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Conservation laws and rational solutions of a coupled Toda equation and its related equation
Chaos, Solitons and Fractals 21 (2004) 29–37 www.elsevier.com/locate/chaos
Conservation laws and rational solutions of a coupled Toda equation and its related equation Jun-Xiao Zhao State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and System Sciences, Academia Sinica, P.O. Box 2719, Beijing 100080, PR China Accepted 11 August 2003 Communicated by Prof. M. Wadati
Abstract Two results are presented in this paper. Firstly, infinitely many conservation laws of a coupled Toda equation and its related equation are obtained from their corresponding Lax pairs in a systematic way. Secondly, rational solutions of a three-coupled bilinear system related to the coupled Toda equation are deduced. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction As is well known, both rational solutions and conservation laws (CLs) are two important ingredients shared by many integrable systems. The existence of an infinite sequence of rational solutions appears to be equivalent to having the Painleve property shared by integrable equations [1], which is greatly helpful to formulate a criterion for integrability of equations under consideration. Meanwhile, the existence of an infinite number of CLs may be viewed as one of the most important algebraic characters for integrable systems [2]. Recently, significant progress has been made in finding rational solutions and CLs, and various approaches have been developed in both aspects (see [3–19]). In this paper we shall consider the following two new integrable differential-difference systems [20], uyyy ðn þ 1Þ þ uyyy ðnÞ þ 2ðuy ðn þ 1Þ uy ðnÞÞðuyy ðn þ 1Þ uyy ðnÞÞ Z y ðeuðnþ3Þþuðnþ1Þ2uðnþ2Þ euðnþ1Þþuðn1Þ2uðnÞ Þ dy 0 ¼ euðnþ2ÞþuðnÞ2uðnþ1Þ Z y euðnþ1Þþuðn1Þ2uðnÞ ðeuðnþ2ÞþuðnÞ2uðnþ1Þ euðnÞþuðn2Þ2uðn1Þ Þ dy 0 ;
ð1:1Þ
Uxx ðnÞ ¼ V ðn þ 1ÞW ðn þ 1ÞeU ðnþ1Þ þ V ðn 1ÞW ðn 1ÞeU ðn1Þ 2V ðnÞW ðnÞeU ðnÞ þ eU ðnþ2ÞþU ðnþ1Þ eU ðnþ1ÞþU ðnÞ eU ðnÞþUðn1Þ þ eU ðn1ÞþU ðn2Þ ; U ðnþ1Þ
Vx ðnÞ ¼ W ðn þ 1Þe
Uðn1Þ
W ðn 1Þe
ð1:2Þ
;
ð1:3Þ
Wx ðnÞ ¼ V ðn þ 1ÞeU ðnþ1Þ V ðn 1ÞeU ðn1Þ ;
ð1:4Þ
which are obtained from the following three-coupled bilinear system [20] ðDx Dy 2Dz eDn Þf ðnÞ f ðnÞ ¼ 0;
ð1:5Þ
Dy Dz f ðnÞ f ðnÞ ¼ ð2eDn 2Þf ðnÞ f ðnÞ; 1 1 Dx e2Dn D2y e2Dn f ðnÞ f ðnÞ ¼ 0:
ð1:6Þ
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.005
ð1:7Þ
30
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
Here the Hirota’s bilinear differential operator Dmy Dkt and the bilinear difference operator expðdDn Þ are defined by [21–23] k o o 0 aðy; tÞbðy 0 ; t0 Þjy 0 ¼y;t0 ¼t ; ot ot o o 0 aðnÞbðn0 Þjn0 ¼n ¼ aðn þ dÞbðn dÞ: expðdDn ÞaðnÞ bðnÞ exp d on on
Dmy Dkt a b
o o oy oy 0
m
System (1.2)–(1.4) is a coupled Toda equation, and Eq. (1.1) is called a system related to the coupled Toda equation [20]. In this paper, we shall firstly derive an infinite number of CLs for the coupled Toda equation (1.2)–(1.4) and its related system (1.1). One key point is to find suitable dependent variable transformations such that the two systems (1.2)–(1.4) and (1.1) are transformed into two other equivalent systems where the method introduced in [12,13] can be used to obtain their CLs. Secondly, we shall present a formula which allows one to deduce rational solutions of the three-coupled bilinear system (1.5)–(1.7) by using its B€ acklund transformation and nonlinear superposition [16–19]. This paper is organized as follows. In Section 2, the coupled Toda equation (1.2)–(1.4) is considered and the corresponding CLs are obtained. In Section 3, we derive the CLs for the system (1.1) in a similar way. Section 4 is devoted to deducing the rational solutions of the three-coupled bilinear system (1.5)–(1.7). Finally, a conclusion is given in Section 5.
2. Conservation laws of the coupled Toda equation (1.2)–(1.4) In this section, we shall deduce the CLs of equations (1.2)–(1.4). A Lax pair for this equation is [20] Z x wn;x ¼ k2 wnþ2 þ kV ðn þ 1Þwnþ1 þ wn V ðn þ 1ÞW ðn þ 1ÞeU ðnþ1Þ V ðnÞW ðnÞeUðnÞ þ eUðnþ2ÞþUðnþ1Þ
eU ðnÞþUðn1Þ dx0 c2 h ; k2 wnþ2 þ kV ðn þ 1Þwnþ1 þ wn UðnÞþU ðn1Þ
e
0
Z
2
dx c h k
x
ð2:1Þ
V ðn þ 1ÞW ðn þ 1ÞeUðnþ1Þ V ðnÞW ðnÞeUðnÞ þ eUðnþ2ÞþUðnþ1Þ þ k1 W ðnÞeUðnÞ wn1 þ k2 eUðnÞþU ðn1Þ wn2 ¼ 0;
ð2:2Þ
where k, c, h and k are arbitrary constants. For the sake of convenience in the calculations, we introduce the following dependent variable transformations U ðnÞ ¼ uðn þ 1Þ 2uðnÞ þ uðn 1Þ;
ð2:3Þ
V ðnÞ ¼ vðn þ 1Þ vðn 1Þ;
ð2:4Þ
W ðnÞ ¼ wðn þ 1Þ wðn 1Þ:
ð2:5Þ
Then, Eqs. (2.1) and (2.2) become wx ðnÞ ¼ wðn þ 2Þ þ ðvðn þ 2Þ vðnÞÞwðn þ 1Þ þ ðrðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1ÞÞwðnÞ;
ð2:6Þ
wðn þ 2Þ þ ðvðn þ 2Þ vðnÞÞwðn þ 1Þ þ ðrðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1ÞÞwðnÞ l2 wðnÞ þ ðwðn þ 1Þ wðn 1ÞÞwðn 1Þeuðnþ1Þ2uðnÞþuðn1Þ þ wðn 2Þeuðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0
ð2:7Þ
and Eqs. (1.2)–(1.4) are transformed into the following system, vx ðn þ 2Þ vx ðnÞ ¼ ðwðn þ 3Þ wðn þ 1ÞÞeuðnþ3Þ2uðnþ2Þþuðnþ1Þ ðwðn þ 1Þ wðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ;
ð2:8Þ
wx ðn þ 1Þ wx ðnÞ ¼ ðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ðvðn þ 1Þ vðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ;
ð2:9Þ
ux ðn þ 1Þ þ ux ðn 1Þ 2ux ðnÞ ¼ rðn þ 1Þ rðn 1Þ þ 2vðnÞvðn 1Þ 2vðnÞvðn þ 1Þ;
