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Generalized double Wronskian solutions of the third-order isospectral AKNS equation
Chaos, Solitons and Fractals 39 (2009) 926–935 www.elsevier.com/locate/chaos
Generalized double Wronskian solutions of the third-order isospectral AKNS equation Fu-mei Yin a
a,b,*
, Peng Chen b, Guang-sheng Wang b, Deng-yuan Chen
b
Institute of Applied Math, Naval Aeronautical Engineering Institute, Yantai 264001, People’s Republic of China b Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China Accepted 22 January 2007
Communicated by Prof. M. Wadati
Abstract The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The study of exact solutions of nonlinear evolution equations plays an important role in soliton theory. For this end, various methods have been developed, such as the inverse scattering transform [1,2], the Darboux transformation [3,4], the Hirota method [5,6] and the Wronskian technique [6–8]. Among these methods, the Wronskian technique is a powerful tool to construct exact solutions for nonlinear evolution equations [9–17]. This technique admits direct and simple verification of the solutions. In this paper, we would like to apply the Wronskian technique to the following third-order isospectral AKNS equation: qt ¼ qxxx 6qrqx ; ð1aÞ rt ¼ rxxx 6qrrx ð1bÞ and verify that it possesses generalized double Wronskian solutions. The Lax pair of Eq. (1) is g q ; /x ¼ M/; M ¼ r g 4g3 þ 2qrg þ qrx rqx 4qg2 2qx g þ qxx 2q2 r ; /t ¼ N /; N ¼ 4rg2 þ 2rx g þ rxx 2qr2 4g3 2qrg qrx þ rqx
ð2aÞ ð2bÞ
where q and r are potential functions, g is a spectral parameter. * Corresponding author. Address: Institute of Applied Math, Naval Aeronautical Engineering Institute, Yantai 264001, People’s Republic of China. E-mail address: [email protected] (F.-m. Yin).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.060
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
927
Through the dependent variable transformation q¼
g ; f
r¼
h : f
Eq. (1) can be transformed into the following bilinear form: D2x f f ¼ 2gh;
ð3aÞ
D3x Þg D3x Þh
f ¼ 0;
ð3bÞ
f ¼ 0;
ð3cÞ
ðDt ðDt
where D is the well-known Hirota bilinear operator [5]. The paper is organized as follows. In Section 2, generalized double Wronsikan solutions of the AKNS equation are verified. In Section 3, soliton solutions and rational solutions of it are obtained. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction. In Section 4, Matveev solutions for the AKNS equation are provided. In Sections 5 and 6, complexitons and interaction solutions are given, respectively.
2. Generalized double Wronskian solutions of Eq. (3) Let us first specify some properties of the Wronskian determinant. As we all know, the double Wronskian determinant [8] is W N ;M ðu; wÞ ¼ detðu; ox u; . . . ; oxN 1 u; w; ox w; . . . ; oxM1 wÞ; where u ¼ ðu1 ; u2 ; . . . ; uN þM ÞT ; w ¼ ðw1 ; w2 ; . . . ; wN þM ÞT . In the most known examples, the two determinantal identities are often used [6]. The one is jQ; a; bkQ; c; dj jQ; a; ckQ; b; dj þ jQ; a; dkQ; b; cj ¼ 0;
ð4Þ
where Q is an N ðN 2Þ matrix and a, b, c and d represent N-dimensional column vectors. The other is N X
ja1 ; . . . ; aj1 ; baj ; ajþ1 ; . . . ; aN j ¼
j¼1
N X
bj ja1 ; . . . ; aN j;
ð5Þ
j¼1
where aj ð1 6 j 6 N Þ are N-dimensional column vectors and baj denotes ðb1 a1j ; b2 a2j ; . . . ; bN aNj ÞT . To obtain generalized double Wronskian solutions, we give the following lemma. Lemma 1. Assume that P ¼ ðpij Þ is an l l operator matrix and its entries pij are differential operators. B ¼ ðbij Þ is an l lfunction matrix with column vector set bi and row vector set b0j ði ¼ 1; 2; . . . ; l; j ¼ 1; 2; . . . ; lÞ, then 0 b1 . .. l l X X p0j b0j ; ð6Þ jb1 ; . . . ; pi bi ; . . . ; bl j ¼ i¼1 j¼1 . . . b0 l where pi bi ¼ ðp1i b1i ; p2i b2i ; . . . ; pli bli ÞT ; p0j b0j ¼ ðpj1 bj1 ; pj2 bj2 ; . . . ; pjl bjl Þ [18]. Theorem 1. The isospectral AKNS equation (3) has the double Wronskian solution f ¼ W N þ1;Mþ1 ðu; wÞ;
g ¼ 2W N þ2;M ðu; wÞ;
h ¼ 2W N ;Mþ2 ðu; wÞ;
ð7Þ
where uj and wj enjoy the conditions uj;x ¼ k j uj ;
wj;x ¼ k j wj ;
uj;t ¼ 4uj;xxx ;
wj;t ¼ 4wj;xxx ;
1 6 j 6 N þ M þ 2;
ð8aÞ
16j6N þM þ2
ð8bÞ
and kj is an arbitrary real constant [19]. Employing the Wronskian technique, we have the following theorem.
