g 3 Þ 2bb 60
243
;
where n ¼ 6; m ¼ 4; a ¼ 4; c ¼ ð2bbÞð2bþbÞð9bþ2bÞ ; g 2 ¼ bð2bbÞ and g 3 ¼ ð13bþ4bÞð2bbÞ and z0 2 C. 6000 120 216000 3
we;5;6 ðzÞ ¼
5b }0 ðz z0 ; g 2 ; g 3 Þ þ 864
2}ðz z0 ; g 2 ; g 3 Þ
7b2 864
þ
b ; 24
6
4
6
2
b b b 5b where n ¼ 6; m ¼ 5; a ¼ 4; c ¼ 11943936 ; g 2 ¼ 497664 and g 3 ¼ 806215680 ; b ¼ 144 and z0 2 C.
" #2 1 }0 ðz z0 ; g 2 ; g 3 Þ b we;3;4 ðzÞ ¼ 2}ðz z0 ; g 2 ; g 3 Þ ; b 4 }ðz z0 ; g 2 ; g 3 Þ þ 60 60 4
2
3
11b b b where n ¼ 4; m ¼ 3; a ¼ 30; c ¼ 303750 ; g 2 ¼ 540 and g 3 ¼ 81000 and z0 2 C. All elliptic solutions of the Eq. (1.9) are of forms
we;9 ðzÞ ¼ 280}2 ðz z0 ; g 2 ; g 3 Þ
140b 31b2 C 0 14g 2 }ðz z0 ; g 2 ; g 3 Þ þ ; 39 3042 6 3
3
31b 2 where g 3 ¼ 7bg 4745520 ; g 2 is a solution of quadratic equation 780
g 22
b2 1457b4 C 20 C1 g2 þ ¼ 0; 1014 662158224 23184 1932
and
! 287 2 1148 41328b4 372b6 5C 20 b2 5b2 C 1 C3 C0C1 g2 þ C0 þ C1 þ þ 0 ; 207 69 656903 4826809 18252 1521 108 6
C2 ¼
and z0 2 C. All elliptic solutions of the Eq. (1.10) are of two forms
we;10;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ 2
b ; 60
3
0 b 1 b where g 2 ¼ 10C180 ; g 3 ¼ 5C 0 bþ450C ; C2 ¼ 5400
we;10;2 ðzÞ ¼
32b4 220C 0 b2 þ125C 20 8100C 1 b 324000
10800}2 ðz z0 ; g 2 ; g 3 Þ 360b}ðz z0 ; g 2 ; g 3 Þ þ 25C 0 b2 ; 180ð60}ðz z0 ; g 2 ; g 3 Þ b
2
2
3
2
0 Þð4b 0 þb 0 bþ2b where g 2 ¼ 5C540 ; g 3 ¼ 25C ; C 1 ¼ 0; C 2 ¼ ð11b 100C 1215000 162000 All elliptic solutions of the Eq. (1.11) are of two forms
we;11;1 ðzÞ ¼ 2
and z0 2 C.
60}0 ðz z0 ; g 2 ; g 3 Þ ; 2ð60}ðz z0 ; g 2 ; g 3 Þ þ bÞ
we;11;2 ðzÞ ¼
3
3
150C 0 b 0 where g 2 ¼ b 10C ; g 3 ¼ 13b108000 ; C 1 ¼ 0; C 2 ¼ 9b 120 Our main results follows.
100C 0 b 3000
25C 0 Þ
and z0 2 C.
60i}0 ðz z0 ; g 2 ; g 3 Þ 2ð60}ðz z0 ; g 2 ; g 3 Þ þ bÞ and z0 2 C.