ð2:10Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
31
rx ðnÞ þ rx ðn þ 1Þ 2ðvðnÞvðn þ 1ÞÞx ¼ euðnþ3Þuðnþ2Þuðnþ1ÞþuðnÞ euðnþ1ÞuðnÞuðn1Þþuðn2Þ ðwðn þ 1Þ wðn 1ÞÞðvðn þ 1Þ vðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ þ ðwðn þ 2Þ wðnÞÞðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ;
ð2:11Þ
where we have chosen k ¼ 1, h ¼ c ¼ 0, k ¼ l2 in (2.1) and (2.2). In order to construct the CLs of system (2.8)–(2.11), let us now take Hn ¼ wðnþ1Þ . In this case, Lax pair (2.6) and (2.7) is rewritten as wðnÞ ðln wðnÞÞx ¼ Hnþ1 Hn þ ðvðn þ 2Þ vðnÞÞHn þ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ;
ð2:12Þ
Hnþ2 Hnþ1 Hn Hn1 þ ðvðn þ 3Þ vðn þ 1ÞÞHnþ1 Hn Hn1 þ ðrðn þ 2Þ þ rðn þ 1Þ 2vðn þ 2Þvðn þ 1ÞÞHn Hn1 l2 Hn Hn1 þ ðwðn þ 2Þ wðnÞÞHn1 euðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ¼ 0:
ð2:13Þ
Thus, based on the following relation [12] lnðHn Þ ¼ ðEþ IÞ lnðwðnÞÞ (where Eþ and I are a shift operator and an identity operator respectively defined by Eþ fn ¼ fnþ1 , Ifn ¼ fn .), Eq. (2.12) becomes ðln Hn Þx ¼ ðEþ IÞ½Hnþ1 Hn þ ðvðn þ 2Þ vðnÞÞHn þ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ ;
ð2:14Þ
which embodies the CLs of equations (2.8)–(2.11) or (1.2)–(1.4), and ln Hn is a conserved density. Similar to the procedure in [12], we now expand Hn into an infinite series of l: Hn ¼
1 X
HnðjÞ lj :
ð2:15Þ
j¼1
Then, by some detailed calculations, we may obtain the following recursion formulae from the Ricatti equation (2.13), ! A 1 ðmþ1Þ ¼ ðI þ E Þ ; ð2:16Þ Rn ð1Þ Hð1Þ n Hn1 Rðmþ1Þ ¼ n
Hðmþ1Þ n ; Hð1Þ n
ð2:17Þ
m ¼ 0; 1; 2; . . . A¼
m3 mi2 X mij1 X X i¼1
j¼1
ðiÞ
ðjÞ
ðmijlÞ
Hnþ2 Hnþ1 HðlÞ n hn1
m1 X i¼1
l¼1
þ ðvðn þ 3Þ vðn þ 1ÞÞ
ðmiþ1Þ
Hðiþ1Þ Hn1 n
m2 mi1 X X i¼1
ðiÞ
ðmijÞ
Hnþ1 HðjÞ n Hn1
þ ðrðn þ 2Þ þ rðn þ 1Þ 2vðn þ 1Þvðn þ 2ÞÞ
j¼1
m1 X
ðmiÞ
HðiÞ n Hn1
i¼1 ðmÞ
þ ð1 dm;0 Þðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ Hn1 þ 2dm;0 euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ;
ð2:18Þ
where the shift operator E is defined by E fn ¼ fn1 , and 1 for m ¼ 0; dm;0 ¼ 0 for m 6¼ 0: The explicit forms of fHðjÞ n g are uðnþ2Þ2uðnþ1ÞþuðnÞ ; Hð1Þ n ¼ e uðnþ2Þ2uðnþ1ÞþuðnÞ Hð2Þ ; n ¼ ðwðn þ 2Þ wðn þ 1ÞÞe 2 1 uðnþ2Þ2uðnþ1ÞþuðnÞ Hð3Þ n ¼ ½rðn þ 2Þ þ wðn þ 1Þ 2ðI þ E Þ ðvðn þ 2Þvðn þ 1Þ þ wðnÞwðn þ 1ÞÞ e
It is remarked that operator ðI þ E Þ1 denotes the inverse operator of I þ E . Finally, substituting (2.15) into (2.14) and comparing the same power of l1 yield an infinite number of CLs of (2.8)–(2.11) or (1.2)–(1.4). Some of the conserved densities are
32
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
qð1Þ n ¼ uðn þ 2Þ 2uðn þ 1Þ þ uðnÞ; qð2Þ n ¼ wðn þ 2Þ wðn þ 1Þ; 1 qð3Þ n ¼ ðI þ E Þ ð2vðn þ 2Þvðn þ 1Þ 2wðnÞwðn þ 1ÞÞ 1 1 þ rðn þ 2Þ þ w2 ðn þ 1Þ w2 ðn þ 2Þ þ wðn þ 1Þwðn þ 2Þ; 2 2
and the corresponding associated fluxes are Fnð1Þ ¼ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ; Fnð2Þ ¼ ðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ; Fnð3Þ ¼ ðvðn þ 2Þ vðnÞÞðwðn þ 2Þ wðn þ 1ÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ3Þuðnþ2Þuðnþ1ÞþuðnÞ ; It is worthy pointing out that qð3Þ n is the first nontrivial conserved density.
3. The CLs of equation (1.1) In the following, we shall show how to derive the CLs of (1.1) in a similar way as in Section 2. From [20], we know that the following equations wn;y þ ðuy ðnÞ uy ðn þ 1ÞÞwn kwnþ1 þ cwn ¼ 0;
ð3:1Þ
k2 wnþ2 þ kðuy ðn þ 2Þ uy ðnÞÞwnþ1 þ ½uyy ðn þ 1Þ þ uyy ðnÞ þ ðuy ðnÞ uy ðn þ 1ÞÞ2 h k c2 wn Z y k1 wn1 euðnþ1Þþuðn1Þ2uðnÞ ðeuðnÞþuðn2Þ2uðn1Þ euðnþ2ÞþuðnÞ2uðnþ1Þ Þ dy 0 þ k2 wn2 euðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0 ð3:2Þ constitute a Lax pair of (1.1). By the dependent variable transformation Z y ðeuðnþ1Þ2uðnÞþuðn1Þ Þ dy 0 þ a wðnÞ ¼ (where a is a arbitrary constant), Lax pair (3.1) and (3.2) become wn;y þ ðuy ðnÞ uy ðn þ 1ÞÞwn wnþ1 ¼ 0;
ð3:3Þ
wnþ2 þ ðuy ðn þ 2Þ uy ðnÞÞwnþ1 þ ½uyy ðn þ 1Þ þ uyy ðnÞ þ ðuy ðnÞ uy ðn þ 1ÞÞ2 l2 wn þ ðwðn þ 1Þ wðn 1ÞÞwn1 euðnþ1Þ2uðnÞþuðn1Þ þ wn2 euðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0
ð3:4Þ
and Eq. (1.1) is transformed into the following system uyyy ðn þ 1Þ þ uyyy ðnÞ þ 2ðuy ðn þ 1Þ uy ðnÞÞðuyy ðn þ 1Þ uyy ðnÞÞ ¼ ðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ðwðn þ 1Þ wðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ; wy ðnÞ ¼ euðnþ1Þ2uðnÞþuðn1Þ ;
ð3:5Þ ð3:6Þ
where the parameters in (3.1) and (3.2) have been taken special values: k ¼ 1;
h ¼ c ¼ 0;
k ¼ l2 :
Similar to Section 2, Lax pair (3.3) and (3.4) can be rewritten as ðln wn Þy ¼ Hn þ uy ðn þ 1Þ uy ðnÞ;
ð3:7Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
33
Hnþ2 Hnþ1 Hn Hn1 þ ðuy ðn þ 3Þ uy ðn þ 1ÞÞHnþ1 Hn Hn1 þ ðuyy ðn þ 2Þ þ uyy ðn þ 1Þ þ ðuy ðn þ 2Þ uy ðn þ 1ÞÞ2 ÞHn Hn1 l2 Hn Hn1 þ ðwðn þ 2Þ wðnÞÞHn1 euðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ¼ 0:
ð3:8Þ
By taking Hn ¼
wnþ1 wn
the CLs of system (3.5) and (3.6) appear to be ðln Hn Þy ¼ ðEþ IÞðHn þ uy ðn þ 1Þ uy ðnÞÞ
ð3:9Þ
and the conserved densities are given by ln Hn . In order to determine the Hn from the Ricatti equation (3.8), we take Hn ¼
1 X
HnðjÞ lj :
ð3:10Þ
j¼1
By some calculations, we may derive a recursion formula of fHðjÞ n g, ! A Rðmþ1Þ ; ¼ ðI þ E Þ1 n ð1Þ Hð1Þ n Hn1 Rðmþ1Þ ¼ n
ð3:11Þ
Hðmþ1Þ n ; Hð1Þ n
ð3:12Þ
m ¼ 0; 1; 2; . . . ; A¼
m3 mi2 X mij1 X X i¼1
j¼1
m1 X
ðiÞ
ðjÞ
ðmijlÞ
Hnþ2 Hnþ1 HðlÞ n Hn1
þ ðuy ðn þ 3Þ uy ðn þ 1ÞÞ
l¼1 ðmiþ1Þ
Hðiþ1Þ Hn1 n
m2 mi1 X X i¼1