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F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
Theorem 2. The isospectral AKNS equation (3) has the generalized double Wronskian solution (7), where u and w satisfy the following conditions: ux ¼ Au; ut ¼ 4uxxx ;
wx ¼ Aw; wt ¼ 4wxxx ;
u ¼ ðu1 ; u2 ; . . . ; uN þMþ2 ÞT ; w ¼ ðw1 ; w2 ; . . . ; wN þMþ2 Þ
ð9aÞ
T
ð9bÞ
and A ¼ ðaij Þ is an ðN þ M þ 2Þ ðN þ M þ 2Þ arbitrary real matrix independent of x and t. Proof. In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [6,7], then we have b; M b j; g ¼ 2j Nd f ¼ jN þ 1; Md 1j; h ¼ 2j Nd 1; Md þ 1j; d d b j þ jN b ; M 1; M þ 1j; fx ¼ j N 1; N þ 1; M 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 d d b b þj N ; M 2; M; M þ 1j þ j N ; M 1; M þ 2j; b j þ 2j Nd b j þ 3j Nd 3; N 1; N ; N þ 1; M 2; N ; N þ 2; M 2; N; N þ 1; fxxx ¼ j Nd d d d d b j þ 3j N 1; N þ 2; M 1; M þ 1j þ 3j Nd M 1; M þ 1j þ j N 1; N þ 3; M 1;
ð10Þ
b ; Md N þ 1; Md 2; M; M þ 1j þ 3j Nd 1; N þ 1; Md 1; M þ 2j þ j N 3; M 1; d d b b M; M þ 1j þ 2j N ; M 2; M; M þ 2j þ j N ; M 1; M þ 3j; b j j Nd b j þ j Nd bj 3; N 1; N ; N þ 1; M 2; N; N þ 2; M 1; N þ 3; M ft ¼ 4ðj Nd d d d b ; M 3; M 1; M; M þ 1j j N b ; M 2; M; M þ 2j þ j N b ; M 1; M þ 3jÞ: þj N Similarly, we can get the various derivatives of g and h. Letting ox 1 6 i 6 N þ M þ 2; 1 6 j 6 N þ 1; pij ¼ ox 1 6 i 6 N þ M þ 2; N þ 2 6 j 6 N þ M þ 2; from (6), we have NX þMþ2
j ox uj ; ox ðox uj Þ; . . . ; ox ðoNx uj Þ; ox wj ; ox ðox wj Þ; . . . ; ox ðoM x wj Þj
j¼1
b j þ jN b ; Md ¼ j Nd 1; N þ 1; M 1; M þ 1j: According to (9a), the left hand side of (11) is equal to NX þMþ2 N þMþ2 N þM N þMþ2 N þMþ2 Xþ2 X X X N M ajl ul ; . . . ; ajl ox ul ; ajl wl ; . . . ; ajl ox wl ¼ j¼1 l¼1 l¼1 l¼1 l¼1
ð11Þ
NX þMþ2
! b; M b j ðtr AÞj N b; M b j: ajj j N
j¼1
i.e. b; M b j ¼ j Nd b j þ jN b ; Md ðtr AÞj N 1; N þ 1; M 1; M þ 1j:
ð12aÞ
Similarly, we can get b ; N þ 2; Md ðtr AÞj Nd þ 1; Md 1j ¼ j N 1j þ j Nd þ 1; Md 2; Mj;
ð12bÞ
b; M b j ¼ j Nd b j þ j Nd bj 2; N ; N þ 1; M 1; N þ 2; M ðtr AÞ j N 2
b ; Md b ; Md 2j 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;
ð12cÞ
b ; N þ 3; Md ðtr AÞ j Nd þ 1; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1j þ j N 1j 2
b ; N þ 2; Md 2j N 2; Mj þ j Nd þ 1; Md 3; M 1; Mj þ j Nd þ 1; Md 2; M þ 1j;
ð12dÞ
b ; N þ 2; Md ðtr AÞj Nd þ 1; Md 3; M 1; Mj ¼ j N 3; M 1; Mj þ j Nd þ 1; Md 4; M 2; M 1; Mj þ j Nd þ 1; Md 3; M 1; M þ 1j;
ð12eÞ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
929
Making use of (4), we have 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 ffxx 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 :
ð13Þ
Noting b; M b jðtr AÞ2 j N b; M b j ¼ ððtr AÞj N b; M b jÞ2 ; jN
ð14aÞ
b; M b jðtr AÞ2 j Nd b; M b jðtr AÞj Nd jN þ 1; Md 1j ¼ ðtr AÞj N þ 1; Md 1j;
ð14bÞ
b; M b j ¼ ðtr AÞj Nd b; M b j; j Nd þ 1; Md 1jðtr AÞ2 j N þ 1; Md 1jðtr AÞj N
ð14cÞ
b; M b j ¼ jN b; M b jðtr AÞj Nd j Nd þ 1; Md 1jðtr AÞj N þ 1; Md 1j;
ð14dÞ
b; M b jðtr AÞ3 j Nd b; M bj jN þ 1; Md 1j ¼ j Nd þ 1; Md 1jðtr AÞ3 j N b; M b jðtr AÞj Nd b; M b jðtr AÞ2 j Nd ¼ ðtr AÞ2 j N þ 1; Md 1j ¼ ðtr AÞj N þ 1; Md 1j;
ð14eÞ
j Nd þ 1; Md 1jðtr AÞj Nd 1; N þ 1; Md 1; M þ 1j ¼ j Nd 1; N þ 1; Md 1; M þ 1jðtr AÞj Nd þ 1; Md 1j;
ð14fÞ
b; M b jðtr AÞj N b ; N þ 2; Md b ; N þ 2; Md b; M b j; jN 2; Mj ¼ j N 2; Mjðtr AÞj N
ð14gÞ
b; M b jðtr AÞj Nd b; M b j; jN þ 1; Md 3; M 1; Mj ¼ j Nd þ 1; Md 3; M 1; Mjðtr AÞj N
ð14hÞ
b; M b jðtr AÞj Nd b; M b j; jN 1; N þ 1; N þ 2; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1jðtr AÞj N
ð14iÞ
b j ¼ j Nd b jðtr AÞj Nd j Nd þ 1; Md 1jðtr AÞj Nd 2; N ; N þ 1; M 2; N ; N þ 1; M þ 1; Md 1j;
ð14jÞ
b ; Md b ; Md j Nd þ 1; Md 1jðtr AÞj N 2; M; M þ 1j ¼ j N 2; M; M þ 1jðtr AÞj Nd þ 1; Md 1j:
ð14kÞ
Utilizing (12c) and (14a), it is easy to see that Eq. (13) is equal to zero. Thus, the proof of Eq. (3a) is completed. h Using (4), (12) and (14), we work out from (3b) f ðgt gxxx Þ gðft gxxx Þ þ 3ðgxx fx gx fxx Þ b; M b jðj N b ; N þ 2; Md ¼ 12½j N 2; M þ 1j j Nd þ 1; Md 3; M 1; M þ 1j þ j Nd 2; N; N þ 1; N þ 2; Md 1j j Nd 1; N þ 1; N þ 2; Md 2; MjÞ bj j Nd þ 1; Md 1jðj Nd 1; N þ 2; Md 1; M þ 1j j Nd 2; N ; N þ 2; M b ; Md j Nd 1; N þ 1; Md 2; M; M þ 1j þ j N 3; M 1; M; M þ 1jÞ b ; N þ 2; Md b ; Md b jÞ þ jN 2; Mjðj N 1; M þ 1j þ j Nd 1; N þ 1; M b ; Md þ jN 1; M þ 1jðj Nd þ 1; Md 3; M 1; Mj þ j Nd 1; N þ 1; N þ 2; Md 1jÞ b jj Nd þ j Nd 1; N þ 1; M þ 1; Md 2; M þ 1j j Nd 1; N þ 1; Md 1; M þ 1jðj Nd þ 1; Md 2; Mj b ; N þ 2; Md b ; N þ 2; Md b j þ jN b ; Md þ jN 1jÞ j N 1jðj Nd 2; N ; N þ 1; M 2; M; M þ 1jÞ b ; N þ 2; M b j: j Nd þ 1; Md 2; Mjj N According to (4), Eq. (15) is equal to zero. Eq. (3c) can be verified similarly.
3. Soliton and rational solutions From (9), we can get u ¼ e4A
3 tAx
C;
3 tþAx
w ¼ e4A
D;
where C ¼ ðc1 ; c2 ; . . . ; cN þMþ2 ÞT and D ¼ ðd 1 ; d 2 ; . . . ; d N þMþ2 ÞT are arbitrary real constant vectors. By the Taylor series expansion, we have
ð15Þ
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F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
2 s 3 ½3 s l l s3l 1 X X ð1Þ 4 t x 4 5As C; u¼e C¼ l!ðs 3lÞ! l¼0 s¼0 2 s 3 ½3 1 l l s3l X X 4 t x 3 4 5As D: w ¼ e4A tþAx D ¼ l!ðs 3lÞ! l¼0 s¼0 4A3 tAx
ð16aÞ
ð16bÞ
If 0 B B A¼B B @
1
0
k1
C C C; C A
k2 ..
.
