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The extension of the Jacobi elliptic function rational expansion method
Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713 www.elsevier.com/locate/cnsns
The extension of the Jacobi elliptic function rational expansion method Yaxuan Yu
a,*
, Qi Wang
a,b
, Hongqing Zhang
a,b
a
b
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China Received 2 January 2005; received in revised form 7 May 2005; accepted 25 May 2005 Available online 11 August 2005
Abstract Using a new ansa¨tz, we extend the Jacobi elliptic function rational expansion method and apply it to the asymmetric Nizhnik–Novikov–Veselov equations and the Davey–Stewartson equations. With the aid of symbolic computation, we construct more new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions. Ó 2005 Elsevier B.V. All rights reserved. PACS: 05.55.+b; 04.20.Jb; 05.45.Yv Keywords: Jacobi elliptic functions; Nonlinear evolution equations; Travelling wave solution; Jacobi elliptic function rational expansion method
1. Introduction There have been a great deal of methods for finding exact solutions of nonlinear evolution equations (NLEEs), such as Ba¨cklund transformation, Darboux transformation, variable separation
*
Corresponding author. E-mail address: [email protected] (Y. Yu).
1007-5704/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.05.009
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
703
approach, various tanh methods, Painleve´ method, generalized hyperbolic-function method, homogeneous balance method, similarity reduction method and so on [1–14]. Noted the relations between the elliptic functions (e.g., Jacobi elliptic functions and Weierstrass elliptic functions) and some NLEEs, many researchers have considered this subject to study whether the elliptic functions can be used to express the exact solutions of more NLEEs. Recently, Liu et al. [15–17] presented the Jacobi elliptic function expansion method. They supposed that uðnÞ ¼
n X
aj snj ðnÞ;
n ¼ kðx ctÞ;
ð1Þ
j¼0
where sn(n) is the Jacobi elliptic function. Fan et al. [18,19] extended the Jacobi elliptic function method and assumed that uðnÞ ¼
n X
aj uðnÞ;
n ¼ kðx ctÞ;
ð2Þ
j¼0
where u is taken as one of the Jacobi elliptic functions sn(n), cn(n), dn(n), etc. In particular, Fan and Zhang [19] had discussed many special-type NLEEs, which cannot be dealt with by directly method and require some kinds of pre-possessing techniques. Yan [20,21] further developed an extended Jacobi elliptic function expansion method by using 12 Jacobi elliptic functions. He gave out uðnÞ ¼ a0 þ
n X
fkj1 ðnÞ½aj fk ðnÞ þ bj gk ðnÞ;
ð3Þ
j¼1
where fk(n) and gk(n) can choose 12 Jacobi elliptic functions. Based on the above ideas, Chen et al. [22–24] presented a Jacobi elliptic function rational expansion method uðnÞ ¼ a0 þ
n X j¼1
snj ðnÞ snj1 ðnÞcnðnÞ þ ð1 þ lsnj ðnÞÞ ð1 þ lsnðnÞÞj
or
a0 þ
n X j¼1
snj ðnÞ snj1 ðnÞcnðnÞ þ ; ð1 þ lcnj ðnÞÞ ð1 þ lcnðnÞÞj ð4Þ
and many new rational solutions are obtained. In this paper, we extend this method by means of a new general ansa¨tz. Our method is more powerful than the above existing Jacobi elliptic function methods [15–24], and we can uniformly construct more new exact doubly periodic solutions in terms of rational formal Jacobi elliptic functions of NLEEs. For illustration, we apply the proposed method to the Nizhnik–Novikov–Veselov (ANNV) equations [25] ut þ uxxx 3ðuvÞx ¼ 0;
ð5:1Þ
ux ¼ vy ;
ð5:2Þ
and the Davey–Stewartson equations [3,19] 2
iU t þ U xx U yy 2jU j U 2UV ¼ 0; 2
V xx þ V yy þ 2ðjU j Þxx ¼ 0.
ð6:1Þ ð6:2Þ
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2. Summary of the method The key steps of our method as follows: Step 1. For a given system of NLEEs for ui(x, y, t) (i = 1, 2, . . .) in three variables x, y, t, F i ðui ; uit ; uix ; uiy ; uitt ; uixt ; uiyt ; uixx ; uiyy ; uixy ; . . .Þ ¼ 0;
ð7Þ
we consider its travelling wave solution ui ðx; y; tÞ ¼ ui ðnÞ;
n ¼ ax þ by þ kt;
ð8Þ
where a, b, k are constants to be determined later. Then Eq. (7) is reduced to a system of nonlinear ordinary differential equations (ODEs): Gi ðui ; u0i ; u00i ; . . .Þ ¼ 0.
ð9Þ
Step 2. We suppose that the system of Eq. (9) has the following formal solutions: ui ðnÞ ¼ ai;0 þ
mi X j¼1
ai;2j1 snj ðnÞ ai;2j snj1 ðnÞcnðnÞ þ ; ðl1 snðnÞ þ l2 cnðnÞ þ 1Þj ðl1 snðnÞ þ l2 cnðnÞ þ 1Þj
ð10Þ
where sn(n), cn(n), dn(n) are the Jacobi elliptic functions. They are double periodic and possess the following properties: cn2 ðnÞ þ sn2 ðnÞ ¼ dn2 ðnÞ þ m2 sn2 ðnÞ ¼ 1; 0
sn ðnÞ ¼ cnðnÞdnðnÞ;
0
cn n ¼ snðnÞdnðnÞ;
ð11:1Þ 0
2
dn ðnÞ ¼ m snðnÞcnðnÞ;
ð11:2Þ
where m (0 < m < 1) is the modulus. Then we have mi X jai;2j1 snj1 ðnÞdnðnÞðl2 þ cnðnÞÞ u0i ðnÞ ¼ ðl1 snðnÞ þ l2 cnðnÞ þ 1Þjþ1 j¼1 þ
ai;2j snj2 ðnÞdnðnÞððj 1Þl2 cnðnÞ l1 snðnÞ jsn2 ðnÞ þ j 1Þ ðl1 snðnÞ þ l2 cnðnÞ þ 1Þjþ1
.
ð12Þ
The Jacobi–Glaisher functions for elliptic function can be found in Refs. [26–28]. Step 3. We define the degree of ui(n) as D[ui(n)] = mi, which gives rise to the degrees of other expressions as ðaÞ
D½ui ¼ ni þ a;
ðaÞ s
D½ubi ðuj Þ ¼ ni b þ ða þ nj Þs.
ð13Þ
By balancing the highest-order linear term with nonlinear terms in Eq. (10) we can obtain values of mi. Step 4. Substitute (10) into (9) along with (11) and (12), and set all coefficients of sni(n)cnj(n) (i = 1, 2, . . .; j = 0, 1) to be zero. We get an over-determined nonlinear algebraic system with respect to ai, 0, ai, j (j = 1–2mi), mu1, mu2, a, b, k. Step 5. Solving the over-determined nonlinear algebraic system by using of Maple, we can obtain the explicit expressions for ai, 0, ai, j (i = 1, 2, . . ., j = 1–2mi), l1, l2, a, b, k. Substituting the values of ai, 0, ai, j (i = 1, 2, . . ., j = 1–2mi), l1, l2, a, b, k to Eqs. (8) and (10), we can get double periodic solutions with Jacobi elliptic function of Eq. (7).
