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A series of new exact solutions for a complex coupled KdV system
Chaos, Solitons and Fractals 19 (2004) 515–525 www.elsevier.com/locate/chaos
A series of new exact solutions for a complex coupled KdV system Y.C. Hon a, E.G. Fan a
b,*
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue 83, Hong Kong SAR, PeopleÕs Republic of China b Institute of Mathematics and Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, PeopleÕs Republic of China Accepted 7 March 2003
Abstract In this paper an algebraic method is devised to uniformly construct a series of exact solutions for general nonlinear equations. Compared with the existing tanh methods and Jacobi function method, the proposed method gives more general exact solutions without much extra effort. More importantly, the method provides a guideline for the classification of the solutions based on the given parameters. For illustration, we apply the proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction The effort in finding exact solution to nonlinear equation is important for the understanding of most nonlinear physical phenomena. For instances, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modelled by the bell shaped sech solutions and the kink shaped tanh solutions. The availability of these exact solutions for those nonlinear equations can greatly facilitate the verification of numerical solvers on the stability analysis of the solutions. In the past decades, there has been significant progression in the development of these methods such as inverse scattering method [1,2], Darboux transformation [3–10], Hirota bilinear method [11–13], and tanh method [14–16]. Among these, the tanh method is considered to be the most effective and direct algebraic method for solving nonlinear equations. Much research work has been concentrated on the various extensions and applications of the tanh method [16–20]. By applying spectral theory, Weierstrass and Theta elliptic functions can be used to find periodic solutions for some equations such as KdV equation, coupled nonlinear Schr€ odinger equation et al. But this method usually is applied in the integrable nonlinear evolution equations admitting Lax pairs representation [21–23]. An alternative method is to transform the equation under study to the Weierstrass equation, Jacobi equation, or more generally, to Painleve-type equations [24,25]. This procedure is in general complicated or impossible, especially for complicated dissipative nonlinear equations and nonlinear coupled systems. Very recently, a Jacobi function expansion method was applied to construct periodic wave solutions for some nonlinear equations. The essential idea of this method is similar to the tanh method by replacing the tanh function with some Jacobi elliptic functions such as cnn, snn and dnn [26,27]. For example, Jacobi periodic solution in terms of snn may be obtained by applying sn-function expansion. To get Jacobi doubly periodic wave solutions in terms of cnn and dnn, many similarly repetitious calculations have to be made, and these efforts will be in vain if a equation does not admit these type solutions at all. Recently, we
*
Corresponding author. E-mail addresses: [email protected] (Y.C. Hon), [email protected] (E.G. Fan).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00099-7
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proposed a new straightforward algebraic method which further exceeds the applicability of the above methods in obtaining a series of exact wave solutions including the soliton, rational, triangular periodic, Jacobi, and Weierstrass doubly periodic solutions [28]. In this paper, we apply the proposed method to solve a complex coupled KdV system constructing a series of new exact solutions including (a) kink-shaped and bell-shaped soliton solutions, and (b) Jacobi and Weierstrass elliptic doubly periodic solutions. In the following Section 2, the detail deviation of the proposed method will be given. The application of the proposed method to a new complex coupled KdV system is illustrated in Section 3. Conclusion is then given in the final Section 4.
2. Methodology Let recall our proposed method whose main steps of the proposed method are outlined as follows [28]: Step 1. For the travelling wave solutions of equation H ðu; ut ; ux ; uxx ; . . .Þ ¼ 0;
ð1Þ
we introduce the wave transformation uðx; tÞ ¼ U ðnÞ, n ¼ x þ ct as follows: o d !c ; ot dn
o d ! ; ox dn
Under the transformation, Eq. (1) becomes an ordinary differential equation (ODE) as: H ðU ; U 0 ; U 00 ; . . .Þ ¼ 0: Step 2. We introduce a new variable u ¼ uðnÞ which is a solution of the following first-order ODE: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u m u 0 ¼ et cj uj ;
ð2Þ
ð3Þ
j¼0
where e ¼ 1. The derivatives with respect to the variable n become the derivatives with respect to the variable u as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # uX m m 2 2 X X u m d d d 1 d d 2 j1 j j ! et ; þ cj u !e jcj u cj u ;... dn du 2 j¼1 du j¼0 du2 dn2 j¼0 Step 3. By virtue of the variable u, we expand the solution of Eqs. (1) or (2) as the following series: uðx; tÞ ¼ U ðnÞ ¼
n X
ai ui :
ð4Þ
i¼0
Balancing the highest derivative term with the nonlinear terms in Eq. (2) will give a relation for the positive integers n and m, from which the different possible values of n and m can be determined. These values lead to the series expansions of the exact solutions for Eq. (1). For example, in the case of KdV equation ut þ 6uux þ uxxx ¼ 0;