ðmijÞ
j¼1
þ ½uyy ðn þ 2Þ þ uyy ðn þ 1Þ þ ðuy ðn þ 1Þ uy ðn þ 2ÞÞ2
i¼1
ðiÞ
Hnþ1 HðjÞ n Hn1
m1 X
ðmiÞ
HðiÞ n Hn1
i¼1 ðmÞ
þ ð1 dm;0 Þðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ Hn1 þ 2dm;0 euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ :
ð3:13Þ
Therefore, we can deduce infinitely many CLs of the nonlinear system (3.5) and (3.6) or (1.1) by substituting (3.10) into (3.9). Some of the conserved densities are qð1Þ n ¼ uðn þ 2Þ 2uðn þ 1Þ þ uðnÞ; qð2Þ n ¼ wðn þ 2Þ wðn þ 1Þ; 1 qð3Þ n ¼ ðI þ E Þ ð2uy ðn þ 1Þuy ðn þ 2Þ 2wðnÞwðn þ 1ÞÞ 1 1 þ uyy ðn þ 2Þ þ u2y ðn þ 2Þ þ w2 ðn þ 1Þ w2 ðn þ 2Þ þ wðn þ 1Þwðn þ 2Þ 2 2
and the corresponding associated fluxes are Fnð1Þ ¼ uy ðn þ 1Þ uy ðnÞ; Fnð2Þ ¼ euðnþ2Þ2uðnþ1ÞþuðnÞ ; Fnð3Þ ¼ ðwðn þ 2Þ wðn þ 1ÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ : ð2Þ ð3Þ We note that qð1Þ n and qn are trivial conserved densities. So qn is the first one nontrivial.
4. Rational solutions of system (1.5)–(1.7) This section is devoted to deriving rational solutions of the three-coupled bilinear system (1.5)–(1.7). We know that a B€ acklund transformation (BT) for this system is given by [20]
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J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
ðDz þ k1 eDn þ lÞf ðnÞ gðnÞ ¼ 0; 1 1 1 Dy e2Dn ke2Dn þ ce2Dn f ðnÞ gðnÞ ¼ 0;
1 l Dz eDn þ eDn k k
Dx þ k f ðnÞ gðnÞ ¼ 0;
ðDx D2y 2cDy þ hÞf ðnÞ gðnÞ ¼ 0;
ð4:1Þ ð4:2Þ ð4:3Þ ð4:4Þ
where k, l, c, k and h are arbitrary constants. In order to derive rational solutions of system (1.5)–(1.7), it is enough to consider a BT with special parameters [19]. We may take k ¼ c ¼ k ¼ 1; l ¼ 1; h ¼ 0: In this case, bilinear equations (4.1)–(4.4) become ðDz þ eDn 1Þf ðnÞ gðnÞ ¼ 0; 1 1 1 Dy e2Dn e2Dn þ e2Dn f ðnÞ gðnÞ ¼ 0;
ð4:5Þ ð4:6Þ
ðDz eDn eDn Dx þ 1Þf ðnÞ gðnÞ ¼ 0;
ð4:7Þ
ðDx D2y 2Dy Þf ðnÞ gðnÞ ¼ 0:
ð4:8Þ
We shall represent the transformations (4.5)–(4.8) symbolically by f ðnÞ ! gðnÞ. Henceforth, we denote that f ðn; x; y; zÞ f ðnÞ f . Now, following the steps described in [19] we begin to construct the rational solutions of (1.5)–(1.7). Firstly, we choose f0 , f1 and f12 to be three solutions of (1.5)–(1.7) and f0 ! f1 ! f12 , with f0 , f1 , f12 6¼ 0. Suppose that f~2 is given by 1 1 Dn f1 f~2 ; c is a nonzero constant: e2Dn f0 f12 ¼ c sinh ð4:9Þ 2 From these assumptions and by means of (A.1)–(A.6) and (4.9), we have that 1 Dn ½cDz f1 f~2 2eDn f0 f12 f12 sinh 2 Dn 1 1 Dn f1 f~2 e 2 f1 f1 2 sinh Dn ðeDn f0 f12 Þ f12 ¼ cDz sinh 2 2 Dn Dn 2 D2n Dn 2 ¼ Dz e f0 f12 e f1 f1 þ e ½ðf0 f1 Þ ðe f1 f12 Þ ðeDn f0 f1 Þ ðf1 f12 Þ Dn
¼ e 2 f½Dz f0 f1 eDn f0 f1 f0 f1 ½f1 f12 ½f0 f1 ½Dz f1 f12 eDn f1 f12 f1 f12 g ¼ 0: Similarly, we can show that the following relations hold, 1 sinh Dn ½cDy f1 f~2 2f0 f12 f12 ¼ 0; 2 1 Dn ½cDx f1 f~2 þ ð2Dz 4ÞeDn f0 f12 f12 ¼ 0; sinh 2
ð4:10Þ
ð4:11Þ ð4:12Þ
which imply that cDz f1 f~2 2eDn f0 f12 ¼ k1 ðx; y; zÞf12 ;
ð4:13Þ
cDy f1 f~2 2f0 f12 ¼ k2 ðx; y; zÞf12 ;
ð4:14Þ
cDx f1 f~2 þ ð2Dz 4ÞeDn f0 f12 ¼ k3 ðx; y; zÞf12 ;
ð4:15Þ
where k1 ðx; y; zÞ, k2 ðx; y; zÞ and k3 ðx; y; zÞ are suitable functions of x, y, z. Furthermore, we assume that f~2 determined by (4.9) can be chosen such that ki ðx; y; zÞ ¼ 0 (i ¼ 1; 2; 3). In this case, we denote f2 ¼ f~2 , so (4.13)–(4.15) obviously become
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
35
cDz f1 f2 2eDn f0 f12 ¼ 0;
ð4:16Þ
cDy f1 f2 2f0 f12 ¼ 0;
ð4:17Þ
cDx f1 f2 þ ð2Dz 4ÞeDn f0 f12 ¼ 0:
ð4:18Þ
Next, by using (4.17), we have Dy ½cDx f1 f2 ð2Dy þ 4Þf0 f12 f12 ¼ 0
ð4:19Þ
from which it follows that cDx f1 f2 ð2Dy þ 4Þf0 f12 ¼ k4 ðn; x; zÞf12 ;
ð4:20Þ
where k4 ðn; x; zÞ is a suitable function of n, x, z. We now assume that k4 ðn; x; zÞ ¼ 0. Therefore we have cDx f1 f2 ð2Dy þ 4Þf0 f12 ¼ 0:
ð4:21Þ
Finally, by means of (4.9), (4.16)–(4.18), (4.21) and (A.7)–(A.10) in Appendix A, we obtain the following relations, ðDz þ eDn 1Þf0 f2 ¼ 0; 1 1 1 Dy e2Dn e2Dn þ e2Dn f0 f2 ¼ 0; ðDz eDn Dx eDn þ 1Þf0 f2 ¼ 0; ðDx D2y 2Dy Þf0 f2 ¼ 0: Using BT (4.5)–(4.8), we know that f2 is a new solution of (1.5)–(1.7) and satisfies f0 ! f2 . Similarly, we can show that f2 ! f12 . In summary, a series of rational solutions of (1.5)–(1.7) can be deduced via the following steps. Firstly, choose a given solution f1 of (1.5)–(1.7). Then from the BT we can obtain solutions f0 and f12 such that f0 ! f1 ! f12 . Thirdly, find a particular solution f~2 of (4.9), therefore f2 ¼ f~2 þ kðx; y; zÞf1 is a general solution of (4.9), where kðx; y; zÞ is a arbitrary function of x, y and z. Finally, if kðx; y; zÞ can be determined such that (4.16)–(4.18) and (4.21) hold, the corresponding f2 is a new solution of (1.5)–(1.7), and f2 satisfies f0 ! f2 ! f12 . As an application of this result, we shall derive some polynomial solutions of (1.5)–(1.7) in the following example. Example: Choose f0 ¼ n þ 2x þ y þ z þ a;
f1 ¼ 1;
f12 ¼ n þ 2x þ y þ z þ b;
where a, b are arbitrary constants, it is easy to verify that n þ 2x þ y þ z þ a ! 1 ! n þ 2x þ y þ z þ b: Furthermore, if we choose b ¼ a 1, then we can prove that 3 1 1 1 ðn þ 2x þ y þ zÞ2 þ 3a2 3a þ ðn þ 2x þ y þ zÞ 4x y z þ c0 f2 ¼ ðn þ 2x þ y þ zÞ3 þ 3a 2 2 2 2 satisfies (4.9) (with c ¼ 23), (4.16)–(4.18) and (4.21), where c0 is a arbitrary constant. Thus f2 is a new solution of (1.5)– (1.7) and the following relations hold, 3 ðn þ 2x þ y þ zÞ2 n þ 2x þ y þ z þ a!ðn þ 2x þ y þ zÞ3 þ 3a 2 1 1 1 þ 3a2 3a þ ðn þ 2x þ y þ zÞ 4x y z þ c0 2 2 2 !n þ 2x þ y þ z þ a 1; from which we can deduce that 3 1 ðn þ 2x þ y þ zÞ3 þ ðn þ 2x þ y þ zÞ2 þ n 3x þ c1 2 2 3 1 3 !n þ 2x þ y þ z!ðn þ 2x þ y þ zÞ ðn þ 2x þ y þ zÞ2 þ n 3x þ c2 ; 2 2