Theorem 1.7. Let p; l; m; n 2 N; deg P½w < n; m be an odd integer and mþ1 be an even positive integer. If the Eq. (1.1) satisfies h mþ1 i22 ^ wÞ ¼ w0 wðmÞ 1 wð 2 Þ awn , then all meromorphic solutions w of the Eq. weak hp; qi condition and the dominant part Eðz; 2 (1.1) belong to the class W. Furthermore, the conclusions of Theorem 1.4 hold. Theorem 1.8. Let p; l; m; n 2 N; deg P½w < n; m and mþ1 be two odd positive integers. If the Eq. (1.1) satisfies weak hp; qi conh mþ1 i22 0 ðmÞ 1 ð Þ ^ dition and the dominant part Eðz; wÞ ¼ w w þ 2 w 2 awn , then all meromorphic solutions w of the Eq. (1.1) belong to the class W. Furthermore, the conclusions of Theorem 1.4 hold. Remark 1.9. By Theorem 1.6 and taking m ¼ 3 in Theorem 1.7, we obtain the following corollary. Corollary 1.10. Consider the equation
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ awn þ PðzÞ ¼ 0; 2 2
ð1:12Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where b and a – 0 are constants, PðzÞ is a polynomial such that deg P < n and n is a positive integer. Then all meromorphic solutions w of the Eq. (1.12) belong to the class W iff n ¼ 6; 4 and 3. Furthermore, without loss of generality, the Eq. (1.12) reduces (1.10),
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ 30w4 þ P1 ðwÞ; 2 2
ð1:13Þ
1 b 2 2 w0 w000 ðw00 Þ ðw0 Þ 4w6 þ P 2 ðwÞ; 2 2
ð1:14Þ
or
where P 1 ðwÞ ¼ C 3 w3 þ C20 w2 þ C 1 w þ C 2 and P2 ðwÞ ¼ C 5 w5 þ C 4 w4 þ C 3 w3 þ C20 w2 þ C 1 w þ C 2 ; b; C 0 ; C 1 ; C 2 ; C 3 ; C 4 and C 5 are constants. All elliptic solutions of the Eq. (1.13) are of
we;13;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ þ C2
C3 b ; 120 60
7C 3
2
7C 2 b
3
2
C2
C0 7C 0 b b 7b b 1 0 C3 3 3 3 where g 2 ¼ 18 þ 720 180 , g 3 ¼ 43200 þ 43200 7C720 þ 1080 7C61 þ 10800 , C 0 ¼ 10 403 ; C 1 ¼ 19200 C 2 and C 3 are three constants, z0 2 C. And
we;13;2 ðzÞ ¼
2304000C 2 þC 43 þ80b4 48C 3 b3 8C 23 b2 þ4C 33 b , C 3 þ3b
b3 216000}3 þ 60ð240} þ C 3 þ 2bÞ 3600}2 108000} þ b2 ; 7200 3600}2 ðz z0 ; g 2 ; g 3 Þ 108000}ðz z0 ; g 2 ; g 3 Þ b2
C2
C3
2
2
4
C4
C 2 b2
b 3b 3 3 3 where C 0 ¼ 403 b5 ; C 1 ¼ 14400 þ C600 ; C 2 ¼ 15000 6912000 144000 ; } ¼ }ðz z0 ; g 2 ; g 3 Þ; b and C 3 are two constants, z0 2 C. All elliptic solutions of the Eq. (1.14) are of
we;14;1 ðzÞ ¼
1 }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24
where g 2 ¼ 0; g 3 ¼ 4A3 B2 , and C 0 ¼ 120 c430 þ 6 c230 b þ 360 c230 A þ 120 Bc30 6 b A; C 1 ¼ 2 b c330 þ 2 b B þ 24 c530 120 c330 A 6 c30 b A; C 2 ¼ 4 c630 10 A3 2 g 3 3b A2 þ 2b c430 2 c30 b B þ 30 c430 A þ 20 Bc330 12 Bc30 A 3 c230 b A; C 3 ¼ 2
60 Bc230 þ 12 BAþ
80 c330 2 c30 b 120 c30 A 20 B; C 4 ¼ 60 c230 þ 2b þ 30 A, and c30 ¼ C245 ; b and C 5 are two given constants, A; B and z0 2 C. Or, some other elliptic solutions of the Eq. (1.14) are of
we;14;2 ðzÞ ¼
i }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24 2
where g 2 ¼ 0; g 3 ¼ 4A3 B2 , and C 0 ¼ 120c440 6bc240 360c240 A 6bA 120ic40 B; C 1 ¼ 24 c540 þ 2 b c340 þ 120 c340 A þ 60 ic40 B þ2ibB þ 12iBA þ 6c40 bA; C 2 ¼ 2 c40 b þ 120 c40 A þ 20 iB; C 4 ¼
3
10A þ 2g 3 þ 3b A2 4c640 2b 2 2 b 2 60 c40 2 30 A; i ¼ 1;
3 30c440 A 20ic40 B 2ic40 bB 12ic40 AB 3c240 bA; C5 c30 ¼ 24 ; b and C 5 are two given constants, A; B and z0