ð5Þ
we have m ¼ n þ 2:
ð6Þ
If we take n ¼ 1 and m ¼ 3 in (6), we obtain the following series expansion of an exact solution of the KdV equation (5) as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ a0 þ a1 u; u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 : Similarly, if we take n ¼ 2, m ¼ 4 in (6), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ a0 þ a1 u þ a2 u2 ; u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 þ c4 u4 :
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517
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm j Step 4. Substituting the expansion (4) into Eq. (2) and setting the coefficients of all powers of ui and ui j¼0 cj u to zeroes, we obtain a system of algebraic equations, from which the constants c, ai , cj (i ¼ 0; 1; . . . ; n; j ¼ 0; 1; . . . ; m) can be found explicitly. Step 5. Substituting the constants cj (j ¼ 0; 1; . . . ; m) obtained in Step 4 into Eq. (3), we can then obtain all the possible solutions. We remark here that the travelling wave solutions of Eq. (2) depend on the explicit solvability of Eq. (3). The solution of the system of algebraic equations will be getting tedious with the increase of the values of n and m. When m ¼ 4, Eq. (3) gives a series of interesting solutions such as soliton, rational, triangular periodic, Jacobi and Weierstrass doubly periodic solutions. We then consider only the case m ¼ 4 in this paper and hence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 þ c4 u4 : ð7Þ We have the following results: (i) If c3 ¼ c1 ¼ c0 ¼ 0, Eq. (7) possesses a bell-shaped soliton solution rffiffiffiffiffiffiffiffiffi pffiffiffiffi c2 u ¼ sechð c2 nÞ; c2 > 0; c4 < 0; c4 a triangular solution rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi c2 u ¼ secð c2 nÞ; c4
ð8Þ
c2 < 0; c4 > 0
and a rational solution e u ¼ pffiffiffiffi ; c2 ¼ 0; c4 > 0: c4 n i(ii) If c3 ¼ c1 ¼ 0; c0 ¼ c22 =4c4 , Eq. (7) possesses a kink-shaped soliton solution rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi c2 c2 tanh n ; c2 < 0; c4 > 0 u¼e 2c4 2
ð9Þ
and a triangular solution rffiffiffiffi rffiffiffiffi c2 c2 tan u¼e n ; c2 > 0; c4 > 0: c4 2 (iii) If c3 ¼ c1 ¼ 0, Eq. (7) admits two Jacobi elliptic doubly periodic solutions sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 k 2 c2 c2 k 2 ð1 k 2 Þ ; u¼ n ; c4 < 0; c2 > 0; c0 ¼ 2 cn 2 2 c4 ð2k 1Þ 2k 1 c4 ð2k 2 1Þ2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi c2 c2 c2 ð1 k 2 Þ dn u¼ n ; c4 < 0; c2 > 0; c0 ¼ 2 2 2 c4 ð2 k Þ 2k c4 ð2 k 2 Þ2
ð10Þ
ð11Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 k 2 c2 u¼e sn 2 n ; c4 ðk 2 þ 1Þ k þ1
c4 > 0; c2 < 0; c0 ¼
c22 k 2 c4 ðk 2 þ 1Þ2
:
ð12Þ
As k ! 1, the Jacobi doubly periodic solutions (10) and (11) degenerate to the soliton solutions (8) and (9) respectively. (iv) If c4 ¼ c0 ¼ c1 ¼ 0, Eq. (7) possesses a bell-shaped soliton solution pffiffiffiffi c2 c2 2 u ¼ sech ð13Þ n ; c2 > 0; c3 2 a triangular solution pffiffiffiffiffiffiffiffi c2 c2 n ; u ¼ sec2 c3 2
c2 < 0
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and a rational solution u¼
1 ; c3 n2
c2 ¼ 0:
(v) If c4 ¼ c2 ¼ 0, c3 > 0, Eq. (7) admits a Weierstrass elliptic doubly periodic solution pffiffiffiffi c3 u¼} n; g2 ; g3 ; 2
ð14Þ
where g2 ¼ 4c1 =c3 , and g3 ¼ 4c0 =c3 . Remark 1. The other types of travelling wave solutions such as cscn, cotn, cschn, and cothn can also be obtained by considering the different values of cj (j ¼ 0; 1; . . . ; 4) in Eq. (7). These solutions appear in pairs of the functions secn, tann, sechn and tanhn. Especially, when c1 ¼ c3 ¼ 0, c0 ¼ 1, c2 ¼ 2, c4 ¼ 1, the Eq. (7) has a solution tanhn which is given by the existing tanh method [14–18]. When c1 ¼ c3 ¼ 0, c0 ¼ b2 , c2 ¼ 2b, c4 ¼ 1, the Eq. (7) degenerates to a Riccati equation. In this case the proposed method is the same as the extended tanh method [19,20]. The cases (10)–(12) readily cover the results of Jacobi function expansion method [26,27]. In conclusion, our proposed method in this paper is a generalization of both the tanh and Jacobi function expansion method. The proposed method not only give an unified formulation to construct various travelling wave solutions, but also provides a rule to classify the types of solutions according to the given parameters. Furthermore, the proposed method is readily computerizable in solving Eq. (1) by using symbolic software like Mathematica or Maple.
3. Application to a new complex coupled KdV system Recently, Wu et al. [29] considered a 4 4 matrix spectral problem with three potentials and derived a new hierarchy of nonlinear evolution equations, in which a particular case will give the following system of complex coupled KdV equations: 1 ut ¼ uxxx 3uux þ 3ðjvj2 Þx ; 2 vt ¼ vxxx þ 3uvx :
ð15Þ
As v ¼ 0 and ImðvÞ ¼ 0, system (15) is reduced to the well-known KdV equation and the Hirota–Satsuma coupled system respectively [11,12]. Two kinds of soliton solutions were given by Fan and Chao [30] using the tanh method. The proposed method in this paper will further give a series of new explicit exact solutions to the system (15) as follows: In order to obtain the travelling wave solutions of system (15), we make the transformation uðx; tÞ ¼ U ðnÞ;
vðx; tÞ ¼ eih V ðnÞ;
where n ¼ x þ ct, h ¼ px þ qt þ r, and reduce system (15) to the following system of ordinary differential equations: 1 cU 0 U 000 þ 3UU 0 3ðV 2 Þ0 ¼ 0; 2 0 cV þ V 000 3p2 V 0 3UV 0 ¼ 0;
ð16Þ