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
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Remark 1. Since lim snðnÞ ¼ tanhðnÞ;
m!1
lim snðnÞ ¼ sinðnÞ;
m!0
lim cnðnÞ ¼ sechðnÞ;
m!1
lim cnðnÞ ¼ cosðnÞ;
m!0
lim dnðnÞ ¼ sechðnÞ;
m!1
lim dnðnÞ ¼ 1;
m!0
ð14:1Þ ð14:2Þ
ui degenerate respectively as follows: 1. Solitary wave solutions: ui ðnÞ ¼ ai;0 þ
k X j¼1
! ai;2j1 tanhj ðnÞ ai;2j tanhj1 ðnÞ sechðnÞ j þ j . ðl1 tanhðnÞ þ l2 sechðnÞ þ 1Þ ðl1 tanhðnÞ þ l2 sechðnÞ þ 1Þ
ð15Þ
2. Triangular function formal solutions: ui ðnÞ ¼ ai;0 þ
k X j¼1
! ai;2j1 sinj ðnÞ ai;2j sinj1 ðnÞ cosðnÞ . þ ðl1 sinðnÞ þ l2 cosðnÞ þ 1Þj ðl1 sinðnÞ þ l2 cosðnÞ þ 1Þj
ð16Þ
Thus, when m ! 1 or m ! 0 the solutions which contain solitary wave solutions, singular solitary solutions and triangular functional formal solutions can be obtained by our method. Remark 2. If we replace the Jacobi elliptic functions sn(n) and cn(n) in ansa¨tz (10) with other pairs of Jacobi elliptic functions such as sn(n) and dn(n), ns(n) and cs(n), ns(n) and ds(n), sc(n) and nc(n), dc(n) and nc(n), sd(n) and nd(n), cd(n) and nd(n) (see Refs. [20–24]), then other new double periodic wave solutions, solitary wave solutions, and triangular functional solutions can be obtained for some systems. For simplicity, we omit them here. Remark 3. We can see that the Jacobi elliptic function rational expansion method is more powerful than the methods by Liu et al. [15–17], Fan and Hon [18], Fan and Zhang [19], and Yan [20,21]. When l1 = 0 or l2 = 0, the solutions of Chen et al. [22–24] also can be obtained by our method. Remark 4. Our method can solve two kinds of equations in Liu et al. [15–17].
3. Exact solutions of the ANNV equations Let us consider the (2 + 1)-dimensional ANNV equations (5). These equations are also known as (2 + 1)-dimensional KdV (Korteweg–de Vries) equations or BLMP (Boiti–Leon–Manna– Pempinelli) equations and had been studied in [25]. According to Section 2, we first make the following transformation uðx; y; tÞ ¼ uðnÞ;
vðx; y; tÞ ¼ vðnÞ;
n ¼ ax þ by þ kt;
where a, b and k are constants to be determined. Thus Eq. (17) become
ð17Þ
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Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
ku0 þ a3 u000 3aðuvÞ0 ¼ 0; 0
ð18:1Þ
0
au ¼ bv .
ð18:2Þ
By balancing u000 and (uv) 0 in Eq. (18.1) and u 0 and v 0 in Eq. (18.2), we suppose that Eq. (18) have the following formal solutions: uðnÞ ¼ a0 þ
a1 snðnÞ þ a2 cnðnÞ a3 sn2 ðnÞ þ a4 snðnÞcnðnÞ ; þ l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2
ð19:1Þ
vðnÞ ¼ b0 þ
b1 snðnÞ þ b2 cnðnÞ b3 sn2 ðnÞ þ b4 snðnÞcnðnÞ þ ; l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2
ð19:2Þ
where a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, l1, l2 are constants to be determined later. With the aid of Maple, substituting (19) along with (11) and (12) into (18), we yield an algebraic equation for sni(n)cnj(n) (i = 1, 2, . . .; j = 0, 1). Setting the coefficients of these terms to zero, a set of over-determined algebraic equations with respect to a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, a, b and k can be obtained. Solving the over-determined algebraic equations, we get the following results. Case 1 l1 ¼ l2 ¼ a1 ¼ a2 ¼ a4 ¼ b1 ¼ b2 ¼ b4 ¼ 0; b3 ¼ 2a2 m2 ;
a0 ¼
b0 ¼ b0 ;
a3 ¼ 2am2 b;
bðk 3ab0 4m2 a3 4a3 Þ . 3a2
ð20Þ
Case 2 l1 ¼ a1 ¼ a2 ¼ a4 ¼ b1 ¼ b4 ¼ 0; a0 ¼
l2 ¼ 1;
bð3ab2 6ab0 8m2 a3 þ 4a3 þ 2kÞ ; 6a2
b0 ¼ b0 ;
b2 ¼ b2 ;
1 a3 ¼ ab; 2
1 b3 ¼ ða2 b2 Þ. 2
ð21Þ
Case 3 l2 ¼ a2 ¼ a4 ¼ b2 ¼ b4 ¼ 0;
pffiffiffiffi l 1 ¼ m;
pffiffiffiffi 2 a1 ¼ 2ab mðm 1Þ ;
b0 ¼ b0 ;
ðk a3 3ab0 þ 18a3 m m2 a3 Þb 2 ; a3 ¼ 2abmðm 1Þ ; 3a2 pffiffiffiffi b1 ¼ 2a2 mðm 1Þ2 ; b3 ¼ 2a2 mðm 1Þ2 . a0 ¼
ð22Þ
Case 4 l2 ¼ a2 ¼ a3 ¼ a4 ¼ b2 ¼ b3 ¼ b4 ¼ 0; a0 ¼
ðk 3ab0 þ 5a3 a3 m2 Þb ; 3a2
l1 ¼ 1; b0 ¼ b0 ;
a1 ¼ abðm2 1Þ;
b1 ¼ a2 ðm2 1Þ.
ð23Þ
Case 5 l2 ¼ a2 ¼ a3 ¼ a4 ¼ b2 ¼ b3 ¼ b4 ¼ 0; a0 ¼
ðk 3ab0 a3 þ 5a3 m2 Þb ; 3a2
l1 ¼ m;
b0 ¼ b0 ;
b1 ¼ ma2 ð1 m2 Þ.
