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N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation
Chaos, Solitons and Fractals 26 (2005) 905–912 www.elsevier.com/locate/chaos
N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation Ye-peng Sun *, Jin-bo Bi, Deng-yuan Chen Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China Accepted 12 January 2005
Communicated by Prof. M. Wadati
Abstract The bilinear equation of the non-isospectral AKNS equation is derived. The N-soliton solutions and the double Wronskian solution are obtained through the Hirota method and the Wronskian technique, respectively. A non-isospectral Schro¨dinger equation and its multi-soliton solutions are given by reducing. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Soliton theory is widely used in mathematics, physics, chemistry, biology and so on. It is one of the most important topics to search for soliton solutions of non-linear evolution equations in the soliton theory. In order to search for the soliton solutions, various methods have been developed. Some of the most important methods are the inverse scattering transform [1,2], the Darboux transformation [3,4], the Hirota method [5], the Wronskian technique [6–8] and so on. The AKNS equation is one of the most important models exhibiting the soliton phenomenon [9]. In this paper, we try to search for N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation through the Hirota method and the Wronskian technique, respectively. The Hirota method and the Wronskian technique are two efficient methods to search for multi-soliton solutions of non-linear evolution equations [10–12]. Both of them are based on the bilinear equation. Hence, it is very important to transform the non-linear evolution equation into the bilinear equation through the dependant variable transformation. However, it is very difficult as well. In this paper, we consider the non-isospectral AKNS equation [13]:
*
qt ¼ xðqxx þ 2q2 rÞ 2qx þ 2qo1 qr;
ð1aÞ
rt ¼ xðrxx 2qr2 Þ þ 2rx 2ro1 qr:
ð1bÞ
Corresponding author. E-mail address: [email protected] (Y.-p. Sun).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.032
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Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
Its Lax pair is
ux ¼ Mðv; gÞu;
Mðv; gÞ ¼
ut ¼ N ðv; gÞu; N ðv; gÞ ¼
g r
q ; g
2g2 x þ xqr þ o1 qr 2xgr þ xrx þ r
ð2aÞ ! 2xgq xqx q ; 2g2 x xqr o1 qr
ð2bÞ
where q and r are potential functions, g is a spectral parameter, o ¼ oxo ; oo1 ¼ o1 o ¼ 1. This paper is organized as follows. In Section 2, we derive the bilinear equation of Eq. (1) and obtain its N-soliton solutions through the Hirota method. In Section 3, the double Wronskian solution is verified. In Section 4, a non-isospectral Schro¨dinger equation and its multi-soliton solutions are given by reducing.
2. The bilinear equation and N-soliton solutions In this section, we derive the bilinear equation of Eq. (1) and obtain its N-soliton solutions through the Hirota method. Through the dependent variable transformation q¼
g ; f
r¼
h : f
ð3Þ
Eq. (1) is transformed into the following bilinear equation: Dt g f ¼ xD2x g f 2gx f ;
ð4aÞ
Dt h f ¼ xD2x h f þ 2hx f ;
ð4bÞ
D2x f f ¼ 2gh;
ð4cÞ
where D is the well-known Hirota bilinear operator defined by Dmt Dnx f g ¼ ðot ot0 Þm ðox ox0 Þn f ðt; xÞgðt0 ; x0 Þjt0 ¼t;x0 ¼x : Expanding f, g and h as the series f ðt; xÞ ¼ 1 þ f ð2Þ e2 þ f ð4Þ e4 þ þ f ð2jÞ e2j þ ;
ð5aÞ
gðt; xÞ ¼ gð1Þ e þ gð3Þ e3 þ þ gð2jþ1Þ e2jþ1 þ ;
ð5bÞ
hðt; xÞ ¼ hð1Þ e þ hð3Þ e3 þ þ hð2jþ1Þ e2jþ1 þ ;