c440
C3 ¼
80c340 þ
2 C.
2. Preliminary lemmas and the complex method In order to prove our results, we need the following three lemmas. Lemma 2.1 ((Clunie Lemma) [2,3]). Let w be a meromorphic solution of an equation wn Pðz; w; w0 ; . . . ; wðmÞ Þ ¼ Q ðz; w; w0 ; . . . ; wðmÞ Þ, where P and Q are two differential polynomials in w and its derivatives w0 ; . . . ; wðmÞ with meromorphic coefficients fak jk 2 Ig, and mðr; ak Þ ¼ Sðr; wÞ. If the total degree of Q ðz; w; w0 ; . . . ; wðmÞ Þ in w and its derivatives w0 ; . . . ; wðmÞ satisfies deg Q ðz; w; w0 ; . . . ; wðmÞ Þ 6 n, then
mðr; Pðz; w; w0 ; . . . ; wðmÞ ÞÞ ¼ Sðr; wÞ: Lemma 2.2 ((Mohon’ko Theorem) [2,3]). Let w be a meromorphic function,
Rðz; wÞ :¼
Pp ai ðzÞwi Pðz; wÞ ¼ Pqi¼0 j Qðz; wÞ j¼0 bj ðzÞw
be an irreducible rational function in w with meromorphic coefficients ai ðzÞ; bj ðzÞ , and Tðr; ai Þ ¼ Sðr; wÞ; Tðr; bj Þ ¼ Sðr; wÞ; i ¼ 0; . . . ; p; j ¼ 0; . . . ; q. Then
Tðr; Rðz; wÞÞ ¼ dTðr; wÞ þ Sðr; wÞ; where d ¼ maxðp; qÞ.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
245
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [9]. 1 Let x1 ; x2 be two given complex numbers such that Im x x2 > 0; L ¼ L½2x1 ; 2x2 be discrete subset L½2x1 ; 2x2 ¼ fxjx ¼ 2nx1 þ 2mx2 ; n; m 2 Zg, which is isomorphic to Z Z. The discriminant D ¼ Dðc1 ; c2 Þ :¼ c31 27c22 and
sn ¼ sn ðLÞ :¼
X
1
xn x2Lnf0g
:
Weierstrass elliptic function }ðzÞ :¼ }ðz; g 2 ; g 3 Þ is a meromorphic function with double periods 2x1 ; 2x2 and satisfying the equation 2
ð}0 ðzÞÞ ¼ 4}ðzÞ3 g 2 }ðzÞ g 3 ;
ð2:1Þ
where g 2 ¼ 60s4 ; g 3 ¼ 140s6 and Dðg 2 ; g 3 Þ – 0. If changing (2.1) to the form as 2
ð}0 ðzÞÞ ¼ 4ð}ðzÞ e1 Þð}ðzÞ e2 Þð}ðzÞ e2 Þ; we have e1 ¼ }ðx1 Þ; e2 ¼ }ðx2 Þ; e3 ¼ }ðx1 þ x2 Þ. Inversely, given two complex numbers g 2 and g 3 such that Dðg 2 ; g 3 Þ – 0, then there exists double periods 2x1 ; 2x2 Weierstrass elliptic function }ðzÞ such that above relations hold. It is easy to see that the set of poles of Weierstrass elliptic function }ðzÞ is L; }ðzÞ has 4 distinct complete multiple values e1 ; e2 ; e3 and infinite, and thus whose any other value must be simple. Lemma 2.3 ([9,10]). Weierstrass elliptic functions }ðzÞ :¼ }ðz; g 2 ; g 3 Þ have two successive degeneracies and addition formula below: (I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential eaz Þ according to
3d 2 coth }ðz; 3d ; d Þ ¼ 2d 2 2
3
rffiffiffiffiffiffi 3d z; 2
ð2:2Þ
if one root ej is double (Dðg 2 ; g 3 Þ ¼ 0Þ. (II) Degeneracy to rational functions of z according to
}ðz; 0; 0Þ ¼
1 z2
if one root ej is triple (g 2 ¼ g 3 ¼ 0). (III) Addition formula
}ðz z0 Þ ¼ }ðzÞ }ðz0 Þ þ
2 1 }0 ðzÞ þ }0 ðz0 Þ : 4 }ðzÞ }ðz0 Þ
ð2:3Þ
By above lemmas, we can give the complex method to find exact solutions of some PDEs. Step 1 Substitute the transform T : uðx; tÞ ! wðzÞ; ðx; tÞ ! z into a given PDE gives an odd order non-linear ordinary differential Eq. (1.1). Step 2 Substitute (1.2) into the Eq. (1.1) to verify that the weak hp; qi condition holds. Step 3 By indeterminant relations (1.3), (1.4) and (1.5) we find the elliptic, rational and simply periodic solutions wðzÞ of the Eq. (1.1) with pole at z ¼ 0, respectively. Step 4 By Theorem 1.4 we obtain all meromorphic solutions wðz z0 Þ. Step 5 Substitute the inverse transform T 1 into these meromorphic solutions wðz z0 Þ, then we get all exact solutions uðx; tÞ of the original given PDE. 3. Proofs of Theorem 1.7 and 1.8