a1 ¼ abmð1 m2 Þ; ð24Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
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Case 6 a2 ¼ a2 ; b2 ¼ b2 ; a4 ¼ a2 ; l1 ¼ 1; l2 ¼ 1; b0 ¼ b0 ; a3 ¼ 2abð1 m2 Þ; a1 ¼ 2abðm2 1Þ a2 ; b1 ¼ 2a2 ðm2 1Þ b2 ; b3 ¼ 2a2 ð1 m2 Þ; b4 ¼ b2 ; bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 a0 ¼ . 6a2 2a Case 7 a2 ¼ a2 ; b2 ¼ b2 ; a4 ¼ a2 ; l1 ¼ 1; l2 ¼ 1; b0 ¼ b0 ; a3 ¼ 2abð1 m2 Þ; a1 ¼ 2abð1 m2 Þ a2 ; b1 ¼ b2 þ 2a2 ðm2 1Þ; b3 ¼ 2a2 ð1 m2 Þ; b4 ¼ b2 ; bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 . a0 ¼ 6a2 2a Thus we can obtain seven families of explicit exact travelling wave solutions as follows. According to case 1, bðk 3ab0 4m2 a3 4a3 Þ u1 ¼ þ 2am2 bsn2 ðax þ by þ ktÞ; 3a2 v1 ¼ b0 þ 2a2 m2 sn2 ðax þ by þ ktÞ;
ð25Þ
ð26Þ
ð27:1Þ ð27:2Þ
where b0, a, b and k are arbitrary constants. According to case 2, bð3ab2 6ab0 8m2 a3 þ 4a3 þ 2kÞ absn2 ðax þ by þ ktÞ þ ; 6a2 2ðcnðax þ by þ ktÞ þ 1Þ2 b2 cnðax þ by þ ktÞ ða2 b2 Þsn2 ðax þ by þ ktÞ þ ; v2 ¼ b0 þ 2ðcnðax þ by þ ktÞ þ 1Þ 2ðcnðax þ by þ ktÞ þ 1Þ2
u2 ¼
ð28:1Þ ð28:2Þ
where b0, b2, a, b and k are arbitrary constants. According to case 3, pffiffiffiffi 2 ðk a3 3ab0 þ 18a3 m m2 a3 Þb 2ab mðm 1Þ snðax þ by þ ktÞ p ffiffiffi ffi u3 ¼ 3a2 msnðax þ by þ ktÞ þ 1 2abmðm 1Þ2 sn2 ðax þ by þ ktÞ ; pffiffiffiffi ð msnðax þ by þ ktÞ þ 1Þ2 pffiffiffiffi 2a2 mðm 1Þ2 snðax þ by þ ktÞ 2a2 mðm 1Þ2 sn2 ðax þ by þ ktÞ pffiffiffiffi v3 ¼ b0 ; pffiffiffiffi msnðax þ by þ ktÞ þ 1 ð msnðax þ by þ ktÞ þ 1Þ2
ð29:1Þ ð29:2Þ
where b0, a, b and k are arbitrary constants. According to case 4, ðk 3ab0 þ 5a3 a3 m2 Þb abðm2 1Þsnðax þ by þ ktÞ ; 3a2 snðax þ by þ ktÞ þ 1 a2 ðm2 1Þsnðax þ by þ ktÞ v4 ¼ b0 ; snðax þ by þ ktÞ þ 1 u4 ¼
where b0, a, b and k are arbitrary constants.
ð30:1Þ ð30:2Þ
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Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
According to case 5, ðk 3ab0 a3 þ 5a3 m2 Þb abmð1 m2 Þsnðax þ by þ ktÞ ; 3a2 msnðax þ by þ ktÞ þ 1 ma2 ð1 m2 Þsnðax þ by þ ktÞ v5 ¼ b0 ; msnðax þ by þ ktÞ þ 1 u5 ¼
ð31:1Þ ð31:2Þ
where b0, a, b and k are arbitrary constants. According to case 6, u6 ¼
bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 ð2abðm2 1Þ a2 Þsnn þ 6a2 snn cnn þ 1 2a 2 2 a2 cnn 2abð1 m Þsn n a2 snðnÞcnn þ þ ; 2 snn cnn þ 1 ðsnn cnn þ 1Þ ðsnn cnn þ 1Þ2
v6 ¼ b0 þ
ð32:1Þ
ðb2 þ 2a2 ð1 m2 ÞÞsnn b2 cnn 2a2 ð1 m2 Þsn2 n þ þ snn cnn þ 1 snn cnn þ 1 ðsnn cnn þ 1Þ2 b2 snncnn
ðsnn cnn þ 1Þ2
ð32:2Þ
;
where n = ax + by + kt, a2, b0, b2, a, b and k are arbitrary constants. According to case 7, u7 ¼
bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 ð2abðm2 1Þ a2 Þsnn 6a2 snn cnn þ 1 2a 2 2 a2 cnn 2abð1 m Þsn n a2 snðnÞcnn þ þ ; þ 2 2 snn cnn þ 1 ðsnn cnn þ 1Þ ðsnn cnn þ 1Þ ðb2 þ 2a2 ðm2 1ÞÞsnn b2 cnn 2a2 ð1 m2 Þsn2 n þ þ snn cnn þ 1 snn cnn þ 1 ðsnn cnn þ 1Þ2 b2 snncnn þ ; ðsnn cnn þ 1Þ2
ð33:1Þ
v7 ¼ b0 þ
ð33:2Þ
where n = ax + by + kt, a2, b0, b2, a, b and k are arbitrary constants. Remark 5. It is easy to see that the above solutions include the solutions obtained by Liu et al. [15–17], Fan and Hon [18], Fan and Zhang [19], Yan [20,21] and Chen [22–24]. To the best of our knowledge, solutions (32) and (33) (Case 6 and Case 7) of have not been found before.
4. Exact solutions of the Davey–Stewartson equations Consider the (2 + 1)-dimensional Davey–Stewartson equations (6) [3,19]. We introduce the following transformations: U ¼ eih uðnÞ;
V ¼ vðnÞ;
h ¼ px þ qy þ rt;
n ¼ ax þ by þ kt;
ð34Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
709
where p, q, r, a, b, and k are real constants. Substituting (34) into (6), we find that k = 2(pa qb), and U and V satisfy the following system: ðq2 p2 rÞu þ ða2 b2 Þu00 2u3 2uv ¼ 0; 2 00
ða2 þ b2 Þv00 þ ðu Þ ¼ 0.
ð35:1Þ ð35:2Þ
By balancing (a2 b2)u00 and 2u3 in Eq. (35.1) and (a2 + b2)v00 and (u2)00 in Eq. (35.2), we suppose that Eqs. (35) have the following formal solutions a1 snðnÞ þ a2 cnðnÞ ; l1 snðnÞ þ l2 cnðnÞ þ 1 b1 snðnÞ þ b2 cnðnÞ b3 sn2 ðnÞ þ b4 snðnÞcnðnÞ þ ; vðnÞ ¼ b0 þ l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2 uðnÞ ¼ a0 þ
ð36:1Þ ð36:2Þ
where a0, a1, a2, b0, b1, b2, b3, b4, l1, l2 are constants to be determined later. Similar to Section 3 and with the aid of Maple we obtain the following solutions. Case 1: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 b4 msnðax þ by 2ðpa qbÞtÞ; ð37:1Þ u1 ¼ a2 þ b2 1 1 ða2 b2 Þm2 sn2 ðax þ by 2ðpa qbÞtÞ v1 ¼ ðq2 r p2 þ ðm2 þ 1Þðb2 a2 ÞÞ ; 2 a2 þ b2 1 where p, q, r, a and b are arbitrary constants. Case 2: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a4 b4 snðnÞ u2 ¼ ; 2 2 2 a þ b 1 cnðnÞ þ 1
ð37:2Þ
ð38:1Þ
1 b2 cnðnÞ v2 ¼ ð2b2 þ 2q2 2r 2p2 þ ð2m2 1Þðb2 a2 ÞÞ þ 4 cnðnÞ þ 1 þ
ð2b2 ðb2 þ a2 1Þ þ b2 a2 Þsn2 ðnÞ 4ða2 þ b2 1ÞðcnðnÞ þ 1Þ2
;
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. Case 3: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðb4 a4 Þðm2 1Þ cnðnÞ ; u3 ¼ 2 a2 þ b2 1 snðnÞ þ 1 v3 ¼ 2ðr2 þ p2 q2 Þ þ
ðm2 þ 1Þðb4 a4 Þ 2ðb2 a2 Þ ða2 b2 Þðm2 1ÞsnðnÞ ; a2 þ b2 1 2ða2 þ b2 1ÞðsnðnÞ þ 1Þ
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð38:2Þ
ð39:1Þ ð39:2Þ
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Case 4: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 ; u4 ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1 ðm2 þ 1Þða4 b4 Þ þ 2ða2 b2 Þ v4 ¼ 2ðq2 b2 r p2 Þ þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ 2 ðm2 1Þða2 b Þsn2 ðnÞ b2 snðnÞcnðnÞ ; þ 2 2 2 ða þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ2 where n = ax + by2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. Case 5: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 ; u5 ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1
ð40:1Þ
ð40:2Þ
ð41:1Þ
ðm2 þ 1Þðb4 a4 Þ þ 2ða2 b2 Þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ 2 ðm2 1Þða2 b Þsn2 ðnÞ b2 snðnÞcnðnÞ þ þ ; ð41:2Þ 2 2 ða2 þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ2 where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. The solutions of Eqs. (6) are as follows. According to Case 1 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 b4 msnðax þ by 2ðpa qbÞtÞeiðpxþqyþrtÞ ; ð42:1Þ U 1 ðx; y; tÞ ¼ a2 þ b2 1 1 ða2 b2 Þm2 sn2 ðax þ by 2ðpa qbÞtÞ V 1 ðx; y; tÞ ¼ ðq2 r p2 þ ðm2 þ 1Þðb2 a2 ÞÞ ; 2 a2 þ b2 1 ð42:2Þ v5 ¼ 2ðq2 b2 þ r þ p2 Þ þ