36
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
where c1 , c2 are arbitrary constants. Thus, we can choose the seed solutions as follows: 3 1 g0 ¼ ðn þ 2x þ y þ zÞ3 þ ðn þ 2x þ y þ zÞ2 þ n 3x þ c1 2 2 g1 ¼ n þ 2x þ y þ z 3 1 g12 ¼ ðn þ 2x þ y þ zÞ3 ðn þ 2x þ y þ zÞ2 þ n 3x þ c2 2 2 and we seek a solution in the form g2 ¼ ðn þ 2x þ y þ zÞ6 þ a1 ðn þ 2x þ y þ zÞ5 þ a2 ðn þ 2x þ y þ zÞ4 þ a3 ðn þ 2x þ y þ zÞ3 þ a4 ðn þ 2x þ y þ zÞ2 þ a5 ðn þ 2x þ y þ zÞ þ a6 such that (4.9), (4.16)–(4.18) and (4.21) hold, where ai ¼ ai ðx; y; zÞ, for i ¼ 1; 2; . . . ; 5; 6. After some detailed calculations, we have 2 c¼ ; 5
c2 ¼ c1 ;
a1 ¼ 0;
5 a2 ¼ ; 4
5 5 a3 ¼ 20x y z þ 5c1 ; 2 2
1 a4 ¼ ; 4
a5 ¼ 17x þ y þ z þ c3 ;
5 5 5 a6 ¼ 80x2 y 2 z2 20xz 20xy yz þ c1 ð40x þ 5y þ 5zÞ 5c21 ; 4 4 2 where c3 , c1 are arbitrary constants. In this way, we may deduce a sequence of polynomial solutions of the bilinear system (1.5)–(1.7). 5. Conclusion In this paper,we have constructed infinitely many CLs of the coupled Toda equation (1.2)–(1.4) and its related Eq. (1.1). In the process, their suitable Lax pairs have played an important role. Besides, based on Hirota’s bilinear operator identities and a B€acklund transformation, rational solutions of three-coupled bilinear system (1.5)–(1.7) have been derived. The key step is the use of the nonlinear superposition formula.
Acknowledgement The author would like to express her thanks to Xing-Biao Hu for his guidance and encouragement. This work was supported by CAS President Grant.
Appendix A. Hirota bilinear operator identities The following bilinear operator identities hold for arbitrary functions a, b, c, and d. h 1 i h 1 i 1 Dt Dz e2Dn aðnÞ aðnÞ e2Dn aðnÞ aðnÞ ¼ sinh Dn ½Dt Dz aðnÞ aðnÞ aðnÞ2 2 h i 1 1 1 sinh Dn ðDt a bÞ a2 ¼ Dt sinh Dn a b e2Dn a a 2 2 1 1 Dn ½eDn a b c2 ¼ e2Dn ½ac ðeDn c bÞ ðeDn a cÞ cb 2 sinh 2 1 1 1 1 1 2 sinh Dn ab c2 ¼ e2Dn a c e2Dn c b e2Dn a c e2Dn c b 2 1 1 Dn ½Dt eDn a b c2 ¼ e2Dn ½ðDt a cÞ ðeDn c bÞ þ ac ðDt eDn c bÞ 2 sinh 2 h 1 i h 1 i 1 Dt e2Dn a b e2Dn c c ¼ e2Dn ½ðDt a cÞ cb ac ðDt c bÞ
ðA:1Þ ðA:2Þ ðA:3Þ ðA:4Þ ðA:5Þ ðA:6Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
37
ðDt a bÞc ðDt a cÞb ¼ aDt b c
ðA:7Þ
ðD2t a
ðA:8Þ
ðD2t a
bÞc cÞb ¼ 2at Dt b c þ aðDt b cÞt h i h i h 1 ih i h 1 i h 1 i 1 1 1 Dt e2Dn a b e2Dn c d e2Dn a b Dt e2Dn c d ¼ Dt e2Dn a d e2Dn c b
½edDn a b ½edDn c d ¼ edDn ad cb
ðA:9Þ ðA:10Þ
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Newell AC. Solitons in mathematics and physics. Philadelphia: SIAM; 1985. Fokas AS. Symmetries and integrability. Stud Appl Math 1987;77:253–99. Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM; 1981. Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. Hirota R. Direct methods in soliton theory. Iwanami; 1992 [in Japanese]. Rogers C, Shadwick WF. In: B€acklund transformations and their applications. Mathematics in science and engineering, vol. 161. New York: Academic; 1982. Bluman GW, Kumei S. In: Symmetries and differential equations. Applied mathematical science, vol. 81. Berlin: Springer; 1989. Olver PJ. In: Applications of Lie groups to differential equations. 2nd ed. Graduate texts in mathematics, vol. 107. Berlin: Springer; 1993. Wadati M. Transformation theories for nonlinear discrete systems. Prog Theo Phys Supp 1976;(59):36–63. Wadati M, Watanabe M. Conservation laws of a Volterra system and nonlinear self-dual network equation. Prog Theo Phys 1977;(57):808–11. Tsuchida T, Ujino H, Wadati M. Integrable semi-discretization of the coupled modified Kdv equations. J Phys A: Math 1998;39:4785–813. Zhang DJ, Chen DY. The conservation laws of some discrete soliton systems. Chaos, Solitons & Fractals 2002;14(4):573–9. Zhu ZN, Wu XN, Xue WM, Zhu ZM. Infinitely many conservation laws for the Blaszak–Marciniak four-field integrable lattice hierarchy. Phys Lett A 2002;296(6):280–8. Toda M, Wadati M. B€acklund transformation for the exponential lattice. J Phys Soc Jpn 1975;39:1196–203. Wadati M, Toda M. Canonical transformation for the exponential lattice. J Phys Soc Jpn 1975;39:1204–11. Hu X-B, Clarkson PA. Rational solutions of a differential-difference Kdv equation, the Toda equation and the discrete KdV equation. J Phys A: Math Gen 1995;28:5009–16. Hu XB, Clarkson PA. Rational solutions of an extended Lotka–Volterra equation. J Nonlinear Math Phys 2002;9(Suppl. 1):75– 86. Hu X-B. Rational solutions of integrable equations via nonlinear superposition formula. J Phys A: Math Gen 1997;30:8225–40. Wu Y-T, Hu X-B. A new integrable differential-difference system and its explicit solutions. J Phys A: Math Gen 1999;32:1515–21. Zhao J-X, Li C-X, Hu X-B. A coupled Toda equation and its related equation via Hirota bilinear approach. J Nonlinear Math Phys 2003;10(Suppl. 2):237–44. Hirota R. Direct methods in soliton theory. In: Bullough RK, Caudrey PJ, editors. Solitons. Berlin: Springer; 1980. Hirota R, Satsuma J. A variety of nonlinear network equations generated from the B€acklund transformation for the Toda lattice. Prog Theor Phys Suppl 1976;59:64–100. Matsuno Y. Bilinear transformation method. New York: Academic; 1984.