ð5cÞ
substituting Eq. (5) into (4) and comparing the coefficients of the same power of e yields ð1Þ ð1Þ gð1Þ t ¼ xg xx 2g x ;
ð6aÞ
ð3Þ ð1Þ ð2Þ gð3Þ f ð2Þ xD2x gð1Þ f ð2Þ 2ðgð1Þ þ gð3Þ t þ xg xx ¼ Dt g x f x Þ; . . . ;
ð6bÞ
ð1Þ ð1Þ hð1Þ t ¼ xhxx þ 2hx ;
ð7aÞ
ð3Þ ð1Þ ð2Þ f ð2Þ þ xD2x hð1Þ f ð2Þ þ 2ðhð1Þ þ hð3Þ hð3Þ t xhxx ¼ Dt h x f x Þ; . . . ;
ð7bÞ
fxxð2Þ ¼ gð1Þ hð1Þ ;
ð8aÞ
2fxxð4Þ ¼ D2x f ð2Þ f ð2Þ 2ðgð1Þ hð3Þ þ gð3Þ hð1Þ Þ; . . . :
ð8bÞ
Taking gð1Þ ¼
n X j¼1
xj ðtÞenj ;
ð0Þ
nj ¼ k j ðtÞx þ nj ;
ð9aÞ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
hð1Þ ¼
n X
907
ð0Þ
rj ðtÞegj ;
gj ¼ lj ðtÞx þ gj ;
ð9bÞ
j¼1
then the N-soliton solution of Eq. (1) is obtained. For n = 1, Eq. (9) becomes hð1Þ ¼ r1 ðtÞeg1 :
gð1Þ ¼ x1 ðtÞen1 ;
Solving Eqs. (6)–(8), we have f ð2Þ ¼
x1 ðtÞr1 ðtÞ ðk 1 ðtÞ þ l1 ðtÞÞ2
en1 þg1 ;
ð10aÞ
f ðmÞ ¼ 0;
m ¼ 4; 6; . . . ;
ð10bÞ
gðnÞ ¼ 0;
hðnÞ ¼ 0; n ¼ 3; 5; . . . ;
ð10cÞ
k 1;t ðtÞ ¼ k 21 ðtÞ;
x1;t ðtÞ ¼ 2x1 ðtÞk 1 ðtÞ;
l1;t ðtÞ ¼ l21 ðtÞ;
r1;t ðtÞ ¼ 2r1 ðtÞl1 ðtÞ;
k 1 ðtÞ ¼
l1 ðtÞ ¼
1 ; t þ a1
1 ; t þ b1
x1 ðtÞ ¼
r1 ðtÞ ¼
1 ðt þ a1 Þ2 1
ðt þ b1 Þ2
;
ð10dÞ
ð10eÞ
;
here a1 and b1 are two arbitrary constants. From (3), we have q¼
x1 ðtÞen1 ; 1 þ x1 ðtÞr1 ðtÞen1 þg1 þh13
where eh13 ¼ ðk
1 2 1 ðtÞþl1 ðtÞÞ
r¼
r1 ðtÞeg1 ; 1 þ x1 ðtÞr1 ðtÞen1 þg1 þh13
ð11Þ
. Thus, the one-soliton solution is given. For n = 2, Eq. (9) becomes hð1Þ ¼ r1 ðtÞeg1 þ r2 ðtÞeg2 :
gð1Þ ¼ x1 ðtÞen1 þ x2 ðtÞen2 ; From Eqs. (6)–(8), we work out
f ð2Þ ¼ x1 ðtÞr1 ðtÞen1 þg1 þh13 þ x1 ðtÞr2 ðtÞen1 þg2 þh14 þ x2 ðtÞr1 ðtÞen2 þg1 þh23 þ x2 ðtÞr2 ðtÞen2 þg2 þh24 ;
ð12aÞ
f ð4Þ ¼ x1 ðtÞx2 ðtÞr1 ðtÞr2 ðtÞen1 þn2 þg1 þg2 þh12 þh13 þh14 þh23 þh24 þh34 ;
ð12bÞ
gð3Þ ¼ x1 ðtÞx2 ðtÞr1 ðtÞen1 þn2 þg1 þh12 þh13 þh23 þ x1 ðtÞx2 ðtÞr2 ðtÞen1 þn2 þg2 þh12 þh14 þh24 ;
ð12cÞ
hð3Þ ¼ x1 ðtÞr1 ðtÞr2 ðtÞen1 þg1 þg2 þh13 þh14 þh34 þ x2 ðtÞr1 ðtÞr2 ðtÞen2 þg1 þg2 þh23 þh24 þh34 ;
ð12dÞ
f ðmÞ ¼ 0;
m ¼ 6; 8; . . . ;
ð12eÞ
gðnÞ ¼ 0;
hðnÞ ¼ 0;
ð12fÞ
n ¼ 5; 7; . . . ;
where k j;t ðtÞ ¼ k 2j ðtÞ;
xj;t ðtÞ ¼ 2xj ðtÞk j ðtÞ;
rq;t ðtÞ ¼ 2rq ðtÞlq ðtÞ;
lq;t ðtÞ ¼ l2q ðtÞ;
eh12 ¼ ðk 1 ðtÞ k 2 ðtÞÞ2 ;
k j ðtÞ ¼
lq ðtÞ ¼
1 ; t þ aj
1 ; t þ bq
eh34 ¼ ðl1 ðtÞ l2 ðtÞÞ2 ;
xj ðtÞ ¼
rq ðtÞ ¼
ehj;qþ2 ¼
1 ðt þ aj Þ2 1
ðt þ bq Þ2
ðj ¼ 1; 2Þ;
;
ðq ¼ 1; 2Þ;
;
1 ðk j ðtÞ þ lq ðtÞÞ2
;
ðj; q ¼ 1; 2Þ:
Setting e = 1, then the series Eq. (5) becomes f2 ¼ 1 þ f ð2Þ þ f ð4Þ ;
g2 ¼ gð1Þ þ gð3Þ ;
h2 ¼ hð1Þ þ hð3Þ :
Substituting Eq. (13) into (3), we obtain the two-soliton solution of Eq. (1).
ð13Þ
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Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
In general, we have fn ¼
X
A1 ðlÞ exp
" 2n X
l¼0;1
gn ¼
X
" A2 ðlÞ exp
l¼0;1
hn ¼
"
X
A3 ðlÞ exp
l¼0;1
lj ðnj þ ln xj ðtÞÞ þ
2n X
j¼1
16j
2n X
2n X
lj ðnj þ ln xj ðtÞÞ þ
j¼1
16j
2n X
2n X
lj ðnj þ ln xj ðtÞÞ þ
# ð14aÞ
lj lq hjq ; #
ð14bÞ
lj lq hjq ; #
ð14cÞ
lj lq hjq ;
16j
j¼1
where k j;t ðtÞ ¼ k 2j ðtÞ;
xj;t ðtÞ ¼ 2xj ðtÞk j ðtÞ;
rq;t ðtÞ ¼ 2rq ðtÞlq ðtÞ;
lq;t ðtÞ ¼ l2q ðtÞ;
ehjq ¼ ðk j ðtÞ k q ðtÞÞ2 ; ehj;nþq ¼
1 ðk j ðtÞ þ lq ðtÞÞ2
k j ðtÞ ¼
lq ðtÞ ¼
1 ; t þ aj
1 ; t þ bq
ehðnþjÞðnþqÞ ¼ ðlj ðtÞ lq ðtÞÞ2 ; ;
nnþj ¼ gj ; xnþj ðtÞ ¼ rj ðtÞ;
xj ðtÞ ¼
rq ðtÞ ¼
1 ðt þ aj Þ2 1
ðt þ bq Þ2
ðj ¼ 1; . . . ; nÞ;
;
;
ðq ¼ 1; . . . ; nÞ;
ðj < q ¼ 2; 3; . . . ; nÞ;
ð15aÞ
ð15bÞ ð15cÞ
ðj; q ¼ 1; 2; . . . ; nÞ;
ð15dÞ
ðj ¼ 1; 2; . . . ; nÞ;
ð15eÞ
A1(l), A2(l) and A3(l) take over all possible combinations of lj = 0, 1 (j = 1, 2, . . ., 2n) and satisfy the following condition n X
lj ¼
j¼1
n X
lnþj ;
j¼1
n X
lj ¼ 1 þ
j¼1
n X
lnþj ;
j¼1
1þ
n X j¼1
lj ¼
n X
lnþj ;
ð16Þ
j¼1
respectively.