0
ðk i –k j ; i–jÞ;
k N þMþ2
we can obtain soliton solutions of Eq. (3), where 3
3
uj ¼ e4k j tkj x cj ; If
wj ¼ e4kj tþkj x d j ;
0
0 B1 B A¼B B @
0
.
..
0
1
. 0
ð17Þ
1 C C C C A
0 ..
ðj ¼ 1; 2; . . . ; N þ M þ 2Þ:
;
ðN þM þ2ÞðN þMþ2Þ
it is easy to know that AN þM þ2 ¼ 0. Thus (16) can be truncated as 2 s 3 2 s 3 ½3 ½3 NX þMþ1 X NX þMþ1 X ð1Þs 4l tl xs3l 5 s 4l tl xs3l 5 s 4 4 A C; w ¼ A D: u¼ l!ðs 3lÞ! l!ðs 3lÞ! l¼0 l¼0 s¼0 s¼0
ð18Þ
So rational solutions of Eq. (3) in double Wronskian form are obtained, where 3 ½j1 3 X x2 x ð1Þj1 4l tl xj13l c1 ; uj ¼ cj xcj1 þ cj2 þ 4t cj3 þ þ 2 3! l!ðj 1 3lÞ! l¼0
ð19aÞ
3 ½j1 3 X x2 x 4l tl xj13l wj ¼ d j þ xd j1 þ d j2 þ þ 4t d j3 þ þ d1: 2 3! l!ðj 1 3lÞ! l¼0
ð19bÞ
In (19), taking c1 ¼ d 1 ¼ 1;
cx ¼ d x ¼ 0;
ðx ¼ 2; 3; . . . ; N þ M þ 2Þ;
ð20Þ
then (19) becomes ½j1 3 X ð1Þj1 4l tl xj13l ; uj ¼ l!ðj 1 3lÞ! l¼0
wj ¼
½j1 3 X l¼0
4l tl xj13l : l!ðj 1 3lÞ!
ð21Þ
Thus, we obtain some special rational solutions of Eq. (1) as follows: 1 2ðx3 þ 6tÞ ; ; r ¼ 2; q ¼ r ¼ 4 2 x x 12xt 2ðx4 12xtÞ 3 3ðx4 þ 24xtÞ q¼ ; r¼ ; q¼ ; r ¼ 2: x3 þ 6t 2ðx3 þ 6tÞ ðx3 12tÞ2 q¼
When q ¼ r ¼ v, Eq. (1) becomes the mKdV equation vt ¼ vxxx 6v2 vx : Similarly, taking q ¼ u; r ¼ 2, Eq. (1) can be reduced to the following KdV equation
ð22Þ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
ut ¼ uxxx 12uux :
ð23Þ
From (21), we know that uj ¼ ð1Þ (I) : When N ¼ M, v¼
931
j1
wj . So we have the following results:
2W N þ2;N ðu; wÞ 2W N ;N þ2 ðu; wÞ ¼ N þ1;N þ1 ðu; wÞ W N þ1;N þ1 ðu; wÞ W
are rational solutions of Eq. (22). (II): When 2f N ¼ 2n þ 1; M ¼ 2n; h¼ 2f N ¼ 2ðn þ 1Þ; M ¼ 2n þ 1; we can obtain rational solutions of Eq. (23), where n is an arbitrary non-negative integer.
4. Matveev solutions Let A be Jordan matrix 0 J ðk1 Þ B J ðk 2 Þ B A¼B .. B @ . 0
1
0
C C C C A J ðk p Þ
:
ð24Þ
ðN þMþ2ÞðN þM þ2Þ
Without loss of generality, we observe the following Jordan block (dropping the subscript of k) 0
0
k
B1 k B J ðkÞ ¼ B .. B @ . 0
.. 1
1
0
C C C C A
. k
¼ kI li þ Eli ;
0
0
B1 0 B E li ¼ B .. B @ . 0
li li
..
C C C C A
.
1
1
0
;
li li
where I li denotes an li li unit matrix. We have J ðkÞs ¼ ðkI li þ Eli Þs ¼ k s I li þ sk s1 Eli þ þ C js k sj Ejli þ þ Esli 1 1 1 ¼ I li þ Eli ok þ E2li o2k þ þ Ejli ojk þ þ Esli osk k s ; 2! j! s! i.e. 0
J ðkÞs ¼ T k k s ;
B B B B B Tk ¼ B B B B B @
1 ok 1 2 o 2 k
0 1 ok
1 3 o 6 k
1 2 o 2 k
..
1 oli 1 ðli 1Þ! k
.
C C C C C C: C C C C A
1 ok .. . 1 3 o 6 k
1 ..
. 1 2 o 2 k
..
. ok
1
1
Substituting (24) into (16), we get c1 3 uj ðkÞ ¼ okj1 þ þ cj1 ok þ cj e4k tkx ; ðj ¼ 1; 2; . . . ; li Þ; ðj 1Þ! d1 3 okj1 þ þ d j1 ok þ d j e4k tþkx ; ðj ¼ 1; 2; . . . ; li Þ: wj ðkÞ ¼ ðj 1Þ! Thus, Matveev solutions of Eq. (3) are constructed, where
ð25aÞ ð25bÞ
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F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
u ¼ ðu1 ðk 1 Þ; . . . ; ul1 ðk 1 Þ; u1 ðk 2 Þ; . . . ; ul2 ðk 2 Þ; ; u1 ðk p Þ; . . . ; ulp ðk p ÞÞT ; ðl1 þ l2 þ þ lp ¼ N þ M þ 2Þ; w ¼ ðw1 ðk 1 Þ; . . . ; wl1 ðk 1 Þ; w1 ðk 2 Þ; ; wl2 ðk 2 Þ; ; w1 ðk p Þ; . . . ; wlp ðk p ÞÞT ; ðl1 þ l2 þ þ lp ¼ N þ M þ 2Þ: Taking ðN ; MÞ ¼ ð0; 0Þ; ð1; 0Þ and ½ð12k 2 t xÞ2 24kt 4k3 tkx ; ð12k t xÞe ; e e 2 !T 2 2 2 4k 3 tþkx 4k 3 tþkx ½ð12k t þ xÞ þ 24kt 4k 3 tþkx ; ð12k t þ xÞe ; ; e e 2 4k 3 tkx
u¼
w¼
!T
4k 3 tkx
2
;
we obtain two special Matveev solutions of Eq. (1) as follows: 3
3
e2ð4k tþkxÞ e2ð4k tþkxÞ ; r ¼ ; 12k 2 t þ x 12k 2 t þ x 3 3 e2ð4k tþkxÞ 2e2ð4k tþkxÞ ½ð12k 2 t þ xÞ2 12kt ; r ¼ : q¼ ð12k 2 t þ xÞ2 þ 12kt ð12k 2 t þ xÞ2 þ 12kt
q¼
Similarly, when ðN ; MÞ ¼ ð1; 0Þ and 3
3
3
T u ¼ ðe4k 1 tk1 x ; ð12k 21 t xÞe4k 1 tk1 x ; e4k2 tk2 x Þ ; 3
3
3
w ¼ ðe4k1 tþk1 x ; ð12k 21 t þ xÞe4k1 tþk1 x ; e4k2 tþk 2 x ÞT ; a direct calculation gives 3
3
3
3
3