ðq p3 ÞV þ 3pV 00 ¼ 0: Suppose that U¼
n1 X i¼0
ai ui ;
V ¼
n2 X
bi ui ;
i¼0
and u satisfies (3). Balancing the highest linear term with the nonlinear terms in (16) gives m ¼ n1 þ 2, n2 6 n1 . By choosing m ¼ 4, n1 ¼ n2 ¼ n3 ¼ 2 we have U ¼ a0 þ a1 u þ a2 u2 ;
V ¼ b0 þ b1 u þ b2 u2 :
ð17Þ
Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
519
Substituting (17) into (16) gives 2eca1 6ea0 a1 12eb0 b1 36e3 a2 c1 e3 a1 c2 ¼ 0; 6ea21 þ 4eca2 þ 12ea0 a2 12eb21 24eb0 b2 8e3 a2 c2 3e3 a1 c3 ¼ 0; 18ea1 a2 36eb1 b2 15e3 a2 c3 6ea1 c4 ¼ 0; 12ea22 24eb22 24e3 a2 c4 ¼ 0; ecb1 3ep2 b1 3ea0 b1 þ 3e3 b2 c1 þ e3 b1 c2 ¼ 0; 3ea1 b1 þ 2ecb2 6ep2 b2 6ea0 b2 þ 8e3 b2 c2 þ 3e3 b1 c3 ¼ 0; 3ea2 b1 6ea1 b2 þ 15e3 b2 c3 þ 6e3 b1 c4 ¼ 0; 6ea2 b2 þ 24e3 b2 c4 ¼ 0; 2p3 b0 þ 2qb0 þ 12e2 pb2 c0 þ 3pe2 b1 c1 ¼ 0; 2p3 b1 þ 2qb1 þ 18e2 pb2 c1 þ 6e2 b1 c2 ¼ 0; 2p3 b2 þ 2qb2 þ 24e2 pb2 c2 þ 9e2 pb1 c3 ¼ 0; 30e2 b2 c3 þ 12e2 pb1 c4 ¼ 0; 36e2 pb2 c4 ¼ 0: Since e ¼ 1 implies that e2 ¼ 1 and e3 ¼ e, we may eliminate the variable e from the above system. With the aid of Mathematica, we obtain a total of four kinds of solutions c3 ¼ c1 ¼ a1 ¼ b1 ¼ p ¼ q ¼ 0; 1 1 2 a0 ¼ ðc þ 4c2 Þ; c4 ¼ a2 ; b0 ¼ ðc þ c2 Þ; 3 4 3
1 b2 ¼ a2 ; 2
ð18Þ
with a2 , c, c0 , c2 being arbitrary constants; c3 ¼ c1 ¼ b1 ¼ b2 ¼ a1 ¼ 0; 1 1 q ¼ p3 ; c4 ¼ a2 ; a0 ¼ ð2c2 cÞ; 2 3
ð19Þ
with a2 , c, p, c0 , c2 being arbitrary constants; c3 ¼ a1 ¼ b2 ¼ p ¼ q ¼ 0; 1 1 ða2 c2 þ b1 Þ; c ¼ ða2 c2 þ 3b21 Þ; a0 ¼ 2a2 2a2
b0 ¼
a2 c1 ; 4b1
1 c4 ¼ a2 ; 2
ð20Þ
with a2 , c, b1 , c0 , c1 , c2 being arbitrary constants; and c4 ¼ a2 ¼ b2 ¼ p ¼ q ¼ 0; 1 c3 ¼ a1 ; a0 ¼ ðc þ c2 Þ; 3
1 b0 ¼ ð4c þ c2 Þ; 6
1 b1 ¼ a1 ; 2
ð21Þ
with a1 , c, c0 , c2 being arbitrary constants. All possible solutions of the complex coupled KdV system are then concluded as follows: (A) By using (8), (10) and (18), we can obtain a solitary wave solution pffiffiffiffi 1 u1 ¼ ðc þ 4c2 Þ 4c2 sech2 ð c2 nÞ; 3 pffiffiffiffi 2 v1 ¼ eir ½c þ c2 3c2 sech2 ð c2 nÞ; 3
ð22Þ c2 > 0
and a Jacobi doubly periodic solution rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4c2 k 2 c2 cn2 n ; u2 ¼ ðc þ 4c2 Þ 2 2k 1 2k 2 1 3 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3c2 k 2 c2 cn2 n ; v2 ¼ eir c þ c2 2 2k 1 2k 2 1 3
ð23Þ c2 > 0;
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
where n ¼ x þ ct, a2 , c and r are arbitrary constants. As k ! 1, the Jacobi doubly periodic solutions (23) degenerates to the soliton solution (22). From (9), (11), (12) and (18), and by simple transformation c2 ! 2c2 , we find that the obtained solutions are the same as (22) and (23). By taking r ¼ 0 in the above solutions, we obtain the solution to the well-known Hirota–Satsuma coupled system. The solutions (22) are a bell-shaped soliton solutions, especially the modulus of v1 is a soliton solution with two-peak. The solution (23) are doubly periodic solution. These properties are illustrated in Figs. 1 and 2 respectively.
u1
Re (v 1 )
1 0 -1
0.2 0 -0.2 -0.4
4 2 0
-4 -2
4 2 0 t
-4
t
-2
-2
0
-2
0 2
x
-4
2
x
4
-4 4
Im (v 1 )
Abs (v 1)
0.25 0 -0.25 -0.5 -0.75
4 2
0 -0.2 -0.4 -0.6 -0.8
0 t
-4 -2 2
0 t
-4 -2
-2
0 x
4 2
-2
0
-4
2
x
4
-4 4
Fig. 1. The soliton solution (22) with c2 ¼ 0:6, c ¼ 0:5, r ¼ 1.
Re (v 2 )
u2
0.965 0.96 0.955 0.95 0.945
10 0
-20
t
1.044 1.042 1.04
10 0
-20
0 -10
x
t
0 -10
x
20
Im (v 2 )
20
Abs (v 2 )
10
1.625 1.6225 1.62 1.6175
0
-20
t
1.9325 1.93 1.9275 1.925 1.9225
10 0
-20 0
0 x
-10 20
x
-10 20
Fig. 2. The Jacobi doubly periodic solution (23) with c2 ¼ 0:6, c ¼ 0:5, r ¼ 1.
t
Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
521
(B) From (8)–(10) and (19), we have pffiffiffiffi 1 u3 ¼ ð2c2 cÞ 2c2 sech2 ð c2 nÞ; 3 v3 ¼ b0 eih ; c2 > 0
ð24Þ
and 1 c2 k 2 cn2 u4 ¼ ð2c2 cÞ 2 3 2k 1 v4 ¼ b0 eih ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 n ; 2k 2 1
ð25Þ
c2 > 0;
where h ¼ px þ p3 t þ r, a2 , c, p, c2 and r are arbitrary constants. As k ! 1, the Jacobi doubly periodic solutions (25) degenerates to the soliton solutions (24). Taking b0 ¼ 0 and r ¼ 0 in this case lead to the exact solutions of the KdV equation and the Hirota–Satsuma coupled system respectively. The solution u3 is bell-shaped. The real part and imaginary part of v3 and v4 are periodic, but their modulus is plane-wave. The solution u4 is doubly periodic. (C) Since c1 ¼ 0 leads to b0 ¼ 0, expressions (8)–(13) and (20) lead to the following solutions ui ¼ a0 þ a2 u2i ;
vi ¼ ðb0 þ b1 uÞeir ;
i ¼ 5; 6; 7; 8; 9;
where n ¼ x þ ct, a0 ; a2 ; b0 and b1 satisfy (20) and ui is given by sffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 2c2 u5 ¼ sechð c2 nÞ; a2 rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi c2 c2 u6 ¼ tanh n ; a2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 k 2 c2 cn n ; u7 ¼ a2 ð2k 2 1Þ 2k 2 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 c2 u8 ¼ dn n ; a2 ð2 k 2 Þ 2 k2
u5
ð26Þ
ð27Þ
ð28Þ
ð29Þ
Re (v 5 )
1
4
0.5 0
2 0
-4 -2
4 2
2
0
-4
t
-2
-2
0 x
0.4 0.2 0
0 2
x
-4
t
-2 -4 4
4
Abs (v 5 )
Im (v 5 )
0.8 0.6 0.4 0.2 0
4 2 0
-4 -2
-2
0 x
2
-4 4
t
1 0.75 0.5 0.25 0
4 2 0 t
-4 -2
-2
0 x
2
-4 4
Fig. 3. The soliton solution (2) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 k 2 c2 sn 2 n ; u9 ¼ a2 ðk 2 þ 1Þ k þ1
ð30Þ
As k ! 1, the Jacobi doubly periodic solutions (28) and (29) degenerate to the soliton solution (26), and (30) degenerates to (27). Taking b0 ¼ 0 and r ¼ 0 leads to the exact solutions of the KdV equation and the Hirota–Satsuma coupled KdV system respectively. Plots for these solutions are given in Figs. 3–6.