Conservation laws and rational solutions of a coupled Toda equation and its related equation Jun-Xiao Zhao State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and System Sciences, Academia Sinica, P.O. Box 2719, Beijing 100080, PR China Accepted 11 August 2003 Communicated by Prof. M. Wadati
Abstract Two results are presented in this paper. Firstly, infinitely many conservation laws of a coupled Toda equation and its related equation are obtained from their corresponding Lax pairs in a systematic way. Secondly, rational solutions of a three-coupled bilinear system related to the coupled Toda equation are deduced. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction As is well known, both rational solutions and conservation laws (CLs) are two important ingredients shared by many integrable systems. The existence of an infinite sequence of rational solutions appears to be equivalent to having the Painleve property shared by integrable equations [1], which is greatly helpful to formulate a criterion for integrability of equations under consideration. Meanwhile, the existence of an infinite number of CLs may be viewed as one of the most important algebraic characters for integrable systems [2]. Recently, significant progress has been made in finding rational solutions and CLs, and various approaches have been developed in both aspects (see [3–19]). In this paper we shall consider the following two new integrable differential-difference systems [20], uyyy ðn þ 1Þ þ uyyy ðnÞ þ 2ðuy ðn þ 1Þ uy ðnÞÞðuyy ðn þ 1Þ uyy ðnÞÞ Z y ðeuðnþ3Þþuðnþ1Þ2uðnþ2Þ euðnþ1Þþuðn1Þ2uðnÞ Þ dy 0 ¼ euðnþ2ÞþuðnÞ2uðnþ1Þ Z y euðnþ1Þþuðn1Þ2uðnÞ ðeuðnþ2ÞþuðnÞ2uðnþ1Þ euðnÞþuðn2Þ2uðn1Þ Þ dy 0 ;
ð1:1Þ
Uxx ðnÞ ¼ V ðn þ 1ÞW ðn þ 1ÞeU ðnþ1Þ þ V ðn 1ÞW ðn 1ÞeU ðn1Þ 2V ðnÞW ðnÞeU ðnÞ þ eU ðnþ2ÞþU ðnþ1Þ eU ðnþ1ÞþU ðnÞ eU ðnÞþUðn1Þ þ eU ðn1ÞþU ðn2Þ ; U ðnþ1Þ
Vx ðnÞ ¼ W ðn þ 1Þe
Uðn1Þ
W ðn 1Þe
ð1:2Þ
;
ð1:3Þ
Wx ðnÞ ¼ V ðn þ 1ÞeU ðnþ1Þ V ðn 1ÞeU ðn1Þ ;
ð1:4Þ
which are obtained from the following three-coupled bilinear system [20] ðDx Dy 2Dz eDn Þf ðnÞ f ðnÞ ¼ 0;
ð1:5Þ
Dy Dz f ðnÞ f ðnÞ ¼ ð2eDn 2Þf ðnÞ f ðnÞ; 1 1 Dx e2Dn D2y e2Dn f ðnÞ f ðnÞ ¼ 0:
ð1:6Þ
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.005
ð1:7Þ
30
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
Here the Hirota’s bilinear differential operator Dmy Dkt and the bilinear difference operator expðdDn Þ are defined by [21–23] k o o 0 aðy; tÞbðy 0 ; t0 Þjy 0 ¼y;t0 ¼t ; ot ot o o 0 aðnÞbðn0 Þjn0 ¼n ¼ aðn þ dÞbðn dÞ: expðdDn ÞaðnÞ bðnÞ exp d on on
Dmy Dkt a b
o o oy oy 0
m
System (1.2)–(1.4) is a coupled Toda equation, and Eq. (1.1) is called a system related to the coupled Toda equation [20]. In this paper, we shall firstly derive an infinite number of CLs for the coupled Toda equation (1.2)–(1.4) and its related system (1.1). One key point is to find suitable dependent variable transformations such that the two systems (1.2)–(1.4) and (1.1) are transformed into two other equivalent systems where the method introduced in [12,13] can be used to obtain their CLs. Secondly, we shall present a formula which allows one to deduce rational solutions of the three-coupled bilinear system (1.5)–(1.7) by using its B€ acklund transformation and nonlinear superposition [16–19]. This paper is organized as follows. In Section 2, the coupled Toda equation (1.2)–(1.4) is considered and the corresponding CLs are obtained. In Section 3, we derive the CLs for the system (1.1) in a similar way. Section 4 is devoted to deducing the rational solutions of the three-coupled bilinear system (1.5)–(1.7). Finally, a conclusion is given in Section 5.