Proof. By Theorem 1.4, we only need to prove that w 2 W. We consider two cases: Case 1 w is an entire function. We will first prove that w is not transcendental. Otherwise, rewrite (1.1) as follows
wn1 aw ¼ w0 wðmÞ
1 h ðmþ1Þ i2 w 2 þ P½w: 2
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
Since deg P½w < n and the Eq. (1.1) satisfies weak hp; qi condition, we see that n P 3, and then, making use of Lemma 2.1, we get
mðr; wÞ ¼ Sðr; wÞ: Combining with Nðr; wÞ ¼ 0, we have
Tðr; wÞ ¼ mðr; wÞ ¼ Sðr; wÞ ¼ oðTðr; wÞÞ: This is impossible. We know that w is a polynomial. Let w be a polynomial of degree kðk P 1Þ, that is
w ¼ ak zk þ ak1 zk1 þ þ a0 ðak – 0Þ: Substituting w into (1.1), we know that the coefficient aank of the highest degree term in z is not zero. Hence w is constant. (I) occurs. w 2 W. Case 2 w is a meromorphic function with poles. We consider two subcases. Subcase 2.1 w has finite poles. using of the logarithmic derivative lemma and properties of Nevanlinna’s characteristic function we have ByðiÞ T r; ww ¼ Oðlog rÞ þ Sðr; wÞ. Change P½w to the form of
P½w ¼
0
n X w wðmÞ wi ; bi ;...; w w i¼0
where each bi ði ¼ 1; . . . ; n 1Þ is its variables polynomial, Lemma 2.2 gives
Tðr; P½wÞ 6 ðn 1ÞTðr; wÞ þ Oðlog rÞ þ Sðr; wÞ: By Lemma 2.2 once more, we have Tðr; awn Þ ¼ nTðr; wÞ þ Oð1:1Þ, and then Tðr; wÞ ¼ Sðr; wÞ þ Oðlog rÞ. Hence w is rational. w 2 W. Subcase 2.2 w has infinite poles. Claim: The Eq. (1.1): Eðz; wÞ ¼ 0 satisfies hp; qi condition. In fact, since the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies weak hp; qi condition and the dominant part h mþ1 i2 ^ Eðz; wÞ ¼ w0 wðmÞ 12 wð 2 Þ awn , we have DðqÞ ¼ m þ 1 þ 2q ¼ nq. Thus
m ¼ ðn 2Þq 1; Noting that limz!0 z
jþq
Dr ðqÞ < m þ 2q þ 1: ðjÞ
ð3:1Þ
j
w ðzÞ ¼ ð1Þ qðq þ 1Þ ðq þ j 1Þcq ,
r r m r limzmþ2qþ1 M r ½w ¼ limzmþ2qþ1Dr ðqÞ zDr ðqÞ M r ½w ¼ limzmþ2qþ1Dr ðqÞ ðzq wÞ 0 zqþ1 w0 1 zqþm wðmÞ ¼ 0; z!0
z!0
z!0
by the Eq. (1.1): Eðz; wÞ ¼ 0, we get 2 acn2 q ¼ q ðq þ 1Þ ðq þ m 1Þ
2 1 mþ1 qðq þ 1Þ q þ 1 : 2 2
ð3:2Þ
h i2 ^ wÞ ¼ w0 wðmÞ 1 wðmþ1 2 Þ Note that Eðz; awn , (1.6) gives 2
! mþ1 m mþ1 d 2 0 q iq q 0 d q ðmÞ d q n1 ð Þ ^ 2 xiq E ðx; cq x Þx ¼ ðcq x Þ m þ ðcq x Þ w naðcq x Þ mþ1 dx dx dx 2 ¼ qði qÞði q 1Þ ði q m þ 1Þ þ ðqÞðq 1Þ ðq m 1Þði qÞ
mþ1 mþ1 cq xðn1Þqþiq : þ 1 ði qÞði q 1Þ i q þ 1 nacn2 qÞðq 1Þ q q 2 2 ð3:3Þ
Hence, by using of (3.1), (3.2), (1.7) and (3.3) and noting that m is an odd positive integer and ger, we deduce that the Fuchs index of the Eq. (1.1): Eðz; wÞ ¼ 0 are zeros of the function
0mþ1 1 mþ1
mþ1 2 2 1 2 2 2 Y Y mþ1 1 Y @ PðiÞ ¼ q ðq i þ jÞ qiþjþ ðq þ 1 þ jÞA þ ðq þ jÞ 2 2 j¼0 j¼0 j¼0 j¼0 0 1 mþ1 mþ1
2 2 2 2 Y Y m þ 1 @ð2q þ m þ 1Þ ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þ j A: 2 j¼0 j¼0 mþ1 1 2
Y
mþ1 2
is an even positive inte-
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
Clearly, Pð1Þ ¼ 0 and for any i 2 N