3. The double Wronskian solution Let us first specify some properties of the Wronskian determinant. As is well-known, the double Wronskian determinant [8] is W N ;M ðu; wÞ ¼ detðu; ox u; . . . ; oNx 1 u; w; ox w; . . . ; oM1 wÞ; x where u = (u1(x), u2(x), . . ., uN+M(x))T and w = (w1(x), w2(x), . . ., wN+M(x))T. In the most known examples, the two determinantal identities are often used [6]. The one is jD; a; bjjD; c; dj jD; a; cjjD; b; dj þ jD; a; djjD; b; cj ¼ 0; where D is an N · (N 2) matrix and a, b, c and d represent N column vectors. The other is ! N N X X ja1 ; . . . ; aj1 ; baj ; ajþ1 ; . . . ; aN j ¼ bj ja1 ; . . . ; aN j; j¼1
ð17Þ
ð18Þ
j¼1
where aj (1 6 j 6 N) are N column vectors and baj denotes (b1a1j, b2a2j, . . ., bNaNj)T. Employing the Wronskian technique, we have Theorem 1. The non-isospectral AKNS equation (4) has the double Wronskian solution g ¼ 2W N þ2;M ðu; wÞ;
f ¼ W N þ1;Mþ1 ðu; wÞ;
h ¼ 2W N ;Mþ2 ðu; wÞ;
ð19Þ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
909
where uj ¼ xj ðtÞ expð 12 kj ðtÞxÞ; wj ¼ rj ðtÞ expð12 kj ðtÞxÞ; kj;t ðtÞ ¼ k2j ; ð1 6 j 6 N þ M þ 2Þ, uj and wj satisfy the conditions uj;x ¼ 12kj uj ;
wj;x ¼ 12kj wj ;
uj;t ¼ 2xuj;xx þ 2Nuj;x ;
ð20Þ
wj;t ¼ 2xwj;xx 2Mwj;x ;
ð21Þ
respectively. Proof. In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [6,7], then Eq. (19) becomes g ¼ 2j Nd þ 1; Md 1j;
b; M b j; f ¼ jN
h ¼ 2j Nd 1; Md þ 1j:
ð22Þ
Note that the lth-order derivatives of uj,t and wj,t with respect to x are ðlÞ
ðlþ2Þ
uj;t ¼ 2xuj ðlÞ
ðlþ2Þ
wj;t ¼ 2xwj
ðlþ1Þ
þ ð2N 2lÞuj
ðlþ1Þ
ð2M 2lÞwj
ð23Þ
;
ð24Þ
:
We have b j þ jN b ; Md 1; N þ 1; M 1; M þ 1j; fx ¼ j Nd b j þ j Nd b j þ 2j Nd 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j fxx ¼ j Nd b ; Md b ; Md þ jN 2; M; M þ 1j þ j N 1; M þ 2j; b ; N þ 2; Md gx ¼ 2j N 1j þ 2j Nd þ 1; Md 2; Mj; b ; N þ 3; Md b ; N þ 2; Md 1; N þ 1; N þ 2; Md 1j þ 2j N 1j þ 4j N 2; Mj gxx ¼ 2j Nd þ 2j Nd þ 1; Md 3; M 1; Mj þ 2j Nd þ 1; Md 2; M þ 1j; b j þ j Nd b jÞ þ 2xðj N b ; Md b ; Md ft ¼ 2xðj Nd 2; N þ 1; N ; M 1; N þ 2; M 2; M þ 1; Mj þ j N 1; M þ 2jÞ; b ; N þ 3; Md b ; N þ 2; Md 1; N þ 2; N þ 1; Md 1j þ j N 1jÞ 4j N 1j gt ¼ 4xðj Nd þ 4xðj Nd þ 1; Md 3; M; M 1j þ j Nd þ 1; Md 2; M þ 1jÞ 4j Nd þ 1; Md 2; Mj: A direct calculation gives b; M b jj Nd 1; N þ 1; N þ 2; Md 1j gt f ft g þ xðgxx f 2gx fx þ gf xx Þ þ 2gx f ¼ 6xj N b; M b jðj N b ; N þ 3; Md b ; N þ 2; Md 2xj N 1j 2j N 2; Mj þ j Nd þ 1; b; M b jj Nd Md 3; M 1; MjÞ þ 6xj N þ 1; Md 2; M þ 1j 2xj Nd 2; N ; b jj Nd N þ 1; M þ 1; Md 1j þ 2xj Nd þ 1; Md 1jð3j Nd 1; N þ 2; b j þ 2j Nd b; M 1; N þ 1; Md 1; M þ 1jÞ þ 2xj Nd þ 1; Md 1jð3j N b ; Md b ; N þ 2; Md 2; M; M þ 1j j N 1; M þ 2jÞ 4xj N b j þ jN b ; Md Md 1jðj Nd 1; N þ 1; M 1; M þ 1jÞ 4xj Nd þ 1; b j þ jN b ; Md Md 2; Mjðj Nd 1; N þ 1; M 1; M þ 1jÞ:
ð25Þ
From (18) and (20), we get X 1 b ; N þ 2; Md b ; N þ 3; Md b ; N þ 2; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1j j N 1j þ j N 2; Mj; kj j N 2 ð26aÞ
910
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
X
1 b ; N þ 2; Md þ 1; Md 2; Mj ¼ j N 2; Mj þ j Nd þ 1; Md 3; M 1; Mj þ j Nd þ 1; Md 2; M þ 1j; kj j Nd 2
X
1 b j ¼ j Nd b j j Nd b j þ j Nd kj j Nd 1; N þ 1; M 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j; 2
ð26bÞ
ð26cÞ X
1 b ; Md b ; Md b ; Md kj j N 1; M þ 1j ¼ j Nd 1; N þ 1; Md 1; M þ 1j þ j N 2; M; M þ 1j þ j N 1; M þ 2j: 2 ð26dÞ
Noting X
X 1 1 b ; N þ 2; Md b; M b j ¼ jN b ; N þ 2; Md b; M b j; kj j N kj j N 1jj N 1j 2 2
ð27aÞ
X
X 1 1 b; M b j ¼ j Nd b; M b j; kj j Nd kj j N þ 1; Md 2; Mjj N þ 1; Md 2; Mj 2 2
ð27bÞ
X
X 1 1 d d d d b b kj j N 1; N þ 1; M jj N þ 1; M 1j ¼ j N 1; N þ 1; M j kj j Nd þ 1; Md 1j; 2 2
ð27cÞ
X
X 1 1 b ; Md b ; Md 1; M þ 1jj Nd þ 1; Md 1j ¼ j N 1; M þ 1j þ 1; Md 1j: kj j N kj j Nd 2 2
ð27dÞ
Using (26) and (27), Eq. (25) becomes b;M bj gt f ft g þ xðgxx f 2gx fx þ gf xx Þ þ 2gx f ¼ 8xj Nd 1; N þ 1; N þ 2; Md 1jj N b ; Md þ 8xj N 2; M; M þ 1jj Nd þ 1; Md 1j b ; N þ 2; Md b j þ 8xj Nd b;M bj 8xj N 1jj Nd 1; N þ 1; M þ 1; Md 2; M þ 1jj N b jj Nd þ 8xj Nd 1; N þ 2; M þ 1; Md 1j b ; Md 8xj Nd þ 1; Md 2; Mjj N 1; M þ 1j:
ð28Þ
It is readily checked that Eq. (28) is equal to zero. So, the proof of Eq. (4a) is completed. Eq. (4b) can be proved similarly. Making use of (17) gives b; M b jj Nd b j þ jN b; M b jðj Nd b j 2j Nd 2; N; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1jÞ fxx f fx2 þ gh ¼ j N b; M b jðj N b ; Md b ; Md bj þ jN 2; M; M þ 1j þ j N 1; M þ 2jÞ ðj Nd 1; N þ 1; M b ; Md jN 1; M þ 1jÞ2 :
ð29Þ
From (18), we obtain X 1 b; M b j ¼ jN b ; Md b j; 1; M þ 1j j Nd 1; N þ 1; M kj j N 2 X
1 kj 2
2
b; M b j ¼ j Nd b j þ j Nd b j 2j Nd jN 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j b ; Md b ; Md þ jN 2; M; M þ 1j þ j N 1; M þ 2j:
Thus, we have b j þ j Nd b j 2j Nd b ; Md ðj Nd 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j þ j N 2; M; M þ 1j b ; Md b; M b j ¼ ðj N b ; Md b jÞ2 : þ jN 1; M þ 2jÞj N 1; M þ 1j j Nd 1; N þ 1; M Then, Eq. (4c) is obtained from Eqs. (29) and (30).