u6
Re (v 6 )
0.5 4
0 -0.5
2 0
-4 -2
0.5 0.25 0 -0.25 -0.5
4 2
0 2
x
0 t
-4
t
-2
-2
-2
0 2
x
-4
-4 4
4
Im (v 6 )
Abs (v 6 )
0.5 0 -0.5
4 2 0
-4 -2
0 -0.25 -0.5 -0.75 -1
t
4 2
0 2
x
0 t
-4 -2
-2
-2
0 x
-4
2
-4
4
4
Fig. 4. The soliton solution (27) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
u7
Re (v 7 )
0.6 0.55 0.5 0.45 0.4
5 0
-10
5
0.2 0 -0.2
t
0
-10
0 -5
x
t
0 -5
x
10
10
Abs (v 7 )
Im (v 7 )
0.5 0.25 0 -0.25 -0.5
5 0
-10
5
0.4 0.2 0 t
0
-10
0 x
0
-5 10
x
-5 10
Fig. 5. The Jacobi periodic solution (28) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
t
Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
u9
523
Re (v 9 )
-0.25 -0.275 -0.3 -0.325 -0.35 -20
10 0
10
0.1 0 -0.1
t
0
-20
0 -10
x
t
0 -10
x
20
Im (v 9 )
20
Abs (v 9 )
10
0.2 0 -0.2 0
-20
10
0.3 0.2 0.1 0
t
0
-20
0
t
0 -10
x
-10
x
20
20
Fig. 6. The Jacobi periodic solution (30) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
(D) From (14) and (21) we obtain a Weierstrass doubly periodic solution as follow: rffiffiffiffiffiffiffiffiffi 1 a1 u10 ¼ c þ a1 } n; g2 ; g3 ; 3 2 rffiffiffiffiffiffiffiffiffi 1 ir a1 n; g2 ; g3 ; c2 ¼ 0; v10 ¼ e 4c þ 3a1 } 6 2
ð31Þ
where g2 ¼ 4c1 =a1 , g3 ¼ 4c0 =a1 , n ¼ x þ ct, a1 , c, c0 , c1 and r are arbitrary constants. Taking r ¼ 0, the above solutions is reducted to the exact solution of the Hirota–Satsuma coupled system. The plot of (31) is given in Fig. 7.
u 10
Re (v 10 )
20
75 50 25 0 20
0
20 100 50 0 -20
10 y
10 0 -10
-10 -10
0 10
x
10
x 20
y
-10
0
-20
20 -20
Im (v 10 )
Abs (v 10 )
20
20
200 150 100 50 0 -20
10 0 -10
10 0 -10
-10
0 x
y
200 100 0 -20
10
x 20 -20
y
-10
0 10
20 -20
Fig. 7. The Weierstrass periodic solution (31) with a1 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
4. Conclusions In this paper we have applied an algebra method to find a series new exact solutions for a complex coupled KdV system. Except to (27), other solutions are new to our knowledge. The proposed method is readily applicable to a large variety of either integrable or nonintegrable nonlinear equations including the classical KdV, MKdV, Jaulent–Miodek, BBM, KP, Kawachra, variant Boussinesq, sine-Gordon, sinh-Gordon, coupled Schr€ odinger–Boussinsq and coupled Ito equations. Furthermore, the proposed method is readily computerizable by using the symbolic software. Using the proposed method, we have shown that the travelling wave solutions of a general nonlinear equation depend on the explicit solvability of a simple system of ordinary differential equations. For illustration purpose, we only considered in details the interesting case when the parameter m is 4. The details for the cases when m > 4 will be investigated in our future works.
Acknowledgements This work has been supported by the City University Strategic Research Grant of project number 7001209, the Chinese Basic Research Plan ‘‘Mathematics Mechanization and a Platform for Automated Reasoning’’ and Shanghai Shuguang Project of China.
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Ablowitz MJ, Segur H. Soliton and the inverse scattering transformation. Philadelphia, PA: SIAM; 1981. Beals R, Coifman RR. Scattering and inverse scattering for 1st order system. Comm pure Appl Math 1984;37:39–90. Matveev VB, Salle MA. Darboux transformation and solitons. Berlin: Springer; 1991. Gu CH, Hu HS, Zhou ZX. Darboux transformations in soliton theory and its geometric applications. Shanghai Sci Tech Publ; 1999. Leble SB, Ustinov NV. Darboux transforms, deep reductions and solitons. J Phys A 1993;26:5007–16. Esteevez PG. Darboux transformation and solutions for an equation in 2 þ 1 dimensions. J Math Phys 1999;40:1406–19. Dubrousky VG, Konopelchenko BG. Delta-dressing and exact solutions for the (2 þ 1)-dimensional Harry Dym equation. J Phys A 1994;27:4719–21. Neugebauer G, Kramerl D. Einstein–Maxwell solitons. J Phys A 1983;16:1927–36. Fan EG. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J Math Phys 2001;42:4327–44. Fan EG. Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation. J Phys A 2000;33:6925–33. Hirota R, Satsuma J. Soliton solution of a coupled KdV equation. Phys Lett A 1981;85:407–8. Satsuma J, Hirota R. A coupled KdV equation is one case of the 4-reduction of the KP hierarchy. J Phys Soc Jpn 1982;51:3390–7. Tam HW, Ma WX, Hu XB, Wang DL. The Hirota–Satsuma coupled KdV equation and a coupled Ito system revisited. J Phys Soc Jpn 2000;69:45–51. Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992;60:650–4. Hereman W. Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA. Comp Phys Comm 1996;65:143–50. Malfliet W, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys Scripta 1996;56:563–8. Parkes EJ. Exact solutions to the two-dimensional Korteweg-de Vries-Burgers equation. J Phys A 1994;27:L497–502. Parkes EJ, Duffy BR. An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Comput Phys Commun 1996;98:288–300. Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8. Fan EG. Travelling wave solutions for two generalized Hirota–Satsuma coupled KdV systems. Z Naturforsch A 2001;56:312–8. Belokolos E, Bobenko A, EnolÕskij V, Its A, Matveev V. Algebro-Geometrical Approach to Nonlinear Integrable Equations. Berlin: Springer; 1994. Alber MS, Fedorov YN. Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Probl 2001;17:1017–42. Cao CW, Geng XG, Wang HY. Algebro-geometric solution of the 2 þ 1 dimensional Burgers equation with a discrete variable. J Math Phys 2002;43:621–43. Previato E. Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrodinger equation. Duke Math J 1985;52:329– 77. Porubov AV, Paeker DF. Some general periodic solutions to coupled nonlinear Schr€ oinger equations. Wave Motion 1999;29:97– 109.