2. Conservation laws of the coupled Toda equation (1.2)–(1.4) In this section, we shall deduce the CLs of equations (1.2)–(1.4). A Lax pair for this equation is [20] Z x wn;x ¼ k2 wnþ2 þ kV ðn þ 1Þwnþ1 þ wn V ðn þ 1ÞW ðn þ 1ÞeU ðnþ1Þ V ðnÞW ðnÞeUðnÞ þ eUðnþ2ÞþUðnþ1Þ
eU ðnÞþUðn1Þ dx0 c2 h ; k2 wnþ2 þ kV ðn þ 1Þwnþ1 þ wn UðnÞþU ðn1Þ
e
0
Z
2
dx c h k
x
ð2:1Þ
V ðn þ 1ÞW ðn þ 1ÞeUðnþ1Þ V ðnÞW ðnÞeUðnÞ þ eUðnþ2ÞþUðnþ1Þ þ k1 W ðnÞeUðnÞ wn1 þ k2 eUðnÞþU ðn1Þ wn2 ¼ 0;
ð2:2Þ
where k, c, h and k are arbitrary constants. For the sake of convenience in the calculations, we introduce the following dependent variable transformations U ðnÞ ¼ uðn þ 1Þ 2uðnÞ þ uðn 1Þ;
ð2:3Þ
V ðnÞ ¼ vðn þ 1Þ vðn 1Þ;
ð2:4Þ
W ðnÞ ¼ wðn þ 1Þ wðn 1Þ:
ð2:5Þ
Then, Eqs. (2.1) and (2.2) become wx ðnÞ ¼ wðn þ 2Þ þ ðvðn þ 2Þ vðnÞÞwðn þ 1Þ þ ðrðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1ÞÞwðnÞ;
ð2:6Þ
wðn þ 2Þ þ ðvðn þ 2Þ vðnÞÞwðn þ 1Þ þ ðrðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1ÞÞwðnÞ l2 wðnÞ þ ðwðn þ 1Þ wðn 1ÞÞwðn 1Þeuðnþ1Þ2uðnÞþuðn1Þ þ wðn 2Þeuðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0
ð2:7Þ
and Eqs. (1.2)–(1.4) are transformed into the following system, vx ðn þ 2Þ vx ðnÞ ¼ ðwðn þ 3Þ wðn þ 1ÞÞeuðnþ3Þ2uðnþ2Þþuðnþ1Þ ðwðn þ 1Þ wðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ;
ð2:8Þ
wx ðn þ 1Þ wx ðnÞ ¼ ðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ðvðn þ 1Þ vðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ;
ð2:9Þ
ux ðn þ 1Þ þ ux ðn 1Þ 2ux ðnÞ ¼ rðn þ 1Þ rðn 1Þ þ 2vðnÞvðn 1Þ 2vðnÞvðn þ 1Þ;
ð2:10Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
31
rx ðnÞ þ rx ðn þ 1Þ 2ðvðnÞvðn þ 1ÞÞx ¼ euðnþ3Þuðnþ2Þuðnþ1ÞþuðnÞ euðnþ1ÞuðnÞuðn1Þþuðn2Þ ðwðn þ 1Þ wðn 1ÞÞðvðn þ 1Þ vðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ þ ðwðn þ 2Þ wðnÞÞðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ;
ð2:11Þ
where we have chosen k ¼ 1, h ¼ c ¼ 0, k ¼ l2 in (2.1) and (2.2). In order to construct the CLs of system (2.8)–(2.11), let us now take Hn ¼ wðnþ1Þ . In this case, Lax pair (2.6) and (2.7) is rewritten as wðnÞ ðln wðnÞÞx ¼ Hnþ1 Hn þ ðvðn þ 2Þ vðnÞÞHn þ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ;
ð2:12Þ
Hnþ2 Hnþ1 Hn Hn1 þ ðvðn þ 3Þ vðn þ 1ÞÞHnþ1 Hn Hn1 þ ðrðn þ 2Þ þ rðn þ 1Þ 2vðn þ 2Þvðn þ 1ÞÞHn Hn1 l2 Hn Hn1 þ ðwðn þ 2Þ wðnÞÞHn1 euðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ¼ 0:
ð2:13Þ
Thus, based on the following relation [12] lnðHn Þ ¼ ðEþ IÞ lnðwðnÞÞ (where Eþ and I are a shift operator and an identity operator respectively defined by Eþ fn ¼ fnþ1 , Ifn ¼ fn .), Eq. (2.12) becomes ðln Hn Þx ¼ ðEþ IÞ½Hnþ1 Hn þ ðvðn þ 2Þ vðnÞÞHn þ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ ;
ð2:14Þ
which embodies the CLs of equations (2.8)–(2.11) or (1.2)–(1.4), and ln Hn is a conserved density. Similar to the procedure in [12], we now expand Hn into an infinite series of l: Hn ¼
1 X
HnðjÞ lj :
ð2:15Þ
j¼1
Then, by some detailed calculations, we may obtain the following recursion formulae from the Ricatti equation (2.13), ! A 1 ðmþ1Þ ¼ ðI þ E Þ ; ð2:16Þ Rn ð1Þ Hð1Þ n Hn1 Rðmþ1Þ ¼ n
Hðmþ1Þ n ; Hð1Þ n
ð2:17Þ
m ¼ 0; 1; 2; . . . A¼
m3 mi2 X mij1 X X i¼1
j¼1
ðiÞ
ðjÞ
ðmijlÞ
Hnþ2 Hnþ1 HðlÞ n hn1
m1 X i¼1
l¼1
þ ðvðn þ 3Þ vðn þ 1ÞÞ
ðmiþ1Þ
Hðiþ1Þ Hn1 n
m2 mi1 X X i¼1
ðiÞ
ðmijÞ
Hnþ1 HðjÞ n Hn1
þ ðrðn þ 2Þ þ rðn þ 1Þ 2vðn þ 1Þvðn þ 2ÞÞ
j¼1
m1 X
ðmiÞ
HðiÞ n Hn1
i¼1 ðmÞ
þ ð1 dm;0 Þðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ Hn1 þ 2dm;0 euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ;
ð2:18Þ
where the shift operator E is defined by E fn ¼ fn1 , and 1 for m ¼ 0; dm;0 ¼ 0 for m 6¼ 0: The explicit forms of fHðjÞ n g are uðnþ2Þ2uðnþ1ÞþuðnÞ ; Hð1Þ n ¼ e uðnþ2Þ2uðnþ1ÞþuðnÞ Hð2Þ ; n ¼ ðwðn þ 2Þ wðn þ 1ÞÞe 2 1 uðnþ2Þ2uðnþ1ÞþuðnÞ Hð3Þ n ¼ ½rðn þ 2Þ þ wðn þ 1Þ 2ðI þ E Þ ðvðn þ 2Þvðn þ 1Þ þ wðnÞwðn þ 1ÞÞ e
It is remarked that operator ðI þ E Þ1 denotes the inverse operator of I þ E . Finally, substituting (2.15) into (2.14) and comparing the same power of l1 yield an infinite number of CLs of (2.8)–(2.11) or (1.2)–(1.4). Some of the conserved densities are
32
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
qð1Þ n ¼ uðn þ 2Þ 2uðn þ 1Þ þ uðnÞ; qð2Þ n ¼ wðn þ 2Þ wðn þ 1Þ; 1 qð3Þ n ¼ ðI þ E Þ ð2vðn þ 2Þvðn þ 1Þ 2wðnÞwðn þ 1ÞÞ 1 1 þ rðn þ 2Þ þ w2 ðn þ 1Þ w2 ðn þ 2Þ þ wðn þ 1Þwðn þ 2Þ; 2 2
and the corresponding associated fluxes are Fnð1Þ ¼ rðn þ 1Þ þ rðnÞ 2vðnÞvðn þ 1Þ; Fnð2Þ ¼ ðvðn þ 2Þ vðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ; Fnð3Þ ¼ ðvðn þ 2Þ vðnÞÞðwðn þ 2Þ wðn þ 1ÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ3Þuðnþ2Þuðnþ1ÞþuðnÞ ; It is worthy pointing out that qð3Þ n is the first nontrivial conserved density.