where p, q, r, a and b are arbitrary constants. According to Case 2 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a4 b4 snðnÞ iðpxþqyþrtÞ ; e U 2 ðx; y; tÞ ¼ 2 2 2 a þ b 1 cnðnÞ þ 1 1 b2 cnðnÞ V 2 ðx; y; tÞ ¼ ð2b2 þ 2q2 2r 2p2 þ ð2m2 1Þðb2 a2 ÞÞ þ 4 cnðnÞ þ 1 2 2 2 2 2 ð2b2 ðb þ a 1Þ þ b a Þsn ðnÞ þ ; 4ða2 þ b2 1ÞðcnðnÞ þ 1Þ2 where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð43:1Þ
ð43:2Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
According to Case 3 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðb4 a4 Þðm2 1Þ cnðnÞ iðpxþqyþrtÞ ; e U 3 ðx; y; tÞ ¼ 2 a2 þ b2 1 snðnÞ þ 1 V 3 ðx; y; tÞ ¼ 2ðr2 þ p2 q2 Þ þ
711
ð44:1Þ
ðm2 þ 1Þðb4 a4 Þ 2ðb2 a2 Þ ða2 b2 Þðm2 1ÞsnðnÞ ; a2 þ b2 1 2ða2 þ b2 1ÞðsnðnÞ þ 1Þ ð44:2Þ
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. According to Case 4 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 eiðpxþqyþrtÞ ; U 4 ðx; y; tÞ ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1
ð45:1Þ
ðm2 þ 1Þða4 b4 Þ þ 2ða2 b2 Þ a2 þ b2 1 ðb2 ða2 þ b2 1Þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ
V 4 ðx; y; tÞ ¼ 2ðq2 b2 r p2 Þ þ
þ
ðm2 1Þða2 b2 Þsn2 ðnÞ
b2 snðnÞcnðnÞ
; ðsnðnÞ cnðnÞ þ 1Þ2 þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. According to Case 5 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 eiðpxþqyþrtÞ ; U 5 ðx; y; tÞ ¼ 2 2 2 ðsnðnÞ cnðnÞ þ 1Þ a þb 1 ða2
2
2
ðm2 þ 1Þðb4 a4 Þ þ 2ða2 b2 Þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ ðm2 1Þða2 b2 Þsn2 ðnÞ b2 snðnÞcnðnÞ þ ; þ 2 2 2 2 ða þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð45:2Þ
ð46:1Þ
V 5 ðx; y; tÞ ¼ 2ðq2 b2 þ r þ p2 Þ þ
ð46:2Þ
Remark 6. The above solutions (45) and (46) (Cases 4 and 5) are new solutions that cannot be obtained by the methods in [15–24].
5. Conclusions In this paper, we have extended the Jacobi elliptic function rational expansion method by a new general ansa¨tz. Our method is more powerful than the method proposed recently in Refs. [15–21]. The asymmetric Nizhnik–Novikov–Veselov equations (ANNV) and the Davey–Stewartsob equa-
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tions are chosen to illustrate the method, and many new Jacobi elliptic function solutions are obtained. When the modulus m ! 1, some of these solutions degenerate as solitary wave solutions. The algorithm can be also applied to many other nonlinear evolution equations in mathematical physics. Acknowledgement This work was supported in part by the National Key Basic Research Project of China (No. 2004CB318000). Many thanks are due to Prof. Zhi Li for his suggestions and help. References [1] Ablowitz MJ, Clarkson PA. Soliton, nonlinear evolution equations and inverse scattering. New York: Cambridge University Press; 1991. [2] Matveev VB, Salle MA. Darbooux transformation and soliton. Berlin: Springer; 1991. [3] Lou SY, Lu JZ. Special solutions from the variable separation approach: the Davey–Stewartson equation. J Phys A 1996;29:4209–15. [4] Lou SY. On the coherent structures of the Nizhnik–Novikov–Veselov equation. Phys Lett A 2000;277:94–100. [5] Lou SY, Ruan HY. Revisitation of the localized excitations. J Phys A 2001;34:305–16. [6] Chen Y, Li B, Zhang HQ. Auto-Backlund transformation and exact solutions for modified nonlinear dispersive mK(m, n) equations. Chaos, Solitons Fract 2003;17:693–8. [7] Li B, Chen Y, Zhang HQ. Auto-Backlund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order. Phys Lett A 2002;305:377–82. [8] Wang ML, Zhou YB, Li ZB. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys Lett A 1996;216:67–75. [9] Parkes EJ, Duffy BR. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput Phys Commun 1996;98:288–300. [10] Parkes EJ, Duffy BR. Travelling solitary wave solutions to a compound KdV–Burgers equation. Phys Lett A 1997;229:217–20. [11] Gao YT, Tian B. Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics. Comput Phys Commun 2001; 133:158–64. [12] Fan EG. A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput Phys Commun 2003;153:17–30. [13] Fan EG. A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos, Solitons Fract 2003;15:559–66. [14] Fan EG. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Phys Lett A 2002;300:243–9. [15] Fu ZT, Liu SK, Liu SD, et al. The JEFE method and periodic solutions of two kinds of nonlinear wave equations. Commun Nonlinear Sci Numer Simulat 2003;8:67–75. [16] Fu Z, Liu SK, Liu SD, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys Lett A 2001;290:72–6. [17] Liu SK et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 2001;289:69–74. [18] Fan EG, Hon Benny YC. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota– Satsuma coupled KdV systems. Phys Lett A 2002;292:335–7. [19] Fan EG, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys Lett A 2002;305:383–92.
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[20] Yan ZY. Abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional integrable Davey– Stewartson-type equation via a new method. Chaos, Solitons Fract 2003;18:299–309. [21] Yan ZY. The extended Jacobian elliptic function expansion method and its application in the generalized Hirota– Satsuma coupled KdV system. Chaos, Solitons Fract 2003;15:575–83. [22] Wang Q, Chen Y, Zhang HQ. A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation. Chaos Solitons Fract 2005;23:477–83. [23] Chen Y, Wang Q, Li B. Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly-periodic solutions of nonlinear evolution equations. Z Naturforsch 2004;59a:529–36. [24] Chen Y, Wang Q. A series of new double periodic solutions to a (2 + 1)-dimensional asymmetric Nizhnik– Novikov–Veselov equation. Chin Phys 2004;13:1796–800. [25] Lou SY. Conformal invariance and integrable models. J Phys A: Math Gen 1997;30:4803–14. [26] Chandrasekharan K. Elliptic function. Berlin: Springer; 1978. [27] Patrick DV. Elliptic function and elliptic curves. Cambridge: Cambridge University Press; 1973. [28] Wang ZX, Xia XJ. Special functions. Singapore: World Scientific; 1989.