0mþ1 1 mþ1
2 2 2 2 Y Y m þ 1 ðq þ jÞ @ ðq þ 1 þ jÞ qþ þ j A < 0: 2 j¼0 j¼0
mþ1 1 2
Y
P1 ðiÞ :¼
j¼0
ð3:4Þ
If i 2 N; i < q, then
0mþ1 1
mþ1 2 2 2 2 Y Y m þ 1 PðiÞ < q ðq þ jÞ @ qþjþ ðq þ 1 þ jÞA 2 j¼0 j¼0 j¼0 0mþ1 1 mþ1 mþ1
1 2 2 2 2 2 Y Y 2q þ m þ 1 Y m þ 1 mþ1 þ ðq þ jÞ @ ðq þ 1 þ jÞ qþ þj A¼ P 1 ðiÞ: 2 2 2 j¼0 j¼0 j¼0 mþ1 1 2
Y
Thus (3.4) gives that PðiÞ < 0. If i 2 N; q 6 i 6 q þ mþ1 2 1, then
0 1 mþ1 mþ1 mþ1
1 2 2 2 2 2 Y Y 1 Y m þ 1 mþ1 P1 ðiÞ PðiÞ ¼ ðq þ jÞ @ð2q þ m þ 1Þ ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A< qþ 2 j¼0 2 2 j¼0 j¼0 < 0:
If i 2 N; q þ mþ1 6 i 6 q þ m 1, noting that 2 mþ1 1 2
PðiÞ ¼ q
Y
mþ1 2
mþ1 2 2
ði q jÞ
Y
j¼0
0
@ð2q þ m þ 1Þ
j¼0
j¼0
1 Y ðq þ jÞ 2 j¼0
1
Y mþ1 ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A 2 j¼0
mþ1 2 2
Y
ðq þ 1 þ jÞ þ
is an even positive integer, then mþ1 1 2
mþ1 2 2
mþ1 P1 ðiÞ < 0: < qþ 2 If i 2 N; i P q þ m, noting that m is an odd positive integer and
mþ1 2
is an even positive integer, then
0mþ1 1 mþ1 mþ1
mþ1 1 2 2 1 2 2 2 2 Y Y Y m þ 1 1 Y þ PðiÞ ¼ q ðq i þ jÞ @ iqj ðq þ 1 þ jÞA þ ðq þ jÞ 2 2 j¼0 j¼0 j¼0 j¼0 0 1 mþ1 mþ1
2 2 2 2 Y Y mþ1 mþ1 @ P1 ðiÞ < 0: ðq þ 1 þ jÞ 2ðq þ i þ m þ 1Þ qþ þj A< qþ ð2q þ m þ 1Þ 2 2 j¼0 j¼0 Therefore, PðiÞ < 0 if i 2 N and m is an odd positive integer and mþ1 is an even positive integer. In other word, the equation 2 is an even positive integer i.e., the Eq. (1.1): PðiÞ ¼ 0 has no positive integer solution if m is an odd positive integer and mþ1 2 Eðz; wÞ ¼ 0 has no positive integer Fuchs index. This means that (Ref. [6], p. 90): The Laurent series of w can be determined by the principle part. Thus, weak hp; qi condition implies that hp; qi condition holds. Claim holds. Assume that z1 , z2 ; . . . ; zpþ1 are of p þ 1 distinct poles of w on the whole C, then wðz þ z1 Þ; . . . ; wðz þ zpþ1 Þ are of solutions of the Eq. (1.1), which have pole at z ¼ 0. By using of our claim, we see that there exist at least two of them are equal, say wðz þ z1 Þ wðz þ zpþ1 Þ, i.e., wðzÞ wðz z1 þ zpþ1 Þ, Hence w is periodic. Hence, there exist l 6 p distinct poles z1 ; . . . ; zl such that the set of poles of w can be expressed as z1 þ C; . . . ; zl þ C, where C is a nontrivial discrete subgroup on ðC; þÞ. Then C isomorphic to Z or Z Z. If C isomorphic to Z , then C=C ¼ C ¼ C n 0, and w is a simple periodic meromorphic function. So w can be expressed by RðexpðazÞÞ, where R is a meromorphic function and has and only has l 6 p distinct poles n1 ; . . . ; nl with multiplicity q on C . Now we will prove that R is a rational function. Set n ¼ expðazÞ, then substituting w ¼ RðnÞ into the Eq. (1.1), we have 2 mþ1 mþ1 1 mþ1 P½RðnÞ þ amþ1 nR0 ½nR0 þ þ nm RðmÞ a 2 fnR0 þ þ n 2 Rð 2 Þ g ¼ aRn ðnÞ: 2
ð3:5Þ
By the similarly arguments in Case 1, if R is transcendental, by using Lemma 2.1 to Eq. (3.5), we get mðr; RÞ ¼ Sðr; RÞ. Noting that R has and only has l 6 p distinct poles, we see that Tðr; RÞ ¼ Sðr; RÞ, a contradiction. Therefore R is a rational function w 2 W. If C isomorphic to Z Z, then w is an elliptic function and has at most lð6 pÞ distinct poles n1 ; ; nl with multiplicity q per parallelogram of periods.