f ¼ e2ð4k1 tþk1 xÞ e4k 2 tk2 x ½2ðk 2 k 1 Þð12k 21 t þ xÞe2ð4k1 tþk1 xÞ þ e2ð4k1 tþk1 xÞ e2ð4k 2 tþk2 xÞ ; 3
3
g ¼ 2e2ð4k1 tþk1 xÞ e4k 2 tk2 x ðk 1 k 2 Þ2 ; 3
3
3
3
3
h ¼ 2e2ð4k1 tþk1 xÞ e4k2 tþk2 x ½2ðk 2 k 1 Þð12k 21 t þ xÞe2ð4k1 tþk 1 xÞ þ e2ð4k2 tþk2 xÞ e2ð4k 1 tþk1 xÞ :
5. Complexitons Firstly, let us consider the simplest case 0 1 a b ; ¼ aI 2 þ br2 ; r2 ¼ A¼ 1 0 b a
ð26Þ
where a; b are real constants. Substituting (26) into (16) yields 3 3ab2 ÞtþaxI
u ¼ e½4ða
2
3 3ab
w ¼ e½4ða
ÞtþaxI 2
2
2 bb3 Þtþbxr
e½4ð3a
2 bb
e½4ð3a
3
Þtþbxr2
2
D;
C;
C ¼ ðc1 ; c2 ÞT ; D ¼ ðd 1 ; d 2 ÞT ;
u ¼ ðu1 ; u2 ÞT ; w ¼ ðw1 ; w2 ÞT :
Expanding the above u and w and making use of r22 ¼ I 2 , we get 3 3ab2 Þtþax
u ¼ e½4ða
½4ða3 3ab2 Þtþax
w¼e
fcos½4ð3a2 b b3 Þt þ bxI 2 sin½4ð3a2 b b3 Þt þ bxr2 gC; 2
3
2
3
fcos½4ð3a b b Þt þ bxI 2 þ sin½4ð3a b b Þt þ bxr2 gD:
Secondly, we consider an real Jordan matrix A 1 0 0 J1 0 Ai C B BI J 2 C B B 2 Ai C; J i ¼ B A¼B .. .. .. C B B A @ @ . . . 0 Jm 0 I2
0
ð27bÞ
1 C C C; C A
Ai
ð27aÞ
Ai ¼
ai bi
bi ai
ð28Þ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
933
where ai ; bi are real constants. Ji can be expressed as J i ¼ A0 þ E0 ; 0 Ai B Ai B A0 ¼ B B @ 0
0 ..
1
0
C C C C A
. Ai
;
0 BI B 2 E0 ¼ B B @
0
.
0
2li 2li
C C C C A
0 ..
..
.
I2
1
0
;
2li 2li
then we have s J si ¼ ðA0 þ E0 Þ ¼ A0s þ sA0s1 E0 þ þ C js A0sj E0j þ þ E0s 1 1 ¼ I 2li þ E0 oai þ þ E0j ojai þ þ E0s osai A0s : j! s!
ð29Þ
n n1 Using oai Ani ¼ oai ðai I 2 þ bi r2 Þ ¼ nðai I 2 þ bi r2 Þ ; ðn ¼ 1; 2; 3; . . . ; sÞ, then (27b) can be written as 1 0 I2 0 C B I2 I 2 oai C B C B 1 2 C B 2 I 2 oai I 2 oai I2 0s s J i ¼ T oai A ; T oai ¼ B C: C B . . . . C B . . . . . . . . A @ 1 I oli 1 ðli 1Þ! 2 ai
...
1 I o2 2 2 ai
I 2 olphai
ð30Þ
I2
Substituting (28) into (16) and taking advantage of (27), we can obtain the following forms: ( ! j X uj1 ðai Þ 1 2 3 uj ðai Þ ¼ e½4ðai 3ai bi Þtþai x : ¼ oajs i ðj sÞ! uj2 ðai Þ s¼1 !) cs1 cos½4ð3a2i bi b3i Þt þ bi x þ cs2 sin½4ð3a2i bi b3i Þt þ bi x ; cs1 sin½4ð3a2i bi b3i Þt þ bi x þ cs2 cos½4ð3a2i bi b3i Þt þ bi x ðj ¼ 1; 2; 3; . . . ; li Þ; ! ( j X wj1 ðai Þ 1 2 3 js ¼ oai e½4ðai 3ai bi Þtþai x : wj ðai Þ ¼ ðj sÞ! wj2 ðai Þ s¼1 d s1 cos½4ð3a2i bi b3i Þt þ bi x d s2 sin½4ð3a2i bi b3i Þt þ bi x d s1 sin½4ð3a2i bi b3i Þt þ bi x þ d s2 cos½4ð3a2i bi b3i Þt þ bi x
ð31aÞ
!) ;
ðj ¼ 1; 2; 3; . . . ; li Þ;
ð31bÞ
where cs1 ; cs2 ; d s1 ; d s2 are real constants. Therefore, we obtain complexitons of Eq. (3) in double Wronskian form, where u ¼ ðu1 ða1 ÞT ; . . . ; ul1 ða1 ÞT ; u1 ða2 ÞT ; . . . ; ul2 ða2 ÞT ; . . . ; u1 ðalm ÞT ; . . . ; ulm ðalm ÞT ÞT ; ðl1 þ l2 þ þ lm ¼ N þ M þ 2Þ; w ¼ ðw1 ða1 ÞT ; . . . ; wl1 ða1 ÞT ; w1 ða2 ÞT ; . . . ; wl2 ða2 ÞT ; . . . ; w1 ðalm ÞT ; . . . ; wlm ðalm ÞT ÞT ; ðl1 þ l2 þ þ lm ¼ N þ M þ 2Þ: For example, letting ðN ; MÞ ¼ ð0; 0Þ and a1 ¼ 0; c11 ¼ 1; c12 ¼ 0; d 11 ¼ d 12 ¼ q¼r¼v¼
2b1 ; sin½2ð4b31 t þ b1 xÞ þ h
pffiffi 2 , 2
then we have
tan h ¼ 1:
The singularity of the solution can be reflected in Fig. 1. In Fig. 1b, white points represent positive infinity, while dark points represent negative infinity. The grey stripes and dark ones denote the alternate changes of positive and negative amplitudes as it is described in Fig. 1a. Because the solution is periodic, so the width of each stripe is unchanged and it moves with constant speed 4b21 .
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F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
a
b
t=0
density graphics 6
30 4 20 2
10 -4
-6
-2
4
2
6
-10
0 -2
-20
-4
-30 -6 -6
-4
-2
0
2
4
6
Fig. 1. (a) t = 0. (b) Density image of v for x 2 ½2p; 2p and t 2 ½6; 6 and plot range of ½25; 25, b1 ¼ 1; h ¼ p4.
On the other hand, for oai Ani ¼ r2 obi Ani ;the partial derivative with respect to ai can be replaced by the partial derivative with respect to bi in (29)–(31).