The extension of the Jacobi elliptic function rational expansion method Yaxuan Yu
a,*
, Qi Wang
a,b
, Hongqing Zhang
a,b
a
b
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China Received 2 January 2005; received in revised form 7 May 2005; accepted 25 May 2005 Available online 11 August 2005
Abstract Using a new ansa¨tz, we extend the Jacobi elliptic function rational expansion method and apply it to the asymmetric Nizhnik–Novikov–Veselov equations and the Davey–Stewartson equations. With the aid of symbolic computation, we construct more new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions. Ó 2005 Elsevier B.V. All rights reserved. PACS: 05.55.+b; 04.20.Jb; 05.45.Yv Keywords: Jacobi elliptic functions; Nonlinear evolution equations; Travelling wave solution; Jacobi elliptic function rational expansion method
1. Introduction There have been a great deal of methods for finding exact solutions of nonlinear evolution equations (NLEEs), such as Ba¨cklund transformation, Darboux transformation, variable separation
*
Corresponding author. E-mail address: [email protected] (Y. Yu).
1007-5704/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.05.009
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
703
approach, various tanh methods, Painleve´ method, generalized hyperbolic-function method, homogeneous balance method, similarity reduction method and so on [1–14]. Noted the relations between the elliptic functions (e.g., Jacobi elliptic functions and Weierstrass elliptic functions) and some NLEEs, many researchers have considered this subject to study whether the elliptic functions can be used to express the exact solutions of more NLEEs. Recently, Liu et al. [15–17] presented the Jacobi elliptic function expansion method. They supposed that uðnÞ ¼
n X
aj snj ðnÞ;
n ¼ kðx ctÞ;
ð1Þ
j¼0
where sn(n) is the Jacobi elliptic function. Fan et al. [18,19] extended the Jacobi elliptic function method and assumed that uðnÞ ¼
n X
aj uðnÞ;
n ¼ kðx ctÞ;
ð2Þ
j¼0
where u is taken as one of the Jacobi elliptic functions sn(n), cn(n), dn(n), etc. In particular, Fan and Zhang [19] had discussed many special-type NLEEs, which cannot be dealt with by directly method and require some kinds of pre-possessing techniques. Yan [20,21] further developed an extended Jacobi elliptic function expansion method by using 12 Jacobi elliptic functions. He gave out uðnÞ ¼ a0 þ
n X
fkj1 ðnÞ½aj fk ðnÞ þ bj gk ðnÞ;
ð3Þ
j¼1
where fk(n) and gk(n) can choose 12 Jacobi elliptic functions. Based on the above ideas, Chen et al. [22–24] presented a Jacobi elliptic function rational expansion method uðnÞ ¼ a0 þ
n X j¼1
snj ðnÞ snj1 ðnÞcnðnÞ þ ð1 þ lsnj ðnÞÞ ð1 þ lsnðnÞÞj
or
a0 þ
n X j¼1
snj ðnÞ snj1 ðnÞcnðnÞ þ ; ð1 þ lcnj ðnÞÞ ð1 þ lcnðnÞÞj ð4Þ
and many new rational solutions are obtained. In this paper, we extend this method by means of a new general ansa¨tz. Our method is more powerful than the above existing Jacobi elliptic function methods [15–24], and we can uniformly construct more new exact doubly periodic solutions in terms of rational formal Jacobi elliptic functions of NLEEs. For illustration, we apply the proposed method to the Nizhnik–Novikov–Veselov (ANNV) equations [25] ut þ uxxx 3ðuvÞx ¼ 0;
ð5:1Þ
ux ¼ vy ;
ð5:2Þ
and the Davey–Stewartson equations [3,19] 2
iU t þ U xx U yy 2jU j U 2UV ¼ 0; 2
V xx þ V yy þ 2ðjU j Þxx ¼ 0.
ð6:1Þ ð6:2Þ
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2. Summary of the method The key steps of our method as follows: Step 1. For a given system of NLEEs for ui(x, y, t) (i = 1, 2, . . .) in three variables x, y, t, F i ðui ; uit ; uix ; uiy ; uitt ; uixt ; uiyt ; uixx ; uiyy ; uixy ; . . .Þ ¼ 0;
ð7Þ
we consider its travelling wave solution ui ðx; y; tÞ ¼ ui ðnÞ;
n ¼ ax þ by þ kt;
ð8Þ
where a, b, k are constants to be determined later. Then Eq. (7) is reduced to a system of nonlinear ordinary differential equations (ODEs): Gi ðui ; u0i ; u00i ; . . .Þ ¼ 0.
ð9Þ
Step 2. We suppose that the system of Eq. (9) has the following formal solutions: ui ðnÞ ¼ ai;0 þ
mi X j¼1
ai;2j1 snj ðnÞ ai;2j snj1 ðnÞcnðnÞ þ ; ðl1 snðnÞ þ l2 cnðnÞ þ 1Þj ðl1 snðnÞ þ l2 cnðnÞ þ 1Þj
ð10Þ
where sn(n), cn(n), dn(n) are the Jacobi elliptic functions. They are double periodic and possess the following properties: cn2 ðnÞ þ sn2 ðnÞ ¼ dn2 ðnÞ þ m2 sn2 ðnÞ ¼ 1; 0
sn ðnÞ ¼ cnðnÞdnðnÞ;
0
cn n ¼ snðnÞdnðnÞ;
ð11:1Þ 0
2
dn ðnÞ ¼ m snðnÞcnðnÞ;
ð11:2Þ
where m (0 < m < 1) is the modulus. Then we have mi X jai;2j1 snj1 ðnÞdnðnÞðl2 þ cnðnÞÞ u0i ðnÞ ¼ ðl1 snðnÞ þ l2 cnðnÞ þ 1Þjþ1 j¼1 þ
ai;2j snj2 ðnÞdnðnÞððj 1Þl2 cnðnÞ l1 snðnÞ jsn2 ðnÞ þ j 1Þ ðl1 snðnÞ þ l2 cnðnÞ þ 1Þjþ1
.
ð12Þ
The Jacobi–Glaisher functions for elliptic function can be found in Refs. [26–28]. Step 3. We define the degree of ui(n) as D[ui(n)] = mi, which gives rise to the degrees of other expressions as ðaÞ
D½ui ¼ ni þ a;
ðaÞ s
D½ubi ðuj Þ ¼ ni b þ ða þ nj Þs.
ð13Þ
By balancing the highest-order linear term with nonlinear terms in Eq. (10) we can obtain values of mi. Step 4. Substitute (10) into (9) along with (11) and (12), and set all coefficients of sni(n)cnj(n) (i = 1, 2, . . .; j = 0, 1) to be zero. We get an over-determined nonlinear algebraic system with respect to ai, 0, ai, j (j = 1–2mi), mu1, mu2, a, b, k. Step 5. Solving the over-determined nonlinear algebraic system by using of Maple, we can obtain the explicit expressions for ai, 0, ai, j (i = 1, 2, . . ., j = 1–2mi), l1, l2, a, b, k. Substituting the values of ai, 0, ai, j (i = 1, 2, . . ., j = 1–2mi), l1, l2, a, b, k to Eqs. (8) and (10), we can get double periodic solutions with Jacobi elliptic function of Eq. (7).