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
The proof of Theorem 1.7 is completed.
h
Proof. The proof of Theorem 1.8 is similar to the proof of Theorem 1.7, we only need to prove the claim below. Claim: The Eq. (1.1): Eðz; wÞ ¼ 0 satisfies hp; qi condition. ^ wÞ ¼ w0 wðmÞ In fact, since the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies weak hp; qi condition and the dominant part Eðz; 2
mþ1
þ 12 ½wð 2 Þ awn , we have that DðqÞ ¼ m þ 1 þ 2q ¼ nq and (3.1) holds. Noting that limz!0 zjþq wðjÞ ðzÞ ¼ ð1Þj qðq þ 1Þ ðq þ j 1Þcq ,
r r m r limzmþ2qþ1 M r ½w ¼ limzmþ2qþ1Dr ðqÞ zDr ðqÞ M r ½w ¼ limzmþ2qþ1Dr ðqÞ ðzq wÞ 0 zqþ1 w0 1 zqþm wðmÞ ¼ 0; z!0
z!0
z!0
by the Eq. (1.1): Eðz; wÞ ¼ 0, we get 2 acn2 q ¼ q ðq þ 1Þ ðq þ m 1Þ þ
2 1 mþ1 qðq þ 1Þ q þ 1 : 2 2
ð3:6Þ
h i2 ^ wÞ ¼ w0 wðmÞ þ 1 wðmþ1 2 Þ Note that Eðz; awn , (1.6) gives 2
! mþ1 m 2 0 q iq q 0 d q ðmÞ d ðmþ1 Þ d q n1 ^ 2 xiq E ðx; cq x Þx ¼ ðcq x Þ m þ ðcq x Þ þw naðcq x Þ mþ1 dx dx dx 2 ¼ qði qÞði q 1Þ ði q m þ 1Þ þ ðqÞðq 1Þ ðq m 1Þði qÞ
mþ1 mþ1 cq xðn1Þqþiq : þ 1 ði qÞði q 1Þ i q þ 1 nacn2 þ qÞðq 1Þ q q 2 2 ð3:7Þ
Hence, by using of (3.1), (3.6), (1.7) and (3.7) and noting that m and Fuchs index of the Eq. (1.1): Eðz; wÞ ¼ 0 satisfies the equation
mþ1 2
are odd positive integers, we deduce that the
0 mþ1 1 mþ1 mþ1 1 1 1 2 2 2 Y Y 1 Y @ PðiÞ ¼ q ðq i þ jÞ ðq þ i þ m þ 1Þ ðq þ jÞ þ ðq þ jÞ 2 ðq i þ jÞ n ðq þ jÞA ¼ 0: 2 j¼0 j¼0 j¼0 j¼0 j¼0 m1 Y
m1 Y
Clearly, Pð1Þ ¼ 0. If i 2 N; i 6 q, then m 1 Y
q
ðq i þ jÞ < ðq þ i þ m þ 1Þ
j¼0
m 1 Y
ðq þ jÞ
j¼0
and mþ1 1 2
2
Y
mþ1 1 2
ðq i þ jÞ < n
j¼0
Y
ðq þ jÞ:
j¼0
Thus pðiÞ < 0. If i 2 N; q < i 6 q þ mþ1 2 1, then
2mþ1 32 1 2 1 4Y PðiÞ ¼ ðq þ i þ m þ 1Þ ðq þ jÞ n ðq þ jÞ5 < 0: 2 j¼0 j¼0 m1 Y
If i 2 N; q þ mþ1 6 i 6 q þ m 1, noting that 2 mþ1 1 2
PðiÞ ¼
Y
ðq þ jÞ
j¼0
8mþ1 1
j¼0
PðiÞ ¼
Y j¼0
ðq þ jÞ
8mþ1 1
j¼0
is an odd positive integer, then
9 2mþ1 32
1 2 Y mþ1 = 1 4 Y ði q jÞ þ ðq þ i þ m þ 1Þ qþjþ n ðq þ jÞ5 < 0: ; 2 2 j¼0 j¼0 mþ1 2 2
If i 2 N; i P q þ m, noting that m and mþ1 1 2
mþ1 2
mþ1 2
are two odd positive integers, then
9 2mþ1 32
1 m1 2 Y Y mþ1 = 1 4Y ði q jÞ þ ðq þ i þ m þ 1Þ qþjþ q ði q jÞ n ðq þ jÞ5 < 0: ; 2 2 j¼0 j¼0 j¼0 mþ1 2 2
Therefore, PðiÞ < 0 if i 2 N and m and mþ1 are two odd positive integers. In other word, the equation PðiÞ ¼ 0 has no posi2 tive integer solution if m and mþ1 are two odd positive integers, i.e., the Eq. (1.1): Eðz; wÞ ¼ 0 has no positive integer Fuchs 2 index. This means that (Ref. [6], p. 90): The Laurent series of w can be determined by the principle part. Thus, weak hp; qi condition implies that hp; qi condition holds. Claim holds.