3. The CLs of equation (1.1) In the following, we shall show how to derive the CLs of (1.1) in a similar way as in Section 2. From [20], we know that the following equations wn;y þ ðuy ðnÞ uy ðn þ 1ÞÞwn kwnþ1 þ cwn ¼ 0;
ð3:1Þ
k2 wnþ2 þ kðuy ðn þ 2Þ uy ðnÞÞwnþ1 þ ½uyy ðn þ 1Þ þ uyy ðnÞ þ ðuy ðnÞ uy ðn þ 1ÞÞ2 h k c2 wn Z y k1 wn1 euðnþ1Þþuðn1Þ2uðnÞ ðeuðnÞþuðn2Þ2uðn1Þ euðnþ2ÞþuðnÞ2uðnþ1Þ Þ dy 0 þ k2 wn2 euðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0 ð3:2Þ constitute a Lax pair of (1.1). By the dependent variable transformation Z y ðeuðnþ1Þ2uðnÞþuðn1Þ Þ dy 0 þ a wðnÞ ¼ (where a is a arbitrary constant), Lax pair (3.1) and (3.2) become wn;y þ ðuy ðnÞ uy ðn þ 1ÞÞwn wnþ1 ¼ 0;
ð3:3Þ
wnþ2 þ ðuy ðn þ 2Þ uy ðnÞÞwnþ1 þ ½uyy ðn þ 1Þ þ uyy ðnÞ þ ðuy ðnÞ uy ðn þ 1ÞÞ2 l2 wn þ ðwðn þ 1Þ wðn 1ÞÞwn1 euðnþ1Þ2uðnÞþuðn1Þ þ wn2 euðnþ1ÞuðnÞuðn1Þþuðn2Þ ¼ 0
ð3:4Þ
and Eq. (1.1) is transformed into the following system uyyy ðn þ 1Þ þ uyyy ðnÞ þ 2ðuy ðn þ 1Þ uy ðnÞÞðuyy ðn þ 1Þ uyy ðnÞÞ ¼ ðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ ðwðn þ 1Þ wðn 1ÞÞeuðnþ1Þ2uðnÞþuðn1Þ ; wy ðnÞ ¼ euðnþ1Þ2uðnÞþuðn1Þ ;
ð3:5Þ ð3:6Þ
where the parameters in (3.1) and (3.2) have been taken special values: k ¼ 1;
h ¼ c ¼ 0;
k ¼ l2 :
Similar to Section 2, Lax pair (3.3) and (3.4) can be rewritten as ðln wn Þy ¼ Hn þ uy ðn þ 1Þ uy ðnÞ;
ð3:7Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
33
Hnþ2 Hnþ1 Hn Hn1 þ ðuy ðn þ 3Þ uy ðn þ 1ÞÞHnþ1 Hn Hn1 þ ðuyy ðn þ 2Þ þ uyy ðn þ 1Þ þ ðuy ðn þ 2Þ uy ðn þ 1ÞÞ2 ÞHn Hn1 l2 Hn Hn1 þ ðwðn þ 2Þ wðnÞÞHn1 euðnþ2Þ2uðnþ1ÞþuðnÞ þ euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ ¼ 0:
ð3:8Þ
By taking Hn ¼
wnþ1 wn
the CLs of system (3.5) and (3.6) appear to be ðln Hn Þy ¼ ðEþ IÞðHn þ uy ðn þ 1Þ uy ðnÞÞ
ð3:9Þ
and the conserved densities are given by ln Hn . In order to determine the Hn from the Ricatti equation (3.8), we take Hn ¼
1 X
HnðjÞ lj :
ð3:10Þ
j¼1
By some calculations, we may derive a recursion formula of fHðjÞ n g, ! A Rðmþ1Þ ; ¼ ðI þ E Þ1 n ð1Þ Hð1Þ n Hn1 Rðmþ1Þ ¼ n
ð3:11Þ
Hðmþ1Þ n ; Hð1Þ n
ð3:12Þ
m ¼ 0; 1; 2; . . . ; A¼
m3 mi2 X mij1 X X i¼1
j¼1
m1 X
ðiÞ
ðjÞ
ðmijlÞ
Hnþ2 Hnþ1 HðlÞ n Hn1
þ ðuy ðn þ 3Þ uy ðn þ 1ÞÞ
l¼1 ðmiþ1Þ
Hðiþ1Þ Hn1 n
m2 mi1 X X i¼1
ðmijÞ
j¼1
þ ½uyy ðn þ 2Þ þ uyy ðn þ 1Þ þ ðuy ðn þ 1Þ uy ðn þ 2ÞÞ2
i¼1
ðiÞ
Hnþ1 HðjÞ n Hn1
m1 X
ðmiÞ
HðiÞ n Hn1
i¼1 ðmÞ
þ ð1 dm;0 Þðwðn þ 2Þ wðnÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ Hn1 þ 2dm;0 euðnþ2Þuðnþ1ÞuðnÞþuðn1Þ :
ð3:13Þ
Therefore, we can deduce infinitely many CLs of the nonlinear system (3.5) and (3.6) or (1.1) by substituting (3.10) into (3.9). Some of the conserved densities are qð1Þ n ¼ uðn þ 2Þ 2uðn þ 1Þ þ uðnÞ; qð2Þ n ¼ wðn þ 2Þ wðn þ 1Þ; 1 qð3Þ n ¼ ðI þ E Þ ð2uy ðn þ 1Þuy ðn þ 2Þ 2wðnÞwðn þ 1ÞÞ 1 1 þ uyy ðn þ 2Þ þ u2y ðn þ 2Þ þ w2 ðn þ 1Þ w2 ðn þ 2Þ þ wðn þ 1Þwðn þ 2Þ 2 2
and the corresponding associated fluxes are Fnð1Þ ¼ uy ðn þ 1Þ uy ðnÞ; Fnð2Þ ¼ euðnþ2Þ2uðnþ1ÞþuðnÞ ; Fnð3Þ ¼ ðwðn þ 2Þ wðn þ 1ÞÞeuðnþ2Þ2uðnþ1ÞþuðnÞ : ð2Þ ð3Þ We note that qð1Þ n and qn are trivial conserved densities. So qn is the first one nontrivial.
4. Rational solutions of system (1.5)–(1.7) This section is devoted to deriving rational solutions of the three-coupled bilinear system (1.5)–(1.7). We know that a B€ acklund transformation (BT) for this system is given by [20]
34
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
ðDz þ k1 eDn þ lÞf ðnÞ gðnÞ ¼ 0; 1 1 1 Dy e2Dn ke2Dn þ ce2Dn f ðnÞ gðnÞ ¼ 0;
1 l Dz eDn þ eDn k k
Dx þ k f ðnÞ gðnÞ ¼ 0;
ðDx D2y 2cDy þ hÞf ðnÞ gðnÞ ¼ 0;
ð4:1Þ ð4:2Þ ð4:3Þ ð4:4Þ
where k, l, c, k and h are arbitrary constants. In order to derive rational solutions of system (1.5)–(1.7), it is enough to consider a BT with special parameters [19]. We may take k ¼ c ¼ k ¼ 1; l ¼ 1; h ¼ 0: In this case, bilinear equations (4.1)–(4.4) become ðDz þ eDn 1Þf ðnÞ gðnÞ ¼ 0; 1 1 1 Dy e2Dn e2Dn þ e2Dn f ðnÞ gðnÞ ¼ 0;
ð4:5Þ ð4:6Þ
ðDz eDn eDn Dx þ 1Þf ðnÞ gðnÞ ¼ 0;
ð4:7Þ
ðDx D2y 2Dy Þf ðnÞ gðnÞ ¼ 0:
ð4:8Þ
We shall represent the transformations (4.5)–(4.8) symbolically by f ðnÞ ! gðnÞ. Henceforth, we denote that f ðn; x; y; zÞ f ðnÞ f . Now, following the steps described in [19] we begin to construct the rational solutions of (1.5)–(1.7). Firstly, we choose f0 , f1 and f12 to be three solutions of (1.5)–(1.7) and f0 ! f1 ! f12 , with f0 , f1 , f12 6¼ 0. Suppose that f~2 is given by 1 1 Dn f1 f~2 ; c is a nonzero constant: e2Dn f0 f12 ¼ c sinh ð4:9Þ 2 From these assumptions and by means of (A.1)–(A.6) and (4.9), we have that 1 Dn ½cDz f1 f~2 2eDn f0 f12 f12 sinh 2 Dn 1 1 Dn f1 f~2 e 2 f1 f1 2 sinh Dn ðeDn f0 f12 Þ f12 ¼ cDz sinh 2 2 Dn Dn 2 D2n Dn 2 ¼ Dz e f0 f12 e f1 f1 þ e ½ðf0 f1 Þ ðe f1 f12 Þ ðeDn f0 f1 Þ ðf1 f12 Þ Dn
¼ e 2 f½Dz f0 f1 eDn f0 f1 f0 f1 ½f1 f12 ½f0 f1 ½Dz f1 f12 eDn f1 f12 f1 f12 g ¼ 0: Similarly, we can show that the following relations hold, 1 sinh Dn ½cDy f1 f~2 2f0 f12 f12 ¼ 0; 2 1 Dn ½cDx f1 f~2 þ ð2Dz 4ÞeDn f0 f12 f12 ¼ 0; sinh 2
ð4:10Þ
ð4:11Þ ð4:12Þ
which imply that cDz f1 f~2 2eDn f0 f12 ¼ k1 ðx; y; zÞf12 ;
ð4:13Þ
cDy f1 f~2 2f0 f12 ¼ k2 ðx; y; zÞf12 ;
ð4:14Þ
cDx f1 f~2 þ ð2Dz 4ÞeDn f0 f12 ¼ k3 ðx; y; zÞf12 ;