6. Interaction solutions In what follows, we give a few double Wronskian interaction solutions of Eq. (1). Letting ðN ; MÞ ¼ ð1; 0Þ; ð0; 1Þ and choosing u ¼ ð1; x; e4k
3
tkx T
w ¼ ð1; x; e4k
Þ ;
3
tþkx T
Þ ;
then we get 3
2k 2 2ð2kx 1Þe2ð4k tþkxÞ þ 2 ; r ¼ ; 3 3 2kx þ 1 e2ð4k tþkxÞ 2kx þ 1 e2ð4k tþkxÞ 3 2ð2kx þ 1Þe2ð4k tþkxÞ 2 2k 2 ; r¼ : q¼ 3 3 2ð4k tþkxÞ 2kx 1 þ e2ð4k tþkxÞ 2kx 1 þ e
q¼
ð32Þ ð33Þ
When ðN; MÞ ¼ ð1; 1Þ, taking u ¼ ð1; x; e4k w ¼ ð1; x; e
3
3
tkx
4k tþkx
; ð12k 2 t xÞe4k 2
; ð12k t þ xÞe
3
3
tkx T
Þ ;
4k tþkx T
Þ ;
we have 3
q ¼ 4k
ðkx þ 1Þe2ð4k tþkxÞ þ 12k 3 t þ kx 1 ; 3 e2ð4k tþkxÞ þ e2ð4k tþkxÞ 2 4k 2 xð12k 2 t þ xÞ 3
ð34aÞ
3
r ¼ 4k
e2ð4k
3
ðkx 1Þe2ð4k tþkxÞ þ 12k 3 t þ kx þ 1 : tþkxÞ þ e2ð4k 3 tþkxÞ 2 4k 2 xð12k 2 t þ xÞ
ð34bÞ
If we choose u ¼ ð1; x; en cos g; en sin gÞT ;
w ¼ ð1; x; en cos g; en sin gÞT ;
then, q¼
ða2 b2 Þ sin 2g 2ab cos 2g þ 2b½ða2 þ b2 Þx þ ae2n ; ða2 þ b2 Þx sin 2g þ bðcos 2g cosh 2nÞ
ð35aÞ
r¼
ða2 b2 Þ sin 2g þ 2ab cos 2g þ 2b½ða2 þ b2 Þx ae2n ; ða2 þ b2 Þx sin 2g þ bðcos 2g cosh 2nÞ
ð35bÞ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
935
where n ¼ 4ða3 3ab2 Þt þ ax; g ¼ 4ð3a2 b b3 Þt þ bx: Of course, we can get more general interaction solutions among three or more kinds of solutions such as soliton solutions, rational solutions, Matveev solutions and complexitons.
7. Conclusions In this paper, by means of the Wronskian technique, we have verified Eq. (3) which possesses generalized double Wronskian solutions. These solutions are rational, Matveev, complexitons and interaction solutions. Further, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.
Acknowledgements The authors are very grateful to Professor M.Wadati for his help. The authors express their sincere thanks to Dr. Sun YP for his valuable advice. This project is supported by National Science Foundation of China (10371070) and the Special Found for Major Specialities of Shanghai Education Committee.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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 transformations and solitons. Berlin: Springer-Verlag; 1991. Hirota R. Phys Lett A 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;99:281. Nimmo JJC, Freeman NC. Phys Lett A 1983;96:443. Matveev VB. Phys Lett A 1992;166:205. Matveev VB. Phys Lett A 1992;166:209. Ma WX. Phys Lett A 2002;301:35. Ma WX. Chaos, Solitons & Fractals 2004;19:163. Ma WX, You YC. Trans Amer Math Soc 2005;357:1753. Zhang DJ. Chaos, Solitons & Fractals 2005;23:1333. Sun YP, Bi JB, Chen DY. Chaos, Solitons & Fractals 2005;26:905. Bi JB, Sun YP, Chen DY. Common Theor Phys 2006;45:398. Zhang DJ. 2006 preprint. Liu QM. J Phys Soc Jpn 1990;59:3520.
Generalized double Wronskian solutions of the third-order isospectral AKNS equation Fu-mei Yin a
a,b,*
, Peng Chen b, Guang-sheng Wang b, Deng-yuan Chen
b
Institute of Applied Math, Naval Aeronautical Engineering Institute, Yantai 264001, People’s Republic of China b Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China Accepted 22 January 2007
Communicated by Prof. M. Wadati
Abstract The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The study of exact solutions of nonlinear evolution equations plays an important role in soliton theory. For this end, various methods have been developed, such as the inverse scattering transform [1,2], the Darboux transformation [3,4], the Hirota method [5,6] and the Wronskian technique [6–8]. Among these methods, the Wronskian technique is a powerful tool to construct exact solutions for nonlinear evolution equations [9–17]. This technique admits direct and simple verification of the solutions. In this paper, we would like to apply the Wronskian technique to the following third-order isospectral AKNS equation: qt ¼ qxxx 6qrqx ; ð1aÞ rt ¼ rxxx 6qrrx ð1bÞ and verify that it possesses generalized double Wronskian solutions. The Lax pair of Eq. (1) is g q ; /x ¼ M/; M ¼ r g 4g3 þ 2qrg þ qrx rqx 4qg2 2qx g þ qxx 2q2 r ; /t ¼ N /; N ¼ 4rg2 þ 2rx g þ rxx 2qr2 4g3 2qrg qrx þ rqx
ð2aÞ ð2bÞ
where q and r are potential functions, g is a spectral parameter. * Corresponding author. Address: Institute of Applied Math, Naval Aeronautical Engineering Institute, Yantai 264001, People’s Republic of China. E-mail address: [email protected] (F.-m. Yin).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.060
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
927
Through the dependent variable transformation q¼
g ; f
r¼
h : f
Eq. (1) can be transformed into the following bilinear form: D2x f f ¼ 2gh;
ð3aÞ
D3x Þg D3x Þh
f ¼ 0;
ð3bÞ
f ¼ 0;
ð3cÞ
ðDt ðDt
where D is the well-known Hirota bilinear operator [5]. The paper is organized as follows. In Section 2, generalized double Wronsikan solutions of the AKNS equation are verified. In Section 3, soliton solutions and rational solutions of it are obtained. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction. In Section 4, Matveev solutions for the AKNS equation are provided. In Sections 5 and 6, complexitons and interaction solutions are given, respectively.