h
ð30Þ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
911
4. Reduction Setting r = q* and replacing t by it in Eq. (1), where * denotes the complex conjugate, i ¼ following non-isospectral Schro¨dinger equation:
pffiffiffiffiffiffiffi 1, we obtain the
iqt þ xðqxx þ 2qjqj2 Þ þ 2qx þ 2qo1 jqj2 ¼ 0:
ð31Þ
Through the dependent variable transformation g q¼ : f Eq. (31) is rewritten as ðiDt þ xD2x Þg f ¼ 2gx f ;
ð32Þ
ð33aÞ
D2x f f ¼ 2gg :
ð33bÞ
Thus, we obtain " # 2n 2n X X X fn ¼ A1 ðlÞ exp lj ðnj þ ln xj ðtÞÞ þ lj ls hjs ; l¼0;1
X
gn ¼
A2 ðlÞ exp
" 2n X
l¼0;1
ð34aÞ
16j
j¼1
lj ðnj þ ln xj ðtÞÞ þ
2n X
# ð34bÞ
lj ls hjs ;
16j
j¼1
where nnþj ¼ nj ; xnþj ðtÞ ¼ xj ðtÞ; xj;t ðtÞ ¼ 2ixj ðtÞk j ðtÞ; ehj;nþs ¼
1 ðk j ðtÞ þ k s ðtÞÞ2
ehjs ¼ ðk j ðtÞ k s ðtÞÞ2 ;
ðj ¼ 1; 2; . . . ; nÞ;
k j;t ðtÞ ¼ ik 2j ðtÞ;
ð35aÞ
ðj ¼ 1; . . . ; nÞ;
ð35bÞ
ðj; s ¼ 1; 2; . . . ; nÞ;
;
ð35cÞ
ehðnþjÞðnþsÞ ¼ ðk j ðtÞ k s ðtÞÞ2 ;
ðj < s ¼ 2; 3; . . . ; nÞ;
ð35dÞ
A1(l) and A2(l) take over all possible combinations of lj = 0,1 (j = 1, 2, . . ., 2n) and satisfy the condition n n n n X X X X lj ¼ lnþj ; lj ¼ 1 þ lnþj ; j¼1
j¼1
j¼1
j¼1
respectively. In the following, we deduce the double Wronskian solution of Eq. (33). Let AN·M and BN·M be two N · M matrices, 0
...
uN
ou2 ... ouN
... ... ...
oM1 u2 C C C; ... A oM 1 uN
w1 Bw B ¼B 2 @... wN
ow1 ow2 ... owN
... ... ... ...
1 oM1 w1 oM1 w2 C C C; ... A oM 1 wN
0
where
1
ou1
Bu B AN M ¼ B 2 @...
BN M
oM1 u1
u1
1 uj ¼ xj ðtÞ exp kj ðtÞx ; 2
wj ¼ rj ðtÞ exp
1 kj ðtÞx ; 2
kj;t ðtÞ ¼ k2j ;
ð36Þ
uj and wj satisfy the following conditions: uj;x ¼ 12kj uj ;
wj;x ¼ 12kj wj ;
ð37Þ
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Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
uj;t ¼ 2ixuj;xx 2iN uj;x ; respectively. We define AðN þ1ÞðN þ1Þ e f ¼ AðN þ1ÞðN þ1Þ
wj;t ¼ 2ixwj;xx þ 2iNwj;x ;
BðN þ1ÞðN þ1Þ b; N b j; jN BðN þ1ÞðN þ1Þ
AðN þ2ÞðN þ2Þ e g ¼ 2 BN ðN þ2Þ
BðN þ2ÞN þ 1; Nd 1j: 2j Nd AN N
In terms of the determinantal properties, we have B AðN þ2ÞðN þ2Þ BðN þ2ÞN AN N ¼ 2ð1Þ3N N ðN þ2Þ e g ¼ 2 AðN þ2ÞðN þ2Þ BðN þ2ÞN BN ðN þ2Þ AN N AN N AN N BN ðN þ2Þ BN ðN þ2Þ 6N 1; Nd þ 1j: ¼ 2ð1Þ ¼ 2 2j Nd BðN þ2ÞN AðN þ2ÞðN þ2Þ BðN þ2ÞN AðN þ2ÞðN þ2Þ
ð38Þ
ð39Þ
ð40Þ
ð41Þ
Letting M = N in Eq. (22) yields g ¼ 2j Nd þ 1; Nd 1j;
b; N b j; f ¼ jN
h ¼ 2j Nd 1; Nd þ 1j:
So, we have the following result. Theorem 2. The non-isospectral Schro¨dinger equation (33) has the double Wronskian solution e g ¼ 2j Nd þ 1; Nd 1j;
b; N b j; fe ¼ j N
e 1; Nd þ 1j; g ¼ 2j Nd
ð42Þ
where uj and wj satisfy the conditions (36)–(38).
5. Conclusions In this paper, we presented the N-soliton solutions and the double Wronskian solution of the non-isospectral AKNS equation (1) through the Hirota method and the Wronskian technique, respectively. Further, the non-isospectral Schro¨dinger equation (31) and its multi-soliton solutions are obtained by reducing.
Acknowledgments The authors are very grateful to Professor M. Wadati for his ardent guidance and help. The authors express their sincere thanks to Dr. Zhang DJ, Zhang Y and Ning TK as well. This project is supported by the National Science Foundation of China (10371070) and the Special Funds for Major Specialities of Shanghai Education Committee.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Garder CS, Green JM, Kruskal MD, Miura RM. Phys Rev Lett 1967;19:1095. Wadati M, Konno K, Ichikawa YH. J Phys Soc Jpn 1979;46:1965. Wadati M, Sanuki H, Konno K. Prog Theor Phys 1975;53:419. Matveev VB, Salle MA. Darboux Transformation and Soliton. Berlin: Springer; 1991. Hirota R. Phys Rev Lett 1971;27:1192. Freeman NC, Nimmo JJC. Phys Lett A 1983;95:1. Nimmo JJC, Freeman NC. Phys Lett A 1983;95:4. Nimmo JJC. Phys Lett A 1983;99:279. Ablowitz MJ, Kaup DJ, Newell AC, Segur H. Stud Appl Math 1974;53:249. Chen DY, Zhang DJ, Deng SF. J Phys Soc Jpn 2002;71:658. Zhang DJ. J Phys Soc Jpn 2002;71:2649. Deng SF, Chen DY. J Phys Soc Jpn 2001;70:3174. Chen DY, Zhang DJ. J Math Phys 1996;37:5524.