ð4:15Þ
where k1 ðx; y; zÞ, k2 ðx; y; zÞ and k3 ðx; y; zÞ are suitable functions of x, y, z. Furthermore, we assume that f~2 determined by (4.9) can be chosen such that ki ðx; y; zÞ ¼ 0 (i ¼ 1; 2; 3). In this case, we denote f2 ¼ f~2 , so (4.13)–(4.15) obviously become
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
35
cDz f1 f2 2eDn f0 f12 ¼ 0;
ð4:16Þ
cDy f1 f2 2f0 f12 ¼ 0;
ð4:17Þ
cDx f1 f2 þ ð2Dz 4ÞeDn f0 f12 ¼ 0:
ð4:18Þ
Next, by using (4.17), we have Dy ½cDx f1 f2 ð2Dy þ 4Þf0 f12 f12 ¼ 0
ð4:19Þ
from which it follows that cDx f1 f2 ð2Dy þ 4Þf0 f12 ¼ k4 ðn; x; zÞf12 ;
ð4:20Þ
where k4 ðn; x; zÞ is a suitable function of n, x, z. We now assume that k4 ðn; x; zÞ ¼ 0. Therefore we have cDx f1 f2 ð2Dy þ 4Þf0 f12 ¼ 0:
ð4:21Þ
Finally, by means of (4.9), (4.16)–(4.18), (4.21) and (A.7)–(A.10) in Appendix A, we obtain the following relations, ðDz þ eDn 1Þf0 f2 ¼ 0; 1 1 1 Dy e2Dn e2Dn þ e2Dn f0 f2 ¼ 0; ðDz eDn Dx eDn þ 1Þf0 f2 ¼ 0; ðDx D2y 2Dy Þf0 f2 ¼ 0: Using BT (4.5)–(4.8), we know that f2 is a new solution of (1.5)–(1.7) and satisfies f0 ! f2 . Similarly, we can show that f2 ! f12 . In summary, a series of rational solutions of (1.5)–(1.7) can be deduced via the following steps. Firstly, choose a given solution f1 of (1.5)–(1.7). Then from the BT we can obtain solutions f0 and f12 such that f0 ! f1 ! f12 . Thirdly, find a particular solution f~2 of (4.9), therefore f2 ¼ f~2 þ kðx; y; zÞf1 is a general solution of (4.9), where kðx; y; zÞ is a arbitrary function of x, y and z. Finally, if kðx; y; zÞ can be determined such that (4.16)–(4.18) and (4.21) hold, the corresponding f2 is a new solution of (1.5)–(1.7), and f2 satisfies f0 ! f2 ! f12 . As an application of this result, we shall derive some polynomial solutions of (1.5)–(1.7) in the following example. Example: Choose f0 ¼ n þ 2x þ y þ z þ a;
f1 ¼ 1;
f12 ¼ n þ 2x þ y þ z þ b;
where a, b are arbitrary constants, it is easy to verify that n þ 2x þ y þ z þ a ! 1 ! n þ 2x þ y þ z þ b: Furthermore, if we choose b ¼ a 1, then we can prove that 3 1 1 1 ðn þ 2x þ y þ zÞ2 þ 3a2 3a þ ðn þ 2x þ y þ zÞ 4x y z þ c0 f2 ¼ ðn þ 2x þ y þ zÞ3 þ 3a 2 2 2 2 satisfies (4.9) (with c ¼ 23), (4.16)–(4.18) and (4.21), where c0 is a arbitrary constant. Thus f2 is a new solution of (1.5)– (1.7) and the following relations hold, 3 ðn þ 2x þ y þ zÞ2 n þ 2x þ y þ z þ a!ðn þ 2x þ y þ zÞ3 þ 3a 2 1 1 1 þ 3a2 3a þ ðn þ 2x þ y þ zÞ 4x y z þ c0 2 2 2 !n þ 2x þ y þ z þ a 1; from which we can deduce that 3 1 ðn þ 2x þ y þ zÞ3 þ ðn þ 2x þ y þ zÞ2 þ n 3x þ c1 2 2 3 1 3 !n þ 2x þ y þ z!ðn þ 2x þ y þ zÞ ðn þ 2x þ y þ zÞ2 þ n 3x þ c2 ; 2 2
36
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
where c1 , c2 are arbitrary constants. Thus, we can choose the seed solutions as follows: 3 1 g0 ¼ ðn þ 2x þ y þ zÞ3 þ ðn þ 2x þ y þ zÞ2 þ n 3x þ c1 2 2 g1 ¼ n þ 2x þ y þ z 3 1 g12 ¼ ðn þ 2x þ y þ zÞ3 ðn þ 2x þ y þ zÞ2 þ n 3x þ c2 2 2 and we seek a solution in the form g2 ¼ ðn þ 2x þ y þ zÞ6 þ a1 ðn þ 2x þ y þ zÞ5 þ a2 ðn þ 2x þ y þ zÞ4 þ a3 ðn þ 2x þ y þ zÞ3 þ a4 ðn þ 2x þ y þ zÞ2 þ a5 ðn þ 2x þ y þ zÞ þ a6 such that (4.9), (4.16)–(4.18) and (4.21) hold, where ai ¼ ai ðx; y; zÞ, for i ¼ 1; 2; . . . ; 5; 6. After some detailed calculations, we have 2 c¼ ; 5
c2 ¼ c1 ;
a1 ¼ 0;
5 a2 ¼ ; 4
5 5 a3 ¼ 20x y z þ 5c1 ; 2 2
1 a4 ¼ ; 4
a5 ¼ 17x þ y þ z þ c3 ;
5 5 5 a6 ¼ 80x2 y 2 z2 20xz 20xy yz þ c1 ð40x þ 5y þ 5zÞ 5c21 ; 4 4 2 where c3 , c1 are arbitrary constants. In this way, we may deduce a sequence of polynomial solutions of the bilinear system (1.5)–(1.7). 5. Conclusion In this paper,we have constructed infinitely many CLs of the coupled Toda equation (1.2)–(1.4) and its related Eq. (1.1). In the process, their suitable Lax pairs have played an important role. Besides, based on Hirota’s bilinear operator identities and a B€acklund transformation, rational solutions of three-coupled bilinear system (1.5)–(1.7) have been derived. The key step is the use of the nonlinear superposition formula.
Acknowledgement The author would like to express her thanks to Xing-Biao Hu for his guidance and encouragement. This work was supported by CAS President Grant.
Appendix A. Hirota bilinear operator identities The following bilinear operator identities hold for arbitrary functions a, b, c, and d. h 1 i h 1 i 1 Dt Dz e2Dn aðnÞ aðnÞ e2Dn aðnÞ aðnÞ ¼ sinh Dn ½Dt Dz aðnÞ aðnÞ aðnÞ2 2 h i 1 1 1 sinh Dn ðDt a bÞ a2 ¼ Dt sinh Dn a b e2Dn a a 2 2 1 1 Dn ½eDn a b c2 ¼ e2Dn ½ac ðeDn c bÞ ðeDn a cÞ cb 2 sinh 2 1 1 1 1 1 2 sinh Dn ab c2 ¼ e2Dn a c e2Dn c b e2Dn a c e2Dn c b 2 1 1 Dn ½Dt eDn a b c2 ¼ e2Dn ½ðDt a cÞ ðeDn c bÞ þ ac ðDt eDn c bÞ 2 sinh 2 h 1 i h 1 i 1 Dt e2Dn a b e2Dn c c ¼ e2Dn ½ðDt a cÞ cb ac ðDt c bÞ
ðA:1Þ ðA:2Þ ðA:3Þ ðA:4Þ ðA:5Þ ðA:6Þ
J.-X. Zhao / Chaos, Solitons and Fractals 21 (2004) 29–37
37
ðDt a bÞc ðDt a cÞb ¼ aDt b c
ðA:7Þ
ðD2t a
ðA:8Þ
ðD2t a
bÞc cÞb ¼ 2at Dt b c þ aðDt b cÞt h i h i h 1 ih i h 1 i h 1 i 1 1 1 Dt e2Dn a b e2Dn c d e2Dn a b Dt e2Dn c d ¼ Dt e2Dn a d e2Dn c b
½edDn a b ½edDn c d ¼ edDn ad cb
ðA:9Þ ðA:10Þ
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