2. Generalized double Wronskian solutions of Eq. (3) Let us first specify some properties of the Wronskian determinant. As we all know, the double Wronskian determinant [8] is W N ;M ðu; wÞ ¼ detðu; ox u; . . . ; oxN 1 u; w; ox w; . . . ; oxM1 wÞ; where u ¼ ðu1 ; u2 ; . . . ; uN þM ÞT ; w ¼ ðw1 ; w2 ; . . . ; wN þM ÞT . In the most known examples, the two determinantal identities are often used [6]. The one is jQ; a; bkQ; c; dj jQ; a; ckQ; b; dj þ jQ; a; dkQ; b; cj ¼ 0;
ð4Þ
where Q is an N ðN 2Þ matrix and a, b, c and d represent N-dimensional column vectors. The other is N X
ja1 ; . . . ; aj1 ; baj ; ajþ1 ; . . . ; aN j ¼
j¼1
N X
bj ja1 ; . . . ; aN j;
ð5Þ
j¼1
where aj ð1 6 j 6 N Þ are N-dimensional column vectors and baj denotes ðb1 a1j ; b2 a2j ; . . . ; bN aNj ÞT . To obtain generalized double Wronskian solutions, we give the following lemma. Lemma 1. Assume that P ¼ ðpij Þ is an l l operator matrix and its entries pij are differential operators. B ¼ ðbij Þ is an l lfunction matrix with column vector set bi and row vector set b0j ði ¼ 1; 2; . . . ; l; j ¼ 1; 2; . . . ; lÞ, then 0 b1 . .. l l X X p0j b0j ; ð6Þ jb1 ; . . . ; pi bi ; . . . ; bl j ¼ i¼1 j¼1 . . . b0 l where pi bi ¼ ðp1i b1i ; p2i b2i ; . . . ; pli bli ÞT ; p0j b0j ¼ ðpj1 bj1 ; pj2 bj2 ; . . . ; pjl bjl Þ [18]. Theorem 1. The isospectral AKNS equation (3) has the double Wronskian solution f ¼ W N þ1;Mþ1 ðu; wÞ;
g ¼ 2W N þ2;M ðu; wÞ;
h ¼ 2W N ;Mþ2 ðu; wÞ;
ð7Þ
where uj and wj enjoy the conditions uj;x ¼ k j uj ;
wj;x ¼ k j wj ;
uj;t ¼ 4uj;xxx ;
wj;t ¼ 4wj;xxx ;
1 6 j 6 N þ M þ 2;
ð8aÞ
16j6N þM þ2
ð8bÞ
and kj is an arbitrary real constant [19]. Employing the Wronskian technique, we have the following theorem.
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F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
Theorem 2. The isospectral AKNS equation (3) has the generalized double Wronskian solution (7), where u and w satisfy the following conditions: ux ¼ Au; ut ¼ 4uxxx ;
wx ¼ Aw; wt ¼ 4wxxx ;
u ¼ ðu1 ; u2 ; . . . ; uN þMþ2 ÞT ; w ¼ ðw1 ; w2 ; . . . ; wN þMþ2 Þ
ð9aÞ
T
ð9bÞ
and A ¼ ðaij Þ is an ðN þ M þ 2Þ ðN þ M þ 2Þ arbitrary real matrix independent of x and t. Proof. In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [6,7], then we have b; M b j; g ¼ 2j Nd f ¼ jN þ 1; Md 1j; h ¼ 2j Nd 1; Md þ 1j; d d b j þ jN b ; M 1; M þ 1j; fx ¼ j N 1; N þ 1; M 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 d d b b þj N ; M 2; M; M þ 1j þ j N ; M 1; M þ 2j; b j þ 2j Nd b j þ 3j Nd 3; N 1; N ; N þ 1; M 2; N ; N þ 2; M 2; N; N þ 1; fxxx ¼ j Nd d d d d b j þ 3j N 1; N þ 2; M 1; M þ 1j þ 3j Nd M 1; M þ 1j þ j N 1; N þ 3; M 1;
ð10Þ
b ; Md N þ 1; Md 2; M; M þ 1j þ 3j Nd 1; N þ 1; Md 1; M þ 2j þ j N 3; M 1; d d b b M; M þ 1j þ 2j N ; M 2; M; M þ 2j þ j N ; M 1; M þ 3j; b j j Nd b j þ j Nd bj 3; N 1; N ; N þ 1; M 2; N; N þ 2; M 1; N þ 3; M ft ¼ 4ðj Nd d d d b ; M 3; M 1; M; M þ 1j j N b ; M 2; M; M þ 2j þ j N b ; M 1; M þ 3jÞ: þj N Similarly, we can get the various derivatives of g and h. Letting ox 1 6 i 6 N þ M þ 2; 1 6 j 6 N þ 1; pij ¼ ox 1 6 i 6 N þ M þ 2; N þ 2 6 j 6 N þ M þ 2; from (6), we have NX þMþ2
j ox uj ; ox ðox uj Þ; . . . ; ox ðoNx uj Þ; ox wj ; ox ðox wj Þ; . . . ; ox ðoM x wj Þj
j¼1
b j þ jN b ; Md ¼ j Nd 1; N þ 1; M 1; M þ 1j: According to (9a), the left hand side of (11) is equal to NX þMþ2 N þMþ2 N þM N þMþ2 N þMþ2 Xþ2 X X X N M ajl ul ; . . . ; ajl ox ul ; ajl wl ; . . . ; ajl ox wl ¼ j¼1 l¼1 l¼1 l¼1 l¼1
ð11Þ
NX þMþ2
! b; M b j ðtr AÞj N b; M b j: ajj j N
j¼1
i.e. b; M b j ¼ j Nd b j þ jN b ; Md ðtr AÞj N 1; N þ 1; M 1; M þ 1j:
ð12aÞ
Similarly, we can get b ; N þ 2; Md ðtr AÞj Nd þ 1; Md 1j ¼ j N 1j þ j Nd þ 1; Md 2; Mj;
ð12bÞ
b; M b j ¼ j Nd b j þ j Nd bj 2; N ; N þ 1; M 1; N þ 2; M ðtr AÞ j N 2
b ; Md b ; Md 2j 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;
ð12cÞ
b ; N þ 3; Md ðtr AÞ j Nd þ 1; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1j þ j N 1j 2
b ; N þ 2; Md 2j N 2; Mj þ j Nd þ 1; Md 3; M 1; Mj þ j Nd þ 1; Md 2; M þ 1j;
ð12dÞ
b ; N þ 2; Md ðtr AÞj Nd þ 1; Md 3; M 1; Mj ¼ j N 3; M 1; Mj þ j Nd þ 1; Md 4; M 2; M 1; Mj þ j Nd þ 1; Md 3; M 1; M þ 1j;
ð12eÞ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
929
Making use of (4), we have 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 ffxx 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 :
ð13Þ
Noting b; M b jðtr AÞ2 j N b; M b j ¼ ððtr AÞj N b; M b jÞ2 ; jN
ð14aÞ
b; M b jðtr AÞ2 j Nd b; M b jðtr AÞj Nd jN þ 1; Md 1j ¼ ðtr AÞj N þ 1; Md 1j;