N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation Ye-peng Sun *, Jin-bo Bi, Deng-yuan Chen Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China Accepted 12 January 2005
Communicated by Prof. M. Wadati
Abstract The bilinear equation of the non-isospectral AKNS equation is derived. The N-soliton solutions and the double Wronskian solution are obtained through the Hirota method and the Wronskian technique, respectively. A non-isospectral Schro¨dinger equation and its multi-soliton solutions are given by reducing. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Soliton theory is widely used in mathematics, physics, chemistry, biology and so on. It is one of the most important topics to search for soliton solutions of non-linear evolution equations in the soliton theory. In order to search for the soliton solutions, various methods have been developed. Some of the most important methods are the inverse scattering transform [1,2], the Darboux transformation [3,4], the Hirota method [5], the Wronskian technique [6–8] and so on. The AKNS equation is one of the most important models exhibiting the soliton phenomenon [9]. In this paper, we try to search for N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation through the Hirota method and the Wronskian technique, respectively. The Hirota method and the Wronskian technique are two efficient methods to search for multi-soliton solutions of non-linear evolution equations [10–12]. Both of them are based on the bilinear equation. Hence, it is very important to transform the non-linear evolution equation into the bilinear equation through the dependant variable transformation. However, it is very difficult as well. In this paper, we consider the non-isospectral AKNS equation [13]:
*
qt ¼ xðqxx þ 2q2 rÞ 2qx þ 2qo1 qr;
ð1aÞ
rt ¼ xðrxx 2qr2 Þ þ 2rx 2ro1 qr:
ð1bÞ
Corresponding author. E-mail address: [email protected] (Y.-p. Sun).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.032
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Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
Its Lax pair is
ux ¼ Mðv; gÞu;
Mðv; gÞ ¼
ut ¼ N ðv; gÞu; N ðv; gÞ ¼
g r
q ; g
2g2 x þ xqr þ o1 qr 2xgr þ xrx þ r
ð2aÞ ! 2xgq xqx q ; 2g2 x xqr o1 qr
ð2bÞ
where q and r are potential functions, g is a spectral parameter, o ¼ oxo ; oo1 ¼ o1 o ¼ 1. This paper is organized as follows. In Section 2, we derive the bilinear equation of Eq. (1) and obtain its N-soliton solutions through the Hirota method. In Section 3, the double Wronskian solution is verified. In Section 4, a non-isospectral Schro¨dinger equation and its multi-soliton solutions are given by reducing.
2. The bilinear equation and N-soliton solutions In this section, we derive the bilinear equation of Eq. (1) and obtain its N-soliton solutions through the Hirota method. Through the dependent variable transformation q¼
g ; f
r¼
h : f
ð3Þ
Eq. (1) is transformed into the following bilinear equation: Dt g f ¼ xD2x g f 2gx f ;
ð4aÞ
Dt h f ¼ xD2x h f þ 2hx f ;
ð4bÞ
D2x f f ¼ 2gh;
ð4cÞ
where D is the well-known Hirota bilinear operator defined by Dmt Dnx f g ¼ ðot ot0 Þm ðox ox0 Þn f ðt; xÞgðt0 ; x0 Þjt0 ¼t;x0 ¼x : Expanding f, g and h as the series f ðt; xÞ ¼ 1 þ f ð2Þ e2 þ f ð4Þ e4 þ þ f ð2jÞ e2j þ ;
ð5aÞ
gðt; xÞ ¼ gð1Þ e þ gð3Þ e3 þ þ gð2jþ1Þ e2jþ1 þ ;
ð5bÞ
hðt; xÞ ¼ hð1Þ e þ hð3Þ e3 þ þ hð2jþ1Þ e2jþ1 þ ;
ð5cÞ
substituting Eq. (5) into (4) and comparing the coefficients of the same power of e yields ð1Þ ð1Þ gð1Þ t ¼ xg xx 2g x ;