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[26] Liu SK, Fu ZT, Liu SD, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 2001;289:69–74. [27] Liu SK, Fu ZT, Liu SD, Zhao Q. New periodic solutions to a kind of nonlinear wave equations. Acta Phys Sin 2002;51:10–4. [28] Fan E, Hon YC. A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos, Solitons & Fractals 2003;15:559–66. [29] Wu YT, Geng XG, Hu XB, Zhu SM. A generalized Hirota Satsuma coupled Kortewege de Vries equation and Miura transformations. Phys Lett A 1999;255:259–64. [30] Fan EG, Chao L. Soliton solutions for the new complex version of a coupled KdV equation and a coupled MKdV equation. Phys Lett A 2001;285:373–6.
A series of new exact solutions for a complex coupled KdV system Y.C. Hon a, E.G. Fan a
b,*
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue 83, Hong Kong SAR, PeopleÕs Republic of China b Institute of Mathematics and Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, PeopleÕs Republic of China Accepted 7 March 2003
Abstract In this paper an algebraic method is devised to uniformly construct a series of exact solutions for general nonlinear equations. Compared with the existing tanh methods and Jacobi function method, the proposed method gives more general exact solutions without much extra effort. More importantly, the method provides a guideline for the classification of the solutions based on the given parameters. For illustration, we apply the proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction The effort in finding exact solution to nonlinear equation is important for the understanding of most nonlinear physical phenomena. For instances, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modelled by the bell shaped sech solutions and the kink shaped tanh solutions. The availability of these exact solutions for those nonlinear equations can greatly facilitate the verification of numerical solvers on the stability analysis of the solutions. In the past decades, there has been significant progression in the development of these methods such as inverse scattering method [1,2], Darboux transformation [3–10], Hirota bilinear method [11–13], and tanh method [14–16]. Among these, the tanh method is considered to be the most effective and direct algebraic method for solving nonlinear equations. Much research work has been concentrated on the various extensions and applications of the tanh method [16–20]. By applying spectral theory, Weierstrass and Theta elliptic functions can be used to find periodic solutions for some equations such as KdV equation, coupled nonlinear Schr€ odinger equation et al. But this method usually is applied in the integrable nonlinear evolution equations admitting Lax pairs representation [21–23]. An alternative method is to transform the equation under study to the Weierstrass equation, Jacobi equation, or more generally, to Painleve-type equations [24,25]. This procedure is in general complicated or impossible, especially for complicated dissipative nonlinear equations and nonlinear coupled systems. Very recently, a Jacobi function expansion method was applied to construct periodic wave solutions for some nonlinear equations. The essential idea of this method is similar to the tanh method by replacing the tanh function with some Jacobi elliptic functions such as cnn, snn and dnn [26,27]. For example, Jacobi periodic solution in terms of snn may be obtained by applying sn-function expansion. To get Jacobi doubly periodic wave solutions in terms of cnn and dnn, many similarly repetitious calculations have to be made, and these efforts will be in vain if a equation does not admit these type solutions at all. Recently, we
*
Corresponding author. E-mail addresses: [email protected] (Y.C. Hon), [email protected] (E.G. Fan).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00099-7
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proposed a new straightforward algebraic method which further exceeds the applicability of the above methods in obtaining a series of exact wave solutions including the soliton, rational, triangular periodic, Jacobi, and Weierstrass doubly periodic solutions [28]. In this paper, we apply the proposed method to solve a complex coupled KdV system constructing a series of new exact solutions including (a) kink-shaped and bell-shaped soliton solutions, and (b) Jacobi and Weierstrass elliptic doubly periodic solutions. In the following Section 2, the detail deviation of the proposed method will be given. The application of the proposed method to a new complex coupled KdV system is illustrated in Section 3. Conclusion is then given in the final Section 4.