ð14bÞ
b; M b j ¼ ðtr AÞj Nd b; M b j; j Nd þ 1; Md 1jðtr AÞ2 j N þ 1; Md 1jðtr AÞj N
ð14cÞ
b; M b j ¼ jN b; M b jðtr AÞj Nd j Nd þ 1; Md 1jðtr AÞj N þ 1; Md 1j;
ð14dÞ
b; M b jðtr AÞ3 j Nd b; M bj jN þ 1; Md 1j ¼ j Nd þ 1; Md 1jðtr AÞ3 j N b; M b jðtr AÞj Nd b; M b jðtr AÞ2 j Nd ¼ ðtr AÞ2 j N þ 1; Md 1j ¼ ðtr AÞj N þ 1; Md 1j;
ð14eÞ
j Nd þ 1; Md 1jðtr AÞj Nd 1; N þ 1; Md 1; M þ 1j ¼ j Nd 1; N þ 1; Md 1; M þ 1jðtr AÞj Nd þ 1; Md 1j;
ð14fÞ
b; M b jðtr AÞj N b ; N þ 2; Md b ; N þ 2; Md b; M b j; jN 2; Mj ¼ j N 2; Mjðtr AÞj N
ð14gÞ
b; M b jðtr AÞj Nd b; M b j; jN þ 1; Md 3; M 1; Mj ¼ j Nd þ 1; Md 3; M 1; Mjðtr AÞj N
ð14hÞ
b; M b jðtr AÞj Nd b; M b j; jN 1; N þ 1; N þ 2; Md 1j ¼ j Nd 1; N þ 1; N þ 2; Md 1jðtr AÞj N
ð14iÞ
b j ¼ j Nd b jðtr AÞj Nd j Nd þ 1; Md 1jðtr AÞj Nd 2; N ; N þ 1; M 2; N ; N þ 1; M þ 1; Md 1j;
ð14jÞ
b ; Md b ; Md j Nd þ 1; Md 1jðtr AÞj N 2; M; M þ 1j ¼ j N 2; M; M þ 1jðtr AÞj Nd þ 1; Md 1j:
ð14kÞ
Utilizing (12c) and (14a), it is easy to see that Eq. (13) is equal to zero. Thus, the proof of Eq. (3a) is completed. h Using (4), (12) and (14), we work out from (3b) f ðgt gxxx Þ gðft gxxx Þ þ 3ðgxx fx gx fxx Þ b; M b jðj N b ; N þ 2; Md ¼ 12½j N 2; M þ 1j j Nd þ 1; Md 3; M 1; M þ 1j þ j Nd 2; N; N þ 1; N þ 2; Md 1j j Nd 1; N þ 1; N þ 2; Md 2; MjÞ bj j Nd þ 1; Md 1jðj Nd 1; N þ 2; Md 1; M þ 1j j Nd 2; N ; N þ 2; M b ; Md j Nd 1; N þ 1; Md 2; M; M þ 1j þ j N 3; M 1; M; M þ 1jÞ b ; N þ 2; Md b ; Md b jÞ þ jN 2; Mjðj N 1; M þ 1j þ j Nd 1; N þ 1; M b ; Md þ jN 1; M þ 1jðj Nd þ 1; Md 3; M 1; Mj þ j Nd 1; N þ 1; N þ 2; Md 1jÞ b jj Nd þ j Nd 1; N þ 1; M þ 1; Md 2; M þ 1j j Nd 1; N þ 1; Md 1; M þ 1jðj Nd þ 1; Md 2; Mj b ; N þ 2; Md b ; N þ 2; Md b j þ jN b ; Md þ jN 1jÞ j N 1jðj Nd 2; N ; N þ 1; M 2; M; M þ 1jÞ b ; N þ 2; M b j: j Nd þ 1; Md 2; Mjj N According to (4), Eq. (15) is equal to zero. Eq. (3c) can be verified similarly.
3. Soliton and rational solutions From (9), we can get u ¼ e4A
3 tAx
C;
3 tþAx
w ¼ e4A
D;
where C ¼ ðc1 ; c2 ; . . . ; cN þMþ2 ÞT and D ¼ ðd 1 ; d 2 ; . . . ; d N þMþ2 ÞT are arbitrary real constant vectors. By the Taylor series expansion, we have
ð15Þ
930
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
2 s 3 ½3 s l l s3l 1 X X ð1Þ 4 t x 4 5As C; u¼e C¼ l!ðs 3lÞ! l¼0 s¼0 2 s 3 ½3 1 l l s3l X X 4 t x 3 4 5As D: w ¼ e4A tþAx D ¼ l!ðs 3lÞ! l¼0 s¼0 4A3 tAx
ð16aÞ
ð16bÞ
If 0 B B A¼B B @
1
0
k1
C C C; C A
k2 ..
.
0
ðk i –k j ; i–jÞ;
k N þMþ2
we can obtain soliton solutions of Eq. (3), where 3
3
uj ¼ e4k j tkj x cj ; If
wj ¼ e4kj tþkj x d j ;
0
0 B1 B A¼B B @
0
.
..
0
1
. 0
ð17Þ
1 C C C C A
0 ..
ðj ¼ 1; 2; . . . ; N þ M þ 2Þ:
;
ðN þM þ2ÞðN þMþ2Þ
it is easy to know that AN þM þ2 ¼ 0. Thus (16) can be truncated as 2 s 3 2 s 3 ½3 ½3 NX þMþ1 X NX þMþ1 X ð1Þs 4l tl xs3l 5 s 4l tl xs3l 5 s 4 4 A C; w ¼ A D: u¼ l!ðs 3lÞ! l!ðs 3lÞ! l¼0 l¼0 s¼0 s¼0
ð18Þ
So rational solutions of Eq. (3) in double Wronskian form are obtained, where 3 ½j1 3 X x2 x ð1Þj1 4l tl xj13l c1 ; uj ¼ cj xcj1 þ cj2 þ 4t cj3 þ þ 2 3! l!ðj 1 3lÞ! l¼0
ð19aÞ
3 ½j1 3 X x2 x 4l tl xj13l wj ¼ d j þ xd j1 þ d j2 þ þ 4t d j3 þ þ d1: 2 3! l!ðj 1 3lÞ! l¼0
ð19bÞ
In (19), taking c1 ¼ d 1 ¼ 1;
cx ¼ d x ¼ 0;
ðx ¼ 2; 3; . . . ; N þ M þ 2Þ;
ð20Þ
then (19) becomes ½j1 3 X ð1Þj1 4l tl xj13l ; uj ¼ l!ðj 1 3lÞ! l¼0
wj ¼
½j1 3 X l¼0
4l tl xj13l : l!ðj 1 3lÞ!
ð21Þ
Thus, we obtain some special rational solutions of Eq. (1) as follows: 1 2ðx3 þ 6tÞ ; ; r ¼ 2; q ¼ r ¼ 4 2 x x 12xt 2ðx4 12xtÞ 3 3ðx4 þ 24xtÞ q¼ ; r¼ ; q¼ ; r ¼ 2: x3 þ 6t 2ðx3 þ 6tÞ ðx3 12tÞ2 q¼
When q ¼ r ¼ v, Eq. (1) becomes the mKdV equation vt ¼ vxxx 6v2 vx : Similarly, taking q ¼ u; r ¼ 2, Eq. (1) can be reduced to the following KdV equation
ð22Þ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
ut ¼ uxxx 12uux :
ð23Þ
From (21), we know that uj ¼ ð1Þ (I) : When N ¼ M, v¼
931
j1
wj . So we have the following results:
2W N þ2;N ðu; wÞ 2W N ;N þ2 ðu; wÞ ¼ N þ1;N þ1 ðu; wÞ W N þ1;N þ1 ðu; wÞ W
are rational solutions of Eq. (22). (II): When 2f N ¼ 2n þ 1; M ¼ 2n; h¼ 2f N ¼ 2ðn þ 1Þ; M ¼ 2n þ 1; we can obtain rational solutions of Eq. (23), where n is an arbitrary non-negative integer.
4. Matveev solutions Let A be Jordan matrix 0 J ðk1 Þ B J ðk 2 Þ B A¼B .. B @ . 0
1
0
C C C C A J ðk p Þ
:
ð24Þ
ðN þMþ2ÞðN þM þ2Þ
Without loss of generality, we observe the following Jordan block (dropping the subscript of k) 0
0
k
B1 k B J ðkÞ ¼ B .. B @ . 0
.. 1
1
0
C C C C A
. k
¼ kI li þ Eli ;
0
0
B1 0 B E li ¼ B .. B @ . 0
li li
..