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Remark 1. Since lim snðnÞ ¼ tanhðnÞ;
m!1
lim snðnÞ ¼ sinðnÞ;
m!0
lim cnðnÞ ¼ sechðnÞ;
m!1
lim cnðnÞ ¼ cosðnÞ;
m!0
lim dnðnÞ ¼ sechðnÞ;
m!1
lim dnðnÞ ¼ 1;
m!0
ð14:1Þ ð14:2Þ
ui degenerate respectively as follows: 1. Solitary wave solutions: ui ðnÞ ¼ ai;0 þ
k X j¼1
! ai;2j1 tanhj ðnÞ ai;2j tanhj1 ðnÞ sechðnÞ j þ j . ðl1 tanhðnÞ þ l2 sechðnÞ þ 1Þ ðl1 tanhðnÞ þ l2 sechðnÞ þ 1Þ
ð15Þ
2. Triangular function formal solutions: ui ðnÞ ¼ ai;0 þ
k X j¼1
! ai;2j1 sinj ðnÞ ai;2j sinj1 ðnÞ cosðnÞ . þ ðl1 sinðnÞ þ l2 cosðnÞ þ 1Þj ðl1 sinðnÞ þ l2 cosðnÞ þ 1Þj
ð16Þ
Thus, when m ! 1 or m ! 0 the solutions which contain solitary wave solutions, singular solitary solutions and triangular functional formal solutions can be obtained by our method. Remark 2. If we replace the Jacobi elliptic functions sn(n) and cn(n) in ansa¨tz (10) with other pairs of Jacobi elliptic functions such as sn(n) and dn(n), ns(n) and cs(n), ns(n) and ds(n), sc(n) and nc(n), dc(n) and nc(n), sd(n) and nd(n), cd(n) and nd(n) (see Refs. [20–24]), then other new double periodic wave solutions, solitary wave solutions, and triangular functional solutions can be obtained for some systems. For simplicity, we omit them here. Remark 3. We can see that the Jacobi elliptic function rational expansion method is more powerful than the methods by Liu et al. [15–17], Fan and Hon [18], Fan and Zhang [19], and Yan [20,21]. When l1 = 0 or l2 = 0, the solutions of Chen et al. [22–24] also can be obtained by our method. Remark 4. Our method can solve two kinds of equations in Liu et al. [15–17].
3. Exact solutions of the ANNV equations Let us consider the (2 + 1)-dimensional ANNV equations (5). These equations are also known as (2 + 1)-dimensional KdV (Korteweg–de Vries) equations or BLMP (Boiti–Leon–Manna– Pempinelli) equations and had been studied in [25]. According to Section 2, we first make the following transformation uðx; y; tÞ ¼ uðnÞ;
vðx; y; tÞ ¼ vðnÞ;
n ¼ ax þ by þ kt;
where a, b and k are constants to be determined. Thus Eq. (17) become
ð17Þ
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Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
ku0 þ a3 u000 3aðuvÞ0 ¼ 0; 0
ð18:1Þ
0
au ¼ bv .
ð18:2Þ
By balancing u000 and (uv) 0 in Eq. (18.1) and u 0 and v 0 in Eq. (18.2), we suppose that Eq. (18) have the following formal solutions: uðnÞ ¼ a0 þ
a1 snðnÞ þ a2 cnðnÞ a3 sn2 ðnÞ þ a4 snðnÞcnðnÞ ; þ l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2
ð19:1Þ
vðnÞ ¼ b0 þ
b1 snðnÞ þ b2 cnðnÞ b3 sn2 ðnÞ þ b4 snðnÞcnðnÞ þ ; l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2
ð19:2Þ
where a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, l1, l2 are constants to be determined later. With the aid of Maple, substituting (19) along with (11) and (12) into (18), we yield an algebraic equation for sni(n)cnj(n) (i = 1, 2, . . .; j = 0, 1). Setting the coefficients of these terms to zero, a set of over-determined algebraic equations with respect to a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, a, b and k can be obtained. Solving the over-determined algebraic equations, we get the following results. Case 1 l1 ¼ l2 ¼ a1 ¼ a2 ¼ a4 ¼ b1 ¼ b2 ¼ b4 ¼ 0; b3 ¼ 2a2 m2 ;
a0 ¼
b0 ¼ b0 ;
a3 ¼ 2am2 b;
bðk 3ab0 4m2 a3 4a3 Þ . 3a2
ð20Þ
Case 2 l1 ¼ a1 ¼ a2 ¼ a4 ¼ b1 ¼ b4 ¼ 0; a0 ¼
l2 ¼ 1;
bð3ab2 6ab0 8m2 a3 þ 4a3 þ 2kÞ ; 6a2
b0 ¼ b0 ;
b2 ¼ b2 ;
1 a3 ¼ ab; 2
1 b3 ¼ ða2 b2 Þ. 2
ð21Þ
Case 3 l2 ¼ a2 ¼ a4 ¼ b2 ¼ b4 ¼ 0;
pffiffiffiffi l 1 ¼ m;
pffiffiffiffi 2 a1 ¼ 2ab mðm 1Þ ;
b0 ¼ b0 ;
ðk a3 3ab0 þ 18a3 m m2 a3 Þb 2 ; a3 ¼ 2abmðm 1Þ ; 3a2 pffiffiffiffi b1 ¼ 2a2 mðm 1Þ2 ; b3 ¼ 2a2 mðm 1Þ2 . a0 ¼
ð22Þ
Case 4 l2 ¼ a2 ¼ a3 ¼ a4 ¼ b2 ¼ b3 ¼ b4 ¼ 0; a0 ¼
ðk 3ab0 þ 5a3 a3 m2 Þb ; 3a2
l1 ¼ 1; b0 ¼ b0 ;
a1 ¼ abðm2 1Þ;
b1 ¼ a2 ðm2 1Þ.
ð23Þ
Case 5 l2 ¼ a2 ¼ a3 ¼ a4 ¼ b2 ¼ b3 ¼ b4 ¼ 0; a0 ¼
ðk 3ab0 a3 þ 5a3 m2 Þb ; 3a2
l1 ¼ m;
b0 ¼ b0 ;
b1 ¼ ma2 ð1 m2 Þ.