2. Methodology Let recall our proposed method whose main steps of the proposed method are outlined as follows [28]: Step 1. For the travelling wave solutions of equation H ðu; ut ; ux ; uxx ; . . .Þ ¼ 0;
ð1Þ
we introduce the wave transformation uðx; tÞ ¼ U ðnÞ, n ¼ x þ ct as follows: o d !c ; ot dn
o d ! ; ox dn
Under the transformation, Eq. (1) becomes an ordinary differential equation (ODE) as: H ðU ; U 0 ; U 00 ; . . .Þ ¼ 0: Step 2. We introduce a new variable u ¼ uðnÞ which is a solution of the following first-order ODE: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u m u 0 ¼ et cj uj ;
ð2Þ
ð3Þ
j¼0
where e ¼ 1. The derivatives with respect to the variable n become the derivatives with respect to the variable u as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # uX m m 2 2 X X u m d d d 1 d d 2 j1 j j ! et ; þ cj u !e jcj u cj u ;... dn du 2 j¼1 du j¼0 du2 dn2 j¼0 Step 3. By virtue of the variable u, we expand the solution of Eqs. (1) or (2) as the following series: uðx; tÞ ¼ U ðnÞ ¼
n X
ai ui :
ð4Þ
i¼0
Balancing the highest derivative term with the nonlinear terms in Eq. (2) will give a relation for the positive integers n and m, from which the different possible values of n and m can be determined. These values lead to the series expansions of the exact solutions for Eq. (1). For example, in the case of KdV equation ut þ 6uux þ uxxx ¼ 0;
ð5Þ
we have m ¼ n þ 2:
ð6Þ
If we take n ¼ 1 and m ¼ 3 in (6), we obtain the following series expansion of an exact solution of the KdV equation (5) as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ a0 þ a1 u; u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 : Similarly, if we take n ¼ 2, m ¼ 4 in (6), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ a0 þ a1 u þ a2 u2 ; u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 þ c4 u4 :
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517
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm j Step 4. Substituting the expansion (4) into Eq. (2) and setting the coefficients of all powers of ui and ui j¼0 cj u to zeroes, we obtain a system of algebraic equations, from which the constants c, ai , cj (i ¼ 0; 1; . . . ; n; j ¼ 0; 1; . . . ; m) can be found explicitly. Step 5. Substituting the constants cj (j ¼ 0; 1; . . . ; m) obtained in Step 4 into Eq. (3), we can then obtain all the possible solutions. We remark here that the travelling wave solutions of Eq. (2) depend on the explicit solvability of Eq. (3). The solution of the system of algebraic equations will be getting tedious with the increase of the values of n and m. When m ¼ 4, Eq. (3) gives a series of interesting solutions such as soliton, rational, triangular periodic, Jacobi and Weierstrass doubly periodic solutions. We then consider only the case m ¼ 4 in this paper and hence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ e c0 þ c1 u þ c2 u2 þ c3 u3 þ c4 u4 : ð7Þ We have the following results: (i) If c3 ¼ c1 ¼ c0 ¼ 0, Eq. (7) possesses a bell-shaped soliton solution rffiffiffiffiffiffiffiffiffi pffiffiffiffi c2 u ¼ sechð c2 nÞ; c2 > 0; c4 < 0; c4 a triangular solution rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi c2 u ¼ secð c2 nÞ; c4
ð8Þ
c2 < 0; c4 > 0
and a rational solution e u ¼ pffiffiffiffi ; c2 ¼ 0; c4 > 0: c4 n i(ii) If c3 ¼ c1 ¼ 0; c0 ¼ c22 =4c4 , Eq. (7) possesses a kink-shaped soliton solution rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi c2 c2 tanh n ; c2 < 0; c4 > 0 u¼e 2c4 2
ð9Þ
and a triangular solution rffiffiffiffi rffiffiffiffi c2 c2 tan u¼e n ; c2 > 0; c4 > 0: c4 2 (iii) If c3 ¼ c1 ¼ 0, Eq. (7) admits two Jacobi elliptic doubly periodic solutions sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 k 2 c2 c2 k 2 ð1 k 2 Þ ; u¼ n ; c4 < 0; c2 > 0; c0 ¼ 2 cn 2 2 c4 ð2k 1Þ 2k 1 c4 ð2k 2 1Þ2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi c2 c2 c2 ð1 k 2 Þ dn u¼ n ; c4 < 0; c2 > 0; c0 ¼ 2 2 2 c4 ð2 k Þ 2k c4 ð2 k 2 Þ2
ð10Þ
ð11Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 k 2 c2 u¼e sn 2 n ; c4 ðk 2 þ 1Þ k þ1
c4 > 0; c2 < 0; c0 ¼
c22 k 2 c4 ðk 2 þ 1Þ2
:
ð12Þ
As k ! 1, the Jacobi doubly periodic solutions (10) and (11) degenerate to the soliton solutions (8) and (9) respectively. (iv) If c4 ¼ c0 ¼ c1 ¼ 0, Eq. (7) possesses a bell-shaped soliton solution pffiffiffiffi c2 c2 2 u ¼ sech ð13Þ n ; c2 > 0; c3 2 a triangular solution pffiffiffiffiffiffiffiffi c2 c2 n ; u ¼ sec2 c3 2
c2 < 0
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and a rational solution u¼
1 ; c3 n2
c2 ¼ 0:
(v) If c4 ¼ c2 ¼ 0, c3 > 0, Eq. (7) admits a Weierstrass elliptic doubly periodic solution pffiffiffiffi c3 u¼} n; g2 ; g3 ; 2
ð14Þ
where g2 ¼ 4c1 =c3 , and g3 ¼ 4c0 =c3 . Remark 1. The other types of travelling wave solutions such as cscn, cotn, cschn, and cothn can also be obtained by considering the different values of cj (j ¼ 0; 1; . . . ; 4) in Eq. (7). These solutions appear in pairs of the functions secn, tann, sechn and tanhn. Especially, when c1 ¼ c3 ¼ 0, c0 ¼ 1, c2 ¼ 2, c4 ¼ 1, the Eq. (7) has a solution tanhn which is given by the existing tanh method [14–18]. When c1 ¼ c3 ¼ 0, c0 ¼ b2 , c2 ¼ 2b, c4 ¼ 1, the Eq. (7) degenerates to a Riccati equation. In this case the proposed method is the same as the extended tanh method [19,20]. The cases (10)–(12) readily cover the results of Jacobi function expansion method [26,27]. In conclusion, our proposed method in this paper is a generalization of both the tanh and Jacobi function expansion method. The proposed method not only give an unified formulation to construct various travelling wave solutions, but also provides a rule to classify the types of solutions according to the given parameters. Furthermore, the proposed method is readily computerizable in solving Eq. (1) by using symbolic software like Mathematica or Maple.