ð6aÞ
ð3Þ ð1Þ ð2Þ gð3Þ f ð2Þ xD2x gð1Þ f ð2Þ 2ðgð1Þ þ gð3Þ t þ xg xx ¼ Dt g x f x Þ; . . . ;
ð6bÞ
ð1Þ ð1Þ hð1Þ t ¼ xhxx þ 2hx ;
ð7aÞ
ð3Þ ð1Þ ð2Þ f ð2Þ þ xD2x hð1Þ f ð2Þ þ 2ðhð1Þ þ hð3Þ hð3Þ t xhxx ¼ Dt h x f x Þ; . . . ;
ð7bÞ
fxxð2Þ ¼ gð1Þ hð1Þ ;
ð8aÞ
2fxxð4Þ ¼ D2x f ð2Þ f ð2Þ 2ðgð1Þ hð3Þ þ gð3Þ hð1Þ Þ; . . . :
ð8bÞ
Taking gð1Þ ¼
n X j¼1
xj ðtÞenj ;
ð0Þ
nj ¼ k j ðtÞx þ nj ;
ð9aÞ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
hð1Þ ¼
n X
907
ð0Þ
rj ðtÞegj ;
gj ¼ lj ðtÞx þ gj ;
ð9bÞ
j¼1
then the N-soliton solution of Eq. (1) is obtained. For n = 1, Eq. (9) becomes hð1Þ ¼ r1 ðtÞeg1 :
gð1Þ ¼ x1 ðtÞen1 ;
Solving Eqs. (6)–(8), we have f ð2Þ ¼
x1 ðtÞr1 ðtÞ ðk 1 ðtÞ þ l1 ðtÞÞ2
en1 þg1 ;
ð10aÞ
f ðmÞ ¼ 0;
m ¼ 4; 6; . . . ;
ð10bÞ
gðnÞ ¼ 0;
hðnÞ ¼ 0; n ¼ 3; 5; . . . ;
ð10cÞ
k 1;t ðtÞ ¼ k 21 ðtÞ;
x1;t ðtÞ ¼ 2x1 ðtÞk 1 ðtÞ;
l1;t ðtÞ ¼ l21 ðtÞ;
r1;t ðtÞ ¼ 2r1 ðtÞl1 ðtÞ;
k 1 ðtÞ ¼
l1 ðtÞ ¼
1 ; t þ a1
1 ; t þ b1
x1 ðtÞ ¼
r1 ðtÞ ¼
1 ðt þ a1 Þ2 1
ðt þ b1 Þ2
;
ð10dÞ
ð10eÞ
;
here a1 and b1 are two arbitrary constants. From (3), we have q¼
x1 ðtÞen1 ; 1 þ x1 ðtÞr1 ðtÞen1 þg1 þh13
where eh13 ¼ ðk
1 2 1 ðtÞþl1 ðtÞÞ
r¼
r1 ðtÞeg1 ; 1 þ x1 ðtÞr1 ðtÞen1 þg1 þh13
ð11Þ
. Thus, the one-soliton solution is given. For n = 2, Eq. (9) becomes hð1Þ ¼ r1 ðtÞeg1 þ r2 ðtÞeg2 :
gð1Þ ¼ x1 ðtÞen1 þ x2 ðtÞen2 ; From Eqs. (6)–(8), we work out
f ð2Þ ¼ x1 ðtÞr1 ðtÞen1 þg1 þh13 þ x1 ðtÞr2 ðtÞen1 þg2 þh14 þ x2 ðtÞr1 ðtÞen2 þg1 þh23 þ x2 ðtÞr2 ðtÞen2 þg2 þh24 ;
ð12aÞ
f ð4Þ ¼ x1 ðtÞx2 ðtÞr1 ðtÞr2 ðtÞen1 þn2 þg1 þg2 þh12 þh13 þh14 þh23 þh24 þh34 ;
ð12bÞ
gð3Þ ¼ x1 ðtÞx2 ðtÞr1 ðtÞen1 þn2 þg1 þh12 þh13 þh23 þ x1 ðtÞx2 ðtÞr2 ðtÞen1 þn2 þg2 þh12 þh14 þh24 ;
ð12cÞ
hð3Þ ¼ x1 ðtÞr1 ðtÞr2 ðtÞen1 þg1 þg2 þh13 þh14 þh34 þ x2 ðtÞr1 ðtÞr2 ðtÞen2 þg1 þg2 þh23 þh24 þh34 ;
ð12dÞ
f ðmÞ ¼ 0;
m ¼ 6; 8; . . . ;
ð12eÞ
gðnÞ ¼ 0;
hðnÞ ¼ 0;
ð12fÞ
n ¼ 5; 7; . . . ;
where k j;t ðtÞ ¼ k 2j ðtÞ;
xj;t ðtÞ ¼ 2xj ðtÞk j ðtÞ;
rq;t ðtÞ ¼ 2rq ðtÞlq ðtÞ;
lq;t ðtÞ ¼ l2q ðtÞ;
eh12 ¼ ðk 1 ðtÞ k 2 ðtÞÞ2 ;
k j ðtÞ ¼
lq ðtÞ ¼
1 ; t þ aj
1 ; t þ bq
eh34 ¼ ðl1 ðtÞ l2 ðtÞÞ2 ;
xj ðtÞ ¼
rq ðtÞ ¼
ehj;qþ2 ¼
1 ðt þ aj Þ2 1
ðt þ bq Þ2
ðj ¼ 1; 2Þ;
;
ðq ¼ 1; 2Þ;
;
1 ðk j ðtÞ þ lq ðtÞÞ2
;
ðj; q ¼ 1; 2Þ:
Setting e = 1, then the series Eq. (5) becomes f2 ¼ 1 þ f ð2Þ þ f ð4Þ ;
g2 ¼ gð1Þ þ gð3Þ ;
h2 ¼ hð1Þ þ hð3Þ :
Substituting Eq. (13) into (3), we obtain the two-soliton solution of Eq. (1).
ð13Þ
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Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
In general, we have fn ¼
X
A1 ðlÞ exp
" 2n X
l¼0;1
gn ¼
X
" A2 ðlÞ exp
l¼0;1
hn ¼
"
X
A3 ðlÞ exp
l¼0;1
lj ðnj þ ln xj ðtÞÞ þ
2n X
j¼1
16j
2n X
2n X
lj ðnj þ ln xj ðtÞÞ þ
j¼1
16j
2n X
2n X
lj ðnj þ ln xj ðtÞÞ þ
# ð14aÞ
lj lq hjq ; #
ð14bÞ
lj lq hjq ; #
ð14cÞ
lj lq hjq ;
16j
j¼1
where k j;t ðtÞ ¼ k 2j ðtÞ;
xj;t ðtÞ ¼ 2xj ðtÞk j ðtÞ;
rq;t ðtÞ ¼ 2rq ðtÞlq ðtÞ;
lq;t ðtÞ ¼ l2q ðtÞ;
ehjq ¼ ðk j ðtÞ k q ðtÞÞ2 ; ehj;nþq ¼
1 ðk j ðtÞ þ lq ðtÞÞ2
k j ðtÞ ¼
lq ðtÞ ¼
1 ; t þ aj
1 ; t þ bq
ehðnþjÞðnþqÞ ¼ ðlj ðtÞ lq ðtÞÞ2 ; ;
nnþj ¼ gj ; xnþj ðtÞ ¼ rj ðtÞ;
xj ðtÞ ¼
rq ðtÞ ¼
1 ðt þ aj Þ2 1
ðt þ bq Þ2
ðj ¼ 1; . . . ; nÞ;
;
;
ðq ¼ 1; . . . ; nÞ;
ðj < q ¼ 2; 3; . . . ; nÞ;
ð15aÞ
ð15bÞ ð15cÞ
ðj; q ¼ 1; 2; . . . ; nÞ;
ð15dÞ
ðj ¼ 1; 2; . . . ; nÞ;
ð15eÞ
A1(l), A2(l) and A3(l) take over all possible combinations of lj = 0, 1 (j = 1, 2, . . ., 2n) and satisfy the following condition n X
lj ¼
j¼1
n X
lnþj ;
j¼1
n X
lj ¼ 1 þ
j¼1
n X
lnþj ;
j¼1
1þ
n X j¼1
lj ¼
n X
lnþj ;
ð16Þ
j¼1
respectively.