W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
The proof of Theorem 1.8 is completed.
249
h
4. Proof of Corollary 1.10 Proof. By Theorem 1.7, relation 3.1 and taking m ¼ 3, noting that 4 ¼ ðn 2Þq has only integer solution pairs of ðn; qÞ ¼ ð6; 1Þ; ð4; 2Þ and ð3; 4Þ, we know that all meromorphic solutions w of the Eq. (1.12) belong to the class W iff n ¼ 6; 4 and 3. Making use of relation (3.2) once more and for the sake of convenient and simplicity, cn2 q ¼ 1 implies that a ¼ 4; 30 and 1 for n ¼ 6; 4 and 3, respectively. Hence the Eq. (1.12) reduces the Eqs. (1.9), (1.13) and (1.14). By Theorem 1.6, we only need to determine all elliptic function solutions of the Eqs. (1.13) and (1.14). Consider the Eq. (1.13). Thus n ¼ 4; q ¼ 2 and c2 ¼ 1. Taking c2 ¼ 1 and substituting the Laurent series (1.2) into the Eq. (1.13), we obtain
wðzÞ ¼
! ! 1 C3 b C0 C 23 b2 C 33 C 23 b C0C3 C0b C1 b3 2 z z4 þ : þ þ þ þ þ þ þ z2 120 60 360 14400 3600 172800 172800 2880 4320 24 43200
ð4:1Þ
By the (1.5) of Theorem 1.4 with l ¼ 1, we should take an elliptic solution in the form
wðzÞ ¼ }ðz; g 2 ; g 3 Þ þ c10 :
ð4:2Þ
Put (4.2) into the Eq. (1.13), and the Laurent series for this function at the point z = 0 is the following
1 g g þ c10 þ 2 z2 þ 3 z4 þ : z2 20 28
wðzÞ ¼
ð4:3Þ
Comparing coefficients of the series (4.3) with coefficients of the series (4.1), we find the parameters of the elliptic solution (4.2)
C3 b ; 120 60
c10 ¼
g2 ¼
C0 C2 b2 þ 3 ; 18 720 180
g3 ¼
7C 33 7C 23 b 7C 0 C 3 7C 0 b 7C 1 7b3 þ þ þ ; 1080 43200 43200 720 6 10800
ð4:4Þ
and the correlation on the parameters of the Eq. (1.13)
C0 ¼
b2 C 23 ; 10 40
C1 ¼
1 2304000C 2 þ C 43 þ 80b4 48C 3 b3 8C 23 b2 þ 4C 33 b : 19200 C 3 þ 3b
ð4:5Þ
Therefore, some elliptic solutions of the Eq. (1.13) are of
we;13;1 ðzÞ ¼ }ðz z0 ; g 2 ; g 3 Þ þ
C3 b ; 120 60
where both (4.4) and (4.5) hold, C 0 and C 1 are given by (4.5), b; C 2 and C 3 are three constants, z0 2 C. By the (1.5) of Theorem 1.4 with l ¼ 2, we should take an elliptic solution in the form
wðzÞ ¼
2 1 }0 ðz; g 2 ; g 3 Þ þ B þ 2}ðz; g 2 ; g 3 Þ þ c20 ; 4 }ðz; g 2 ; g 3 Þ A
ð4:6Þ
where B2 ¼ 4A3 g 2 A g 3 . Put (4.6) into the Eq. (1.13), and the Laurent series for this function at the point z = 0 is the following
1 3g 2 2g 3 g2 A 4 2 3 2 z z þ : 2A þ c þ 3A þ 4A þ 20 z2 2 10 7
wðzÞ ¼
ð4:7Þ
Comparing coefficients of the series (4.7) with coefficients of the series (4.1), we find the parameters of the elliptic solution (4.6)
c20 ¼
C3 b þ ; 120 60
A¼
b ; 60
B ¼ 0; ð4:8Þ
3
g 2 ¼ 0;
g3 ¼
b ; 54000
and the correlation on the parameters of the Eq. (1.13)
C0 ¼
C 23 b2 ; 40 5
C1 ¼
C 33 C 3 b2 þ ; 14400 600
C2 ¼
b4 C 43 C 23 b2 : 15000 6912000 144000
Therefore, some other elliptic solutions of the Eq. (1.13) are of
we;13;2 ðzÞ ¼
b3 216000}3 þ 60ð240} þ C 3 þ 2bÞ 3600}2 108000} þ b2 ; 7200 3600}2 ðz z0 ; g 2 ; g 3 Þ 108000}ðz z0 ; g 2 ; g 3 Þ b2