3. Application to a new complex coupled KdV system Recently, Wu et al. [29] considered a 4 4 matrix spectral problem with three potentials and derived a new hierarchy of nonlinear evolution equations, in which a particular case will give the following system of complex coupled KdV equations: 1 ut ¼ uxxx 3uux þ 3ðjvj2 Þx ; 2 vt ¼ vxxx þ 3uvx :
ð15Þ
As v ¼ 0 and ImðvÞ ¼ 0, system (15) is reduced to the well-known KdV equation and the Hirota–Satsuma coupled system respectively [11,12]. Two kinds of soliton solutions were given by Fan and Chao [30] using the tanh method. The proposed method in this paper will further give a series of new explicit exact solutions to the system (15) as follows: In order to obtain the travelling wave solutions of system (15), we make the transformation uðx; tÞ ¼ U ðnÞ;
vðx; tÞ ¼ eih V ðnÞ;
where n ¼ x þ ct, h ¼ px þ qt þ r, and reduce system (15) to the following system of ordinary differential equations: 1 cU 0 U 000 þ 3UU 0 3ðV 2 Þ0 ¼ 0; 2 0 cV þ V 000 3p2 V 0 3UV 0 ¼ 0;
ð16Þ
ðq p3 ÞV þ 3pV 00 ¼ 0: Suppose that U¼
n1 X i¼0
ai ui ;
V ¼
n2 X
bi ui ;
i¼0
and u satisfies (3). Balancing the highest linear term with the nonlinear terms in (16) gives m ¼ n1 þ 2, n2 6 n1 . By choosing m ¼ 4, n1 ¼ n2 ¼ n3 ¼ 2 we have U ¼ a0 þ a1 u þ a2 u2 ;
V ¼ b0 þ b1 u þ b2 u2 :
ð17Þ
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519
Substituting (17) into (16) gives 2eca1 6ea0 a1 12eb0 b1 36e3 a2 c1 e3 a1 c2 ¼ 0; 6ea21 þ 4eca2 þ 12ea0 a2 12eb21 24eb0 b2 8e3 a2 c2 3e3 a1 c3 ¼ 0; 18ea1 a2 36eb1 b2 15e3 a2 c3 6ea1 c4 ¼ 0; 12ea22 24eb22 24e3 a2 c4 ¼ 0; ecb1 3ep2 b1 3ea0 b1 þ 3e3 b2 c1 þ e3 b1 c2 ¼ 0; 3ea1 b1 þ 2ecb2 6ep2 b2 6ea0 b2 þ 8e3 b2 c2 þ 3e3 b1 c3 ¼ 0; 3ea2 b1 6ea1 b2 þ 15e3 b2 c3 þ 6e3 b1 c4 ¼ 0; 6ea2 b2 þ 24e3 b2 c4 ¼ 0; 2p3 b0 þ 2qb0 þ 12e2 pb2 c0 þ 3pe2 b1 c1 ¼ 0; 2p3 b1 þ 2qb1 þ 18e2 pb2 c1 þ 6e2 b1 c2 ¼ 0; 2p3 b2 þ 2qb2 þ 24e2 pb2 c2 þ 9e2 pb1 c3 ¼ 0; 30e2 b2 c3 þ 12e2 pb1 c4 ¼ 0; 36e2 pb2 c4 ¼ 0: Since e ¼ 1 implies that e2 ¼ 1 and e3 ¼ e, we may eliminate the variable e from the above system. With the aid of Mathematica, we obtain a total of four kinds of solutions c3 ¼ c1 ¼ a1 ¼ b1 ¼ p ¼ q ¼ 0; 1 1 2 a0 ¼ ðc þ 4c2 Þ; c4 ¼ a2 ; b0 ¼ ðc þ c2 Þ; 3 4 3
1 b2 ¼ a2 ; 2
ð18Þ
with a2 , c, c0 , c2 being arbitrary constants; c3 ¼ c1 ¼ b1 ¼ b2 ¼ a1 ¼ 0; 1 1 q ¼ p3 ; c4 ¼ a2 ; a0 ¼ ð2c2 cÞ; 2 3
ð19Þ
with a2 , c, p, c0 , c2 being arbitrary constants; c3 ¼ a1 ¼ b2 ¼ p ¼ q ¼ 0; 1 1 ða2 c2 þ b1 Þ; c ¼ ða2 c2 þ 3b21 Þ; a0 ¼ 2a2 2a2
b0 ¼
a2 c1 ; 4b1
1 c4 ¼ a2 ; 2
ð20Þ
with a2 , c, b1 , c0 , c1 , c2 being arbitrary constants; and c4 ¼ a2 ¼ b2 ¼ p ¼ q ¼ 0; 1 c3 ¼ a1 ; a0 ¼ ðc þ c2 Þ; 3
1 b0 ¼ ð4c þ c2 Þ; 6
1 b1 ¼ a1 ; 2
ð21Þ
with a1 , c, c0 , c2 being arbitrary constants. All possible solutions of the complex coupled KdV system are then concluded as follows: (A) By using (8), (10) and (18), we can obtain a solitary wave solution pffiffiffiffi 1 u1 ¼ ðc þ 4c2 Þ 4c2 sech2 ð c2 nÞ; 3 pffiffiffiffi 2 v1 ¼ eir ½c þ c2 3c2 sech2 ð c2 nÞ; 3
ð22Þ c2 > 0
and a Jacobi doubly periodic solution rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4c2 k 2 c2 cn2 n ; u2 ¼ ðc þ 4c2 Þ 2 2k 1 2k 2 1 3 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3c2 k 2 c2 cn2 n ; v2 ¼ eir c þ c2 2 2k 1 2k 2 1 3
ð23Þ c2 > 0;
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
where n ¼ x þ ct, a2 , c and r are arbitrary constants. As k ! 1, the Jacobi doubly periodic solutions (23) degenerates to the soliton solution (22). From (9), (11), (12) and (18), and by simple transformation c2 ! 2c2 , we find that the obtained solutions are the same as (22) and (23). By taking r ¼ 0 in the above solutions, we obtain the solution to the well-known Hirota–Satsuma coupled system. The solutions (22) are a bell-shaped soliton solutions, especially the modulus of v1 is a soliton solution with two-peak. The solution (23) are doubly periodic solution. These properties are illustrated in Figs. 1 and 2 respectively.
u1
Re (v 1 )
1 0 -1
0.2 0 -0.2 -0.4
4 2 0
-4 -2
4 2 0 t
-4
t
-2
-2
0
-2
0 2
x
-4
2
x
4
-4 4
Im (v 1 )
Abs (v 1)
0.25 0 -0.25 -0.5 -0.75
4 2
0 -0.2 -0.4 -0.6 -0.8
0 t
-4 -2 2
0 t
-4 -2
-2
0 x
4 2
-2
0
-4
2
x
4
-4 4
Fig. 1. The soliton solution (22) with c2 ¼ 0:6, c ¼ 0:5, r ¼ 1.