3. The double Wronskian solution Let us first specify some properties of the Wronskian determinant. As is well-known, the double Wronskian determinant [8] is W N ;M ðu; wÞ ¼ detðu; ox u; . . . ; oNx 1 u; w; ox w; . . . ; oM1 wÞ; x where u = (u1(x), u2(x), . . ., uN+M(x))T and w = (w1(x), w2(x), . . ., wN+M(x))T. In the most known examples, the two determinantal identities are often used [6]. The one is jD; a; bjjD; c; dj jD; a; cjjD; b; dj þ jD; a; djjD; b; cj ¼ 0; where D is an N · (N 2) matrix and a, b, c and d represent N column vectors. The other is ! N N X X ja1 ; . . . ; aj1 ; baj ; ajþ1 ; . . . ; aN j ¼ bj ja1 ; . . . ; aN j; j¼1
ð17Þ
ð18Þ
j¼1
where aj (1 6 j 6 N) are N column vectors and baj denotes (b1a1j, b2a2j, . . ., bNaNj)T. Employing the Wronskian technique, we have Theorem 1. The non-isospectral AKNS equation (4) has the double Wronskian solution g ¼ 2W N þ2;M ðu; wÞ;
f ¼ W N þ1;Mþ1 ðu; wÞ;
h ¼ 2W N ;Mþ2 ðu; wÞ;
ð19Þ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
909
where uj ¼ xj ðtÞ expð 12 kj ðtÞxÞ; wj ¼ rj ðtÞ expð12 kj ðtÞxÞ; kj;t ðtÞ ¼ k2j ; ð1 6 j 6 N þ M þ 2Þ, uj and wj satisfy the conditions uj;x ¼ 12kj uj ;
wj;x ¼ 12kj wj ;
uj;t ¼ 2xuj;xx þ 2Nuj;x ;
ð20Þ
wj;t ¼ 2xwj;xx 2Mwj;x ;
ð21Þ
respectively. Proof. In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [6,7], then Eq. (19) becomes g ¼ 2j Nd þ 1; Md 1j;
b; M b j; f ¼ jN
h ¼ 2j Nd 1; Md þ 1j:
ð22Þ
Note that the lth-order derivatives of uj,t and wj,t with respect to x are ðlÞ
ðlþ2Þ
uj;t ¼ 2xuj ðlÞ
ðlþ2Þ
wj;t ¼ 2xwj
ðlþ1Þ
þ ð2N 2lÞuj
ðlþ1Þ
ð2M 2lÞwj
ð23Þ
;
ð24Þ
:
We have b j þ jN b ; Md 1; N þ 1; M 1; M þ 1j; fx ¼ j Nd b j þ j Nd b j þ 2j Nd 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j fxx ¼ j Nd b ; Md b ; Md þ jN 2; M; M þ 1j þ j N 1; M þ 2j; b ; N þ 2; Md gx ¼ 2j N 1j þ 2j Nd þ 1; Md 2; Mj; b ; N þ 3; Md b ; N þ 2; Md 1; N þ 1; N þ 2; Md 1j þ 2j N 1j þ 4j N 2; Mj gxx ¼ 2j Nd þ 2j Nd þ 1; Md 3; M 1; Mj þ 2j Nd þ 1; Md 2; M þ 1j; b j þ j Nd b jÞ þ 2xðj N b ; Md b ; Md ft ¼ 2xðj Nd 2; N þ 1; N ; M 1; N þ 2; M 2; M þ 1; Mj þ j N 1; M þ 2jÞ; b ; N þ 3; Md b ; N þ 2; Md 1; N þ 2; N þ 1; Md 1j þ j N 1jÞ 4j N 1j gt ¼ 4xðj Nd þ 4xðj Nd þ 1; Md 3; M; M 1j þ j Nd þ 1; Md 2; M þ 1jÞ 4j Nd þ 1; Md 2; Mj: A direct calculation gives b; M b jj Nd 1; N þ 1; N þ 2; Md 1j gt f ft g þ xðgxx f 2gx fx þ gf xx Þ þ 2gx f ¼ 6xj N b; M b jðj N b ; N þ 3; Md b ; N þ 2; Md 2xj N 1j 2j N 2; Mj þ j Nd þ 1; b; M b jj Nd Md 3; M 1; MjÞ þ 6xj N þ 1; Md 2; M þ 1j 2xj Nd 2; N ; b jj Nd N þ 1; M þ 1; Md 1j þ 2xj Nd þ 1; Md 1jð3j Nd 1; N þ 2; b j þ 2j Nd b; M 1; N þ 1; Md 1; M þ 1jÞ þ 2xj Nd þ 1; Md 1jð3j N b ; Md b ; N þ 2; Md 2; M; M þ 1j j N 1; M þ 2jÞ 4xj N b j þ jN b ; Md Md 1jðj Nd 1; N þ 1; M 1; M þ 1jÞ 4xj Nd þ 1; b j þ jN b ; Md Md 2; Mjðj Nd 1; N þ 1; M 1; M þ 1jÞ:
ð25Þ
From (18) and (20), we get X 1 b ; N þ 2; Md b ; N þ 3; Md b ; N þ 2; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1j j N 1j þ j N 2; Mj; kj j N 2 ð26aÞ
910
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
X
1 b ; N þ 2; Md þ 1; Md 2; Mj ¼ j N 2; Mj þ j Nd þ 1; Md 3; M 1; Mj þ j Nd þ 1; Md 2; M þ 1j; kj j Nd 2
X
1 b j ¼ j Nd b j j Nd b j þ j Nd kj j Nd 1; N þ 1; M 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j; 2
ð26bÞ
ð26cÞ X
1 b ; Md b ; Md b ; Md kj j N 1; M þ 1j ¼ j Nd 1; N þ 1; Md 1; M þ 1j þ j N 2; M; M þ 1j þ j N 1; M þ 2j: 2 ð26dÞ
Noting X
X 1 1 b ; N þ 2; Md b; M b j ¼ jN b ; N þ 2; Md b; M b j; kj j N kj j N 1jj N 1j 2 2
ð27aÞ
X
X 1 1 b; M b j ¼ j Nd b; M b j; kj j Nd kj j N þ 1; Md 2; Mjj N þ 1; Md 2; Mj 2 2
ð27bÞ
X
X 1 1 d d d d b b kj j N 1; N þ 1; M jj N þ 1; M 1j ¼ j N 1; N þ 1; M j kj j Nd þ 1; Md 1j; 2 2
ð27cÞ
X
X 1 1 b ; Md b ; Md 1; M þ 1jj Nd þ 1; Md 1j ¼ j N 1; M þ 1j þ 1; Md 1j: kj j N kj j Nd 2 2
ð27dÞ
Using (26) and (27), Eq. (25) becomes b;M bj gt f ft g þ xðgxx f 2gx fx þ gf xx Þ þ 2gx f ¼ 8xj Nd 1; N þ 1; N þ 2; Md 1jj N b ; Md þ 8xj N 2; M; M þ 1jj Nd þ 1; Md 1j b ; N þ 2; Md b j þ 8xj Nd b;M bj 8xj N 1jj Nd 1; N þ 1; M þ 1; Md 2; M þ 1jj N b jj Nd þ 8xj Nd 1; N þ 2; M þ 1; Md 1j b ; Md 8xj Nd þ 1; Md 2; Mjj N 1; M þ 1j:
ð28Þ
It is readily checked that Eq. (28) is equal to zero. So, the proof of Eq. (4a) is completed. Eq. (4b) can be proved similarly. Making use of (17) gives b; M b jj Nd b j þ jN b; M b jðj Nd b j 2j Nd 2; N; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1jÞ fxx f fx2 þ gh ¼ j N b; M b jðj N b ; Md b ; Md bj þ jN 2; M; M þ 1j þ j N 1; M þ 2jÞ ðj Nd 1; N þ 1; M b ; Md jN 1; M þ 1jÞ2 :
ð29Þ
From (18), we obtain X 1 b; M b j ¼ jN b ; Md b j; 1; M þ 1j j Nd 1; N þ 1; M kj j N 2 X
1 kj 2
2
b; M b j ¼ j Nd b j þ j Nd b j 2j Nd jN 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j b ; Md b ; Md þ jN 2; M; M þ 1j þ j N 1; M þ 2j:
Thus, we have b j þ j Nd b j 2j Nd b ; Md ðj Nd 2; N ; N þ 1; M 1; N þ 2; M 1; N þ 1; Md 1; M þ 1j þ j N 2; M; M þ 1j b ; Md b; M b j ¼ ðj N b ; Md b jÞ2 : þ jN 1; M þ 2jÞj N 1; M þ 1j j Nd 1; N þ 1; M Then, Eq. (4c) is obtained from Eqs. (29) and (30).