ð4:9Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where both (4.8) and (4.9) hold, } ¼ }ðz z0 ; g 2 ; g 3 Þ; b and C 3 are two constants, z0 2 C. Consider the Eq. (1.14). Thus n ¼ 6; q ¼ 1 and c1 ¼ 1; i. Taking c1 ¼ 1 or i and substituting the Laurent series (1.2) into the Eq. (1.14), we obtain
! ! 1 C5 C52 b C4 C3 C4 C5 C 35 wðzÞ ¼ þ þ þ þ þ zþ z2 þ ; z 24 288 60 30 40 240 3456
ð4:10Þ
or
! ! i C5 C52 b C4 C3 C4C5 C 35 wðzÞ ¼ þ þ þ þ þ iz z2 þ : z 24 288 60 30 40 240 3456
ð4:11Þ
By the (1.5) of Theorem 1.4 with l ¼ 2, we should take an elliptic solution in the form
we;14;1 ðzÞ ¼
1 }0 ðz; g 2 ; g 3 Þ þ B þ c30 ; 2 }ðz; g 2 ; g 3 Þ A
ð4:12Þ
we;14;2 ðzÞ ¼
i }0 ðz; g 2 ; g 3 Þ þ B þ c40 ; 2 }ðz; g 2 ; g 3 Þ A
ð4:13Þ
or
2
where i ¼ 1; B2 ¼ 4A3 g 2 A g 3 . Put (4.12) or (4.13) into the Eq. (1.14), and the Laurent series for this function at the point z = 0 is the following
wðzÞ ¼
1 B AB 4 3 A2 B 6 Ag þ c30 þ Az z2 þ A2 z3 z þ A3 g 3 z5 z þ A4 3 z 7 þ ; z 2 2 28 2 7
ð4:14Þ
or
i Bi ABi 4 33 A2 Bi 6 3960A 3 5 7 wðzÞ ¼ þ c40 þ Aiz z2 þ A2 iz z þ A3 g 3 iz z þ A4 g 3 iz þ ; z 2 2 308 2 27720
ð4:15Þ
Comparing coefficients of the series (4.14) or (4.15) with coefficients of the series (4.10) or (4.11), we find the parameters of the elliptic solution (4.12)
c30 ¼
C5 ; 24
g 2 ¼ 0;
g 3 ¼ 4A3 B2 ;
ð4:16Þ
and the correlation on the parameters of the Eq. (1.14)
C 0 ¼ 120 c430 þ 6 c230 b þ 360 c230 A þ 120 Bc30 6 b A; C 1 ¼ 2 b c330 þ 2 b B þ 24 c530 120 c330 A 60 Bc230 þ 12 BA þ 6 c30 b A; 3b 2 b 4 A þ c30 2 c30 b B þ 30 c430 A þ 20 Bc330 12 Bc30 A 3 c230 b A; C 2 ¼ 4 c630 10 A3 2 g 3 2 2 C 3 ¼ 80 c330 2 c30 b 120 c30 A 20 B; b C 4 ¼ 60 c230 þ þ 30 A: 2
ð4:17Þ
Or we find the parameters of the elliptic solution (4.13)
c40 ¼
C5 ; 24
g 2 ¼ 0;
g 3 ¼ 4A3 B2 ;
ð4:18Þ
and the correlation on the parameters of the Eq. (1.14)
C 0 ¼ 120 c440 6 b c240 360 c240 A 6 b A 120 ic40 B; 2
C 1 ¼ 24 c540 þ 2 b c340 þ 120 c340 A þ 60 ic40 B þ 2 ib B þ 12 iBA þ 6 c40 b A; 3b 2 b 3 A 4 c640 c440 30 c440 A 20 ic40 B 2 ic40 b B 12 ic40 AB 3 c240 b A; C 2 ¼ 10 A3 þ 2 g 3 þ 2 2 C 3 ¼ 80 c340 þ 2 c40 b þ 120 c40 A þ 20 iB; b C 4 ¼ 60 c240 30 A: 2 Therefore, some elliptic solutions of the Eq. (1.14) are of
we;14;1 ðzÞ ¼
1 }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24
ð4:19Þ
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W. Yuan et al. / Applied Mathematics and Computation 240 (2014) 240–251
where both (4.16) and (4.17) hold, b and C 5 are two given constants, A; B and z0 2 C. Or, some other elliptic solutions of the Eq. (1.14) are of
we;14;2 ðzÞ ¼
i }0 ðz z0 ; 0; g 3 Þ þ B C 5 þ ; 2 }ðz z0 ; 0; g 3 Þ A 24 2
where both (4.18) and (4.19) hold, i ¼ 1; b and C 5 are two given constants, A; B and z0 2 C.
h
Acknowledgment This work was supported by the Visiting Scholar Program of Department of Mathematics and Statistics at Curtin University of Technology - Australia when the first author worked as a visiting scholar (200001807894). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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