Re (v 2 )
u2
0.965 0.96 0.955 0.95 0.945
10 0
-20
t
1.044 1.042 1.04
10 0
-20
0 -10
x
t
0 -10
x
20
Im (v 2 )
20
Abs (v 2 )
10
1.625 1.6225 1.62 1.6175
0
-20
t
1.9325 1.93 1.9275 1.925 1.9225
10 0
-20 0
0 x
-10 20
x
-10 20
Fig. 2. The Jacobi doubly periodic solution (23) with c2 ¼ 0:6, c ¼ 0:5, r ¼ 1.
t
Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
521
(B) From (8)–(10) and (19), we have pffiffiffiffi 1 u3 ¼ ð2c2 cÞ 2c2 sech2 ð c2 nÞ; 3 v3 ¼ b0 eih ; c2 > 0
ð24Þ
and 1 c2 k 2 cn2 u4 ¼ ð2c2 cÞ 2 3 2k 1 v4 ¼ b0 eih ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 n ; 2k 2 1
ð25Þ
c2 > 0;
where h ¼ px þ p3 t þ r, a2 , c, p, c2 and r are arbitrary constants. As k ! 1, the Jacobi doubly periodic solutions (25) degenerates to the soliton solutions (24). Taking b0 ¼ 0 and r ¼ 0 in this case lead to the exact solutions of the KdV equation and the Hirota–Satsuma coupled system respectively. The solution u3 is bell-shaped. The real part and imaginary part of v3 and v4 are periodic, but their modulus is plane-wave. The solution u4 is doubly periodic. (C) Since c1 ¼ 0 leads to b0 ¼ 0, expressions (8)–(13) and (20) lead to the following solutions ui ¼ a0 þ a2 u2i ;
vi ¼ ðb0 þ b1 uÞeir ;
i ¼ 5; 6; 7; 8; 9;
where n ¼ x þ ct, a0 ; a2 ; b0 and b1 satisfy (20) and ui is given by sffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 2c2 u5 ¼ sechð c2 nÞ; a2 rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi c2 c2 u6 ¼ tanh n ; a2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 k 2 c2 cn n ; u7 ¼ a2 ð2k 2 1Þ 2k 2 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 c2 u8 ¼ dn n ; a2 ð2 k 2 Þ 2 k2
u5
ð26Þ
ð27Þ
ð28Þ
ð29Þ
Re (v 5 )
1
4
0.5 0
2 0
-4 -2
4 2
2
0
-4
t
-2
-2
0 x
0.4 0.2 0
0 2
x
-4
t
-2 -4 4
4
Abs (v 5 )
Im (v 5 )
0.8 0.6 0.4 0.2 0
4 2 0
-4 -2
-2
0 x
2
-4 4
t
1 0.75 0.5 0.25 0
4 2 0 t
-4 -2
-2
0 x
2
-4 4
Fig. 3. The soliton solution (2) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c2 k 2 c2 sn 2 n ; u9 ¼ a2 ðk 2 þ 1Þ k þ1
ð30Þ
As k ! 1, the Jacobi doubly periodic solutions (28) and (29) degenerate to the soliton solution (26), and (30) degenerates to (27). Taking b0 ¼ 0 and r ¼ 0 leads to the exact solutions of the KdV equation and the Hirota–Satsuma coupled KdV system respectively. Plots for these solutions are given in Figs. 3–6.
u6
Re (v 6 )
0.5 4
0 -0.5
2 0
-4 -2
0.5 0.25 0 -0.25 -0.5
4 2
0 2
x
0 t
-4
t
-2
-2
-2
0 2
x
-4
-4 4
4
Im (v 6 )
Abs (v 6 )
0.5 0 -0.5
4 2 0
-4 -2
0 -0.25 -0.5 -0.75 -1
t
4 2
0 2
x
0 t
-4 -2
-2
-2
0 x
-4
2
-4
4
4
Fig. 4. The soliton solution (27) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
u7
Re (v 7 )
0.6 0.55 0.5 0.45 0.4
5 0
-10
5
0.2 0 -0.2
t
0
-10
0 -5
x
t
0 -5
x
10
10
Abs (v 7 )
Im (v 7 )
0.5 0.25 0 -0.25 -0.5
5 0
-10
5
0.4 0.2 0 t
0
-10
0 x
0
-5 10
x
-5 10
Fig. 5. The Jacobi periodic solution (28) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
t
Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
u9
523
Re (v 9 )
-0.25 -0.275 -0.3 -0.325 -0.35 -20
10 0
10
0.1 0 -0.1
t
0
-20
0 -10
x
t
0 -10
x
20
Im (v 9 )
20
Abs (v 9 )
10
0.2 0 -0.2 0
-20
10
0.3 0.2 0.1 0
t
0
-20
0
t
0 -10
x
-10
x
20
20
Fig. 6. The Jacobi periodic solution (30) with c2 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
(D) From (14) and (21) we obtain a Weierstrass doubly periodic solution as follow: rffiffiffiffiffiffiffiffiffi 1 a1 u10 ¼ c þ a1 } n; g2 ; g3 ; 3 2 rffiffiffiffiffiffiffiffiffi 1 ir a1 n; g2 ; g3 ; c2 ¼ 0; v10 ¼ e 4c þ 3a1 } 6 2
ð31Þ
where g2 ¼ 4c1 =a1 , g3 ¼ 4c0 =a1 , n ¼ x þ ct, a1 , c, c0 , c1 and r are arbitrary constants. Taking r ¼ 0, the above solutions is reducted to the exact solution of the Hirota–Satsuma coupled system. The plot of (31) is given in Fig. 7.
u 10
Re (v 10 )
20
75 50 25 0 20
0
20 100 50 0 -20
10 y
10 0 -10
-10 -10
0 10
x
10
x 20
y
-10
0
-20
20 -20
Im (v 10 )
Abs (v 10 )
20
20
200 150 100 50 0 -20
10 0 -10
10 0 -10
-10
0 x
y
200 100 0 -20
10
x 20 -20
y
-10
0 10
20 -20
Fig. 7. The Weierstrass periodic solution (31) with a1 ¼ 0:6, c ¼ 0:5, p ¼ 1:2.
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Y.C. Hon, E.G. Fan / Chaos, Solitons and Fractals 19 (2004) 515–525
4. Conclusions In this paper we have applied an algebra method to find a series new exact solutions for a complex coupled KdV system. Except to (27), other solutions are new to our knowledge. The proposed method is readily applicable to a large variety of either integrable or nonintegrable nonlinear equations including the classical KdV, MKdV, Jaulent–Miodek, BBM, KP, Kawachra, variant Boussinesq, sine-Gordon, sinh-Gordon, coupled Schr€ odinger–Boussinsq and coupled Ito equations. Furthermore, the proposed method is readily computerizable by using the symbolic software. Using the proposed method, we have shown that the travelling wave solutions of a general nonlinear equation depend on the explicit solvability of a simple system of ordinary differential equations. For illustration purpose, we only considered in details the interesting case when the parameter m is 4. The details for the cases when m > 4 will be investigated in our future works.
Acknowledgements This work has been supported by the City University Strategic Research Grant of project number 7001209, the Chinese Basic Research Plan ‘‘Mathematics Mechanization and a Platform for Automated Reasoning’’ and Shanghai Shuguang Project of China.
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