h
ð30Þ
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
911
4. Reduction Setting r = q* and replacing t by it in Eq. (1), where * denotes the complex conjugate, i ¼ following non-isospectral Schro¨dinger equation:
pffiffiffiffiffiffiffi 1, we obtain the
iqt þ xðqxx þ 2qjqj2 Þ þ 2qx þ 2qo1 jqj2 ¼ 0:
ð31Þ
Through the dependent variable transformation g q¼ : f Eq. (31) is rewritten as ðiDt þ xD2x Þg f ¼ 2gx f ;
ð32Þ
ð33aÞ
D2x f f ¼ 2gg :
ð33bÞ
Thus, we obtain " # 2n 2n X X X fn ¼ A1 ðlÞ exp lj ðnj þ ln xj ðtÞÞ þ lj ls hjs ; l¼0;1
X
gn ¼
A2 ðlÞ exp
" 2n X
l¼0;1
ð34aÞ
16j
j¼1
lj ðnj þ ln xj ðtÞÞ þ
2n X
# ð34bÞ
lj ls hjs ;
16j
j¼1
where nnþj ¼ nj ; xnþj ðtÞ ¼ xj ðtÞ; xj;t ðtÞ ¼ 2ixj ðtÞk j ðtÞ; ehj;nþs ¼
1 ðk j ðtÞ þ k s ðtÞÞ2
ehjs ¼ ðk j ðtÞ k s ðtÞÞ2 ;
ðj ¼ 1; 2; . . . ; nÞ;
k j;t ðtÞ ¼ ik 2j ðtÞ;
ð35aÞ
ðj ¼ 1; . . . ; nÞ;
ð35bÞ
ðj; s ¼ 1; 2; . . . ; nÞ;
;
ð35cÞ
ehðnþjÞðnþsÞ ¼ ðk j ðtÞ k s ðtÞÞ2 ;
ðj < s ¼ 2; 3; . . . ; nÞ;
ð35dÞ
A1(l) and A2(l) take over all possible combinations of lj = 0,1 (j = 1, 2, . . ., 2n) and satisfy the condition n n n n X X X X lj ¼ lnþj ; lj ¼ 1 þ lnþj ; j¼1
j¼1
j¼1
j¼1
respectively. In the following, we deduce the double Wronskian solution of Eq. (33). Let AN·M and BN·M be two N · M matrices, 0
...
uN
ou2 ... ouN
... ... ...
oM1 u2 C C C; ... A oM 1 uN
w1 Bw B ¼B 2 @... wN
ow1 ow2 ... owN
... ... ... ...
1 oM1 w1 oM1 w2 C C C; ... A oM 1 wN
0
where
1
ou1
Bu B AN M ¼ B 2 @...
BN M
oM1 u1
u1
1 uj ¼ xj ðtÞ exp kj ðtÞx ; 2
wj ¼ rj ðtÞ exp
1 kj ðtÞx ; 2
kj;t ðtÞ ¼ k2j ;
ð36Þ
uj and wj satisfy the following conditions: uj;x ¼ 12kj uj ;
wj;x ¼ 12kj wj ;
ð37Þ
912
Y.-p. Sun et al. / Chaos, Solitons and Fractals 26 (2005) 905–912
uj;t ¼ 2ixuj;xx 2iN uj;x ; respectively. We define AðN þ1ÞðN þ1Þ e f ¼ AðN þ1ÞðN þ1Þ
wj;t ¼ 2ixwj;xx þ 2iNwj;x ;
BðN þ1ÞðN þ1Þ b; N b j; jN BðN þ1ÞðN þ1Þ
AðN þ2ÞðN þ2Þ e g ¼ 2 BN ðN þ2Þ
BðN þ2ÞN þ 1; Nd 1j: 2j Nd AN N
In terms of the determinantal properties, we have B AðN þ2ÞðN þ2Þ BðN þ2ÞN AN N ¼ 2ð1Þ3N N ðN þ2Þ e g ¼ 2 AðN þ2ÞðN þ2Þ BðN þ2ÞN BN ðN þ2Þ AN N AN N AN N BN ðN þ2Þ BN ðN þ2Þ 6N 1; Nd þ 1j: ¼ 2ð1Þ ¼ 2 2j Nd BðN þ2ÞN AðN þ2ÞðN þ2Þ BðN þ2ÞN AðN þ2ÞðN þ2Þ
ð38Þ
ð39Þ
ð40Þ
ð41Þ
Letting M = N in Eq. (22) yields g ¼ 2j Nd þ 1; Nd 1j;
b; N b j; f ¼ jN
h ¼ 2j Nd 1; Nd þ 1j:
So, we have the following result. Theorem 2. The non-isospectral Schro¨dinger equation (33) has the double Wronskian solution e g ¼ 2j Nd þ 1; Nd 1j;
b; N b j; fe ¼ j N
e 1; Nd þ 1j; g ¼ 2j Nd
ð42Þ
where uj and wj satisfy the conditions (36)–(38).
5. Conclusions In this paper, we presented the N-soliton solutions and the double Wronskian solution of the non-isospectral AKNS equation (1) through the Hirota method and the Wronskian technique, respectively. Further, the non-isospectral Schro¨dinger equation (31) and its multi-soliton solutions are obtained by reducing.
Acknowledgments The authors are very grateful to Professor M. Wadati for his ardent guidance and help. The authors express their sincere thanks to Dr. Zhang DJ, Zhang Y and Ning TK as well. This project is supported by the National Science Foundation of China (10371070) and the Special Funds for Major Specialities of Shanghai Education Committee.
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