a1 ¼ abmð1 m2 Þ; ð24Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
707
Case 6 a2 ¼ a2 ; b2 ¼ b2 ; a4 ¼ a2 ; l1 ¼ 1; l2 ¼ 1; b0 ¼ b0 ; a3 ¼ 2abð1 m2 Þ; a1 ¼ 2abðm2 1Þ a2 ; b1 ¼ 2a2 ðm2 1Þ b2 ; b3 ¼ 2a2 ð1 m2 Þ; b4 ¼ b2 ; bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 a0 ¼ . 6a2 2a Case 7 a2 ¼ a2 ; b2 ¼ b2 ; a4 ¼ a2 ; l1 ¼ 1; l2 ¼ 1; b0 ¼ b0 ; a3 ¼ 2abð1 m2 Þ; a1 ¼ 2abð1 m2 Þ a2 ; b1 ¼ b2 þ 2a2 ðm2 1Þ; b3 ¼ 2a2 ð1 m2 Þ; b4 ¼ b2 ; bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 . a0 ¼ 6a2 2a Thus we can obtain seven families of explicit exact travelling wave solutions as follows. According to case 1, bðk 3ab0 4m2 a3 4a3 Þ u1 ¼ þ 2am2 bsn2 ðax þ by þ ktÞ; 3a2 v1 ¼ b0 þ 2a2 m2 sn2 ðax þ by þ ktÞ;
ð25Þ
ð26Þ
ð27:1Þ ð27:2Þ
where b0, a, b and k are arbitrary constants. According to case 2, bð3ab2 6ab0 8m2 a3 þ 4a3 þ 2kÞ absn2 ðax þ by þ ktÞ þ ; 6a2 2ðcnðax þ by þ ktÞ þ 1Þ2 b2 cnðax þ by þ ktÞ ða2 b2 Þsn2 ðax þ by þ ktÞ þ ; v2 ¼ b0 þ 2ðcnðax þ by þ ktÞ þ 1Þ 2ðcnðax þ by þ ktÞ þ 1Þ2
u2 ¼
ð28:1Þ ð28:2Þ
where b0, b2, a, b and k are arbitrary constants. According to case 3, pffiffiffiffi 2 ðk a3 3ab0 þ 18a3 m m2 a3 Þb 2ab mðm 1Þ snðax þ by þ ktÞ p ffiffiffi ffi u3 ¼ 3a2 msnðax þ by þ ktÞ þ 1 2abmðm 1Þ2 sn2 ðax þ by þ ktÞ ; pffiffiffiffi ð msnðax þ by þ ktÞ þ 1Þ2 pffiffiffiffi 2a2 mðm 1Þ2 snðax þ by þ ktÞ 2a2 mðm 1Þ2 sn2 ðax þ by þ ktÞ pffiffiffiffi v3 ¼ b0 ; pffiffiffiffi msnðax þ by þ ktÞ þ 1 ð msnðax þ by þ ktÞ þ 1Þ2
ð29:1Þ ð29:2Þ
where b0, a, b and k are arbitrary constants. According to case 4, ðk 3ab0 þ 5a3 a3 m2 Þb abðm2 1Þsnðax þ by þ ktÞ ; 3a2 snðax þ by þ ktÞ þ 1 a2 ðm2 1Þsnðax þ by þ ktÞ v4 ¼ b0 ; snðax þ by þ ktÞ þ 1 u4 ¼
where b0, a, b and k are arbitrary constants.
ð30:1Þ ð30:2Þ
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According to case 5, ðk 3ab0 a3 þ 5a3 m2 Þb abmð1 m2 Þsnðax þ by þ ktÞ ; 3a2 msnðax þ by þ ktÞ þ 1 ma2 ð1 m2 Þsnðax þ by þ ktÞ v5 ¼ b0 ; msnðax þ by þ ktÞ þ 1 u5 ¼
ð31:1Þ ð31:2Þ
where b0, a, b and k are arbitrary constants. According to case 6, u6 ¼
bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 ð2abðm2 1Þ a2 Þsnn þ 6a2 snn cnn þ 1 2a 2 2 a2 cnn 2abð1 m Þsn n a2 snðnÞcnn þ þ ; 2 snn cnn þ 1 ðsnn cnn þ 1Þ ðsnn cnn þ 1Þ2
v6 ¼ b0 þ
ð32:1Þ
ðb2 þ 2a2 ð1 m2 ÞÞsnn b2 cnn 2a2 ð1 m2 Þsn2 n þ þ snn cnn þ 1 snn cnn þ 1 ðsnn cnn þ 1Þ2 b2 snncnn
ðsnn cnn þ 1Þ2
ð32:2Þ
;
where n = ax + by + kt, a2, b0, b2, a, b and k are arbitrary constants. According to case 7, u7 ¼
bð2k 6ab0 þ 10a3 2a3 m2 Þ a2 a þ b2 ð2abðm2 1Þ a2 Þsnn 6a2 snn cnn þ 1 2a 2 2 a2 cnn 2abð1 m Þsn n a2 snðnÞcnn þ þ ; þ 2 2 snn cnn þ 1 ðsnn cnn þ 1Þ ðsnn cnn þ 1Þ ðb2 þ 2a2 ðm2 1ÞÞsnn b2 cnn 2a2 ð1 m2 Þsn2 n þ þ snn cnn þ 1 snn cnn þ 1 ðsnn cnn þ 1Þ2 b2 snncnn þ ; ðsnn cnn þ 1Þ2
ð33:1Þ
v7 ¼ b0 þ
ð33:2Þ
where n = ax + by + kt, a2, b0, b2, a, b and k are arbitrary constants. Remark 5. It is easy to see that the above solutions include the solutions obtained by Liu et al. [15–17], Fan and Hon [18], Fan and Zhang [19], Yan [20,21] and Chen [22–24]. To the best of our knowledge, solutions (32) and (33) (Case 6 and Case 7) of have not been found before.
4. Exact solutions of the Davey–Stewartson equations Consider the (2 + 1)-dimensional Davey–Stewartson equations (6) [3,19]. We introduce the following transformations: U ¼ eih uðnÞ;
V ¼ vðnÞ;
h ¼ px þ qy þ rt;
n ¼ ax þ by þ kt;
ð34Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
709
where p, q, r, a, b, and k are real constants. Substituting (34) into (6), we find that k = 2(pa qb), and U and V satisfy the following system: ðq2 p2 rÞu þ ða2 b2 Þu00 2u3 2uv ¼ 0; 2 00
ða2 þ b2 Þv00 þ ðu Þ ¼ 0.
ð35:1Þ ð35:2Þ
By balancing (a2 b2)u00 and 2u3 in Eq. (35.1) and (a2 + b2)v00 and (u2)00 in Eq. (35.2), we suppose that Eqs. (35) have the following formal solutions a1 snðnÞ þ a2 cnðnÞ ; l1 snðnÞ þ l2 cnðnÞ þ 1 b1 snðnÞ þ b2 cnðnÞ b3 sn2 ðnÞ þ b4 snðnÞcnðnÞ þ ; vðnÞ ¼ b0 þ l1 snðnÞ þ l2 cnðnÞ þ 1 ðl1 snðnÞ þ l2 cnðnÞ þ 1Þ2 uðnÞ ¼ a0 þ
ð36:1Þ ð36:2Þ
where a0, a1, a2, b0, b1, b2, b3, b4, l1, l2 are constants to be determined later. Similar to Section 3 and with the aid of Maple we obtain the following solutions. Case 1: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 b4 msnðax þ by 2ðpa qbÞtÞ; ð37:1Þ u1 ¼ a2 þ b2 1 1 ða2 b2 Þm2 sn2 ðax þ by 2ðpa qbÞtÞ v1 ¼ ðq2 r p2 þ ðm2 þ 1Þðb2 a2 ÞÞ ; 2 a2 þ b2 1 where p, q, r, a and b are arbitrary constants. Case 2: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a4 b4 snðnÞ u2 ¼ ; 2 2 2 a þ b 1 cnðnÞ þ 1
ð37:2Þ
ð38:1Þ
1 b2 cnðnÞ v2 ¼ ð2b2 þ 2q2 2r 2p2 þ ð2m2 1Þðb2 a2 ÞÞ þ 4 cnðnÞ þ 1 þ
ð2b2 ðb2 þ a2 1Þ þ b2 a2 Þsn2 ðnÞ 4ða2 þ b2 1ÞðcnðnÞ þ 1Þ2
;
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. Case 3: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðb4 a4 Þðm2 1Þ cnðnÞ ; u3 ¼ 2 a2 þ b2 1 snðnÞ þ 1 v3 ¼ 2ðr2 þ p2 q2 Þ þ
ðm2 þ 1Þðb4 a4 Þ 2ðb2 a2 Þ ða2 b2 Þðm2 1ÞsnðnÞ ; a2 þ b2 1 2ða2 þ b2 1ÞðsnðnÞ þ 1Þ
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð38:2Þ
ð39:1Þ ð39:2Þ
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Case 4: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 ; u4 ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1 ðm2 þ 1Þða4 b4 Þ þ 2ða2 b2 Þ v4 ¼ 2ðq2 b2 r p2 Þ þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ 2 ðm2 1Þða2 b Þsn2 ðnÞ b2 snðnÞcnðnÞ ; þ 2 2 2 ða þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ2 where n = ax + by2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. Case 5: The solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 ; u5 ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1
ð40:1Þ
ð40:2Þ
ð41:1Þ
ðm2 þ 1Þðb4 a4 Þ þ 2ða2 b2 Þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ 2 ðm2 1Þða2 b Þsn2 ðnÞ b2 snðnÞcnðnÞ þ þ ; ð41:2Þ 2 2 ða2 þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ2 where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. The solutions of Eqs. (6) are as follows. According to Case 1 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 b4 msnðax þ by 2ðpa qbÞtÞeiðpxþqyþrtÞ ; ð42:1Þ U 1 ðx; y; tÞ ¼ a2 þ b2 1 1 ða2 b2 Þm2 sn2 ðax þ by 2ðpa qbÞtÞ V 1 ðx; y; tÞ ¼ ðq2 r p2 þ ðm2 þ 1Þðb2 a2 ÞÞ ; 2 a2 þ b2 1 ð42:2Þ v5 ¼ 2ðq2 b2 þ r þ p2 Þ þ
where p, q, r, a and b are arbitrary constants. According to Case 2 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a4 b4 snðnÞ iðpxþqyþrtÞ ; e U 2 ðx; y; tÞ ¼ 2 2 2 a þ b 1 cnðnÞ þ 1 1 b2 cnðnÞ V 2 ðx; y; tÞ ¼ ð2b2 þ 2q2 2r 2p2 þ ð2m2 1Þðb2 a2 ÞÞ þ 4 cnðnÞ þ 1 2 2 2 2 2 ð2b2 ðb þ a 1Þ þ b a Þsn ðnÞ þ ; 4ða2 þ b2 1ÞðcnðnÞ þ 1Þ2 where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð43:1Þ
ð43:2Þ
Y. Yu et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 702–713
According to Case 3 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðb4 a4 Þðm2 1Þ cnðnÞ iðpxþqyþrtÞ ; e U 3 ðx; y; tÞ ¼ 2 a2 þ b2 1 snðnÞ þ 1 V 3 ðx; y; tÞ ¼ 2ðr2 þ p2 q2 Þ þ
711
ð44:1Þ
ðm2 þ 1Þðb4 a4 Þ 2ðb2 a2 Þ ða2 b2 Þðm2 1ÞsnðnÞ ; a2 þ b2 1 2ða2 þ b2 1ÞðsnðnÞ þ 1Þ ð44:2Þ
where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. According to Case 4 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 eiðpxþqyþrtÞ ; U 4 ðx; y; tÞ ¼ 2 ðsnðnÞ cnðnÞ þ 1Þ a2 þ b2 1
ð45:1Þ
ðm2 þ 1Þða4 b4 Þ þ 2ða2 b2 Þ a2 þ b2 1 ðb2 ða2 þ b2 1Þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ
V 4 ðx; y; tÞ ¼ 2ðq2 b2 r p2 Þ þ
þ
ðm2 1Þða2 b2 Þsn2 ðnÞ
b2 snðnÞcnðnÞ
; ðsnðnÞ cnðnÞ þ 1Þ2 þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants. According to Case 5 the solutions are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðm2 1Þðb4 a4 Þ 2snðnÞ 1 eiðpxþqyþrtÞ ; U 5 ðx; y; tÞ ¼ 2 2 2 ðsnðnÞ cnðnÞ þ 1Þ a þb 1 ða2
2
2
ðm2 þ 1Þðb4 a4 Þ þ 2ða2 b2 Þ a2 þ b2 1 2 ðb2 ða2 þ b 1Þ þ ðm2 1Þða2 b2 ÞÞsnðnÞ b2 cnðnÞ þ þ 2 2 snðnÞ cnðnÞ þ 1 ða þ b 1ÞðsnðnÞ cnðnÞ þ 1Þ ðm2 1Þða2 b2 Þsn2 ðnÞ b2 snðnÞcnðnÞ þ ; þ 2 2 2 2 ða þ b 1ÞðsnðnÞ þ cnðnÞ þ 1Þ ðsnðnÞ cnðnÞ þ 1Þ where n = ax + by 2(pa qb)t, b2, p, q, r, a and b are arbitrary constants.
ð45:2Þ
ð46:1Þ
V 5 ðx; y; tÞ ¼ 2ðq2 b2 þ r þ p2 Þ þ
ð46:2Þ
Remark 6. The above solutions (45) and (46) (Cases 4 and 5) are new solutions that cannot be obtained by the methods in [15–24].
5. Conclusions In this paper, we have extended the Jacobi elliptic function rational expansion method by a new general ansa¨tz. Our method is more powerful than the method proposed recently in Refs. [15–21]. The asymmetric Nizhnik–Novikov–Veselov equations (ANNV) and the Davey–Stewartsob equa-
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tions are chosen to illustrate the method, and many new Jacobi elliptic function solutions are obtained. When the modulus m ! 1, some of these solutions degenerate as solitary wave solutions. The algorithm can be also applied to many other nonlinear evolution equations in mathematical physics. Acknowledgement This work was supported in part by the National Key Basic Research Project of China (No. 2004CB318000). Many thanks are due to Prof. Zhi Li for his suggestions and help. References [1] Ablowitz MJ, Clarkson PA. Soliton, nonlinear evolution equations and inverse scattering. New York: Cambridge University Press; 1991. [2] Matveev VB, Salle MA. Darbooux transformation and soliton. Berlin: Springer; 1991. [3] Lou SY, Lu JZ. Special solutions from the variable separation approach: the Davey–Stewartson equation. J Phys A 1996;29:4209–15. [4] Lou SY. On the coherent structures of the Nizhnik–Novikov–Veselov equation. Phys Lett A 2000;277:94–100. [5] Lou SY, Ruan HY. Revisitation of the localized excitations. J Phys A 2001;34:305–16. [6] Chen Y, Li B, Zhang HQ. Auto-Backlund transformation and exact solutions for modified nonlinear dispersive mK(m, n) equations. Chaos, Solitons Fract 2003;17:693–8. [7] Li B, Chen Y, Zhang HQ. Auto-Backlund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order. Phys Lett A 2002;305:377–82. [8] Wang ML, Zhou YB, Li ZB. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys Lett A 1996;216:67–75. [9] Parkes EJ, Duffy BR. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput Phys Commun 1996;98:288–300. [10] Parkes EJ, Duffy BR. Travelling solitary wave solutions to a compound KdV–Burgers equation. Phys Lett A 1997;229:217–20. [11] Gao YT, Tian B. Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics. Comput Phys Commun 2001; 133:158–64. [12] Fan EG. A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput Phys Commun 2003;153:17–30. [13] Fan EG. A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos, Solitons Fract 2003;15:559–66. [14] Fan EG. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Phys Lett A 2002;300:243–9. [15] Fu ZT, Liu SK, Liu SD, et al. The JEFE method and periodic solutions of two kinds of nonlinear wave equations. Commun Nonlinear Sci Numer Simulat 2003;8:67–75. [16] Fu Z, Liu SK, Liu SD, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys Lett A 2001;290:72–6. [17] Liu SK et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 2001;289:69–74. [18] Fan EG, Hon Benny YC. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota– Satsuma coupled KdV systems. Phys Lett A 2002;292:335–7. [19] Fan EG, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys Lett A 2002;305:383–92.
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