C C C C A
.
1
1
0
;
li li
where I li denotes an li li unit matrix. We have J ðkÞs ¼ ðkI li þ Eli Þs ¼ k s I li þ sk s1 Eli þ þ C js k sj Ejli þ þ Esli 1 1 1 ¼ I li þ Eli ok þ E2li o2k þ þ Ejli ojk þ þ Esli osk k s ; 2! j! s! i.e. 0
J ðkÞs ¼ T k k s ;
B B B B B Tk ¼ B B B B B @
1 ok 1 2 o 2 k
0 1 ok
1 3 o 6 k
1 2 o 2 k
..
1 oli 1 ðli 1Þ! k
.
C C C C C C: C C C C A
1 ok .. . 1 3 o 6 k
1 ..
. 1 2 o 2 k
..
. ok
1
1
Substituting (24) into (16), we get c1 3 uj ðkÞ ¼ okj1 þ þ cj1 ok þ cj e4k tkx ; ðj ¼ 1; 2; . . . ; li Þ; ðj 1Þ! d1 3 okj1 þ þ d j1 ok þ d j e4k tþkx ; ðj ¼ 1; 2; . . . ; li Þ: wj ðkÞ ¼ ðj 1Þ! Thus, Matveev solutions of Eq. (3) are constructed, where
ð25aÞ ð25bÞ
932
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
u ¼ ðu1 ðk 1 Þ; . . . ; ul1 ðk 1 Þ; u1 ðk 2 Þ; . . . ; ul2 ðk 2 Þ; ; u1 ðk p Þ; . . . ; ulp ðk p ÞÞT ; ðl1 þ l2 þ þ lp ¼ N þ M þ 2Þ; w ¼ ðw1 ðk 1 Þ; . . . ; wl1 ðk 1 Þ; w1 ðk 2 Þ; ; wl2 ðk 2 Þ; ; w1 ðk p Þ; . . . ; wlp ðk p ÞÞT ; ðl1 þ l2 þ þ lp ¼ N þ M þ 2Þ: Taking ðN ; MÞ ¼ ð0; 0Þ; ð1; 0Þ and ½ð12k 2 t xÞ2 24kt 4k3 tkx ; ð12k t xÞe ; e e 2 !T 2 2 2 4k 3 tþkx 4k 3 tþkx ½ð12k t þ xÞ þ 24kt 4k 3 tþkx ; ð12k t þ xÞe ; ; e e 2 4k 3 tkx
u¼
w¼
!T
4k 3 tkx
2
;
we obtain two special Matveev solutions of Eq. (1) as follows: 3
3
e2ð4k tþkxÞ e2ð4k tþkxÞ ; r ¼ ; 12k 2 t þ x 12k 2 t þ x 3 3 e2ð4k tþkxÞ 2e2ð4k tþkxÞ ½ð12k 2 t þ xÞ2 12kt ; r ¼ : q¼ ð12k 2 t þ xÞ2 þ 12kt ð12k 2 t þ xÞ2 þ 12kt
q¼
Similarly, when ðN ; MÞ ¼ ð1; 0Þ and 3
3
3
T u ¼ ðe4k 1 tk1 x ; ð12k 21 t xÞe4k 1 tk1 x ; e4k2 tk2 x Þ ; 3
3
3
w ¼ ðe4k1 tþk1 x ; ð12k 21 t þ xÞe4k1 tþk1 x ; e4k2 tþk 2 x ÞT ; a direct calculation gives 3
3
3
3
3
f ¼ e2ð4k1 tþk1 xÞ e4k 2 tk2 x ½2ðk 2 k 1 Þð12k 21 t þ xÞe2ð4k1 tþk1 xÞ þ e2ð4k1 tþk1 xÞ e2ð4k 2 tþk2 xÞ ; 3
3
g ¼ 2e2ð4k1 tþk1 xÞ e4k 2 tk2 x ðk 1 k 2 Þ2 ; 3
3
3
3
3
h ¼ 2e2ð4k1 tþk1 xÞ e4k2 tþk2 x ½2ðk 2 k 1 Þð12k 21 t þ xÞe2ð4k1 tþk 1 xÞ þ e2ð4k2 tþk2 xÞ e2ð4k 1 tþk1 xÞ :
5. Complexitons Firstly, let us consider the simplest case 0 1 a b ; ¼ aI 2 þ br2 ; r2 ¼ A¼ 1 0 b a
ð26Þ
where a; b are real constants. Substituting (26) into (16) yields 3 3ab2 ÞtþaxI
u ¼ e½4ða
2
3 3ab
w ¼ e½4ða
ÞtþaxI 2
2
2 bb3 Þtþbxr
e½4ð3a
2 bb
e½4ð3a
3
Þtþbxr2
2
D;
C;
C ¼ ðc1 ; c2 ÞT ; D ¼ ðd 1 ; d 2 ÞT ;
u ¼ ðu1 ; u2 ÞT ; w ¼ ðw1 ; w2 ÞT :
Expanding the above u and w and making use of r22 ¼ I 2 , we get 3 3ab2 Þtþax
u ¼ e½4ða
½4ða3 3ab2 Þtþax
w¼e
fcos½4ð3a2 b b3 Þt þ bxI 2 sin½4ð3a2 b b3 Þt þ bxr2 gC; 2
3
2
3
fcos½4ð3a b b Þt þ bxI 2 þ sin½4ð3a b b Þt þ bxr2 gD:
Secondly, we consider an real Jordan matrix A 1 0 0 J1 0 Ai C B BI J 2 C B B 2 Ai C; J i ¼ B A¼B .. .. .. C B B A @ @ . . . 0 Jm 0 I2
0
ð27bÞ
1 C C C; C A
Ai
ð27aÞ
Ai ¼
ai bi
bi ai
ð28Þ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
933
where ai ; bi are real constants. Ji can be expressed as J i ¼ A0 þ E0 ; 0 Ai B Ai B A0 ¼ B B @ 0
0 ..
1
0
C C C C A
. Ai
;
0 BI B 2 E0 ¼ B B @
0
.
0
2li 2li
C C C C A
0 ..
..
.
I2
1
0
;
2li 2li
then we have s J si ¼ ðA0 þ E0 Þ ¼ A0s þ sA0s1 E0 þ þ C js A0sj E0j þ þ E0s 1 1 ¼ I 2li þ E0 oai þ þ E0j ojai þ þ E0s osai A0s : j! s!
ð29Þ
n n1 Using oai Ani ¼ oai ðai I 2 þ bi r2 Þ ¼ nðai I 2 þ bi r2 Þ ; ðn ¼ 1; 2; 3; . . . ; sÞ, then (27b) can be written as 1 0 I2 0 C B I2 I 2 oai C B C B 1 2 C B 2 I 2 oai I 2 oai I2 0s s J i ¼ T oai A ; T oai ¼ B C: C B . . . . C B . . . . . . . . A @ 1 I oli 1 ðli 1Þ! 2 ai
...
1 I o2 2 2 ai
I 2 olphai
ð30Þ
I2
Substituting (28) into (16) and taking advantage of (27), we can obtain the following forms: ( ! j X uj1 ðai Þ 1 2 3 uj ðai Þ ¼ e½4ðai 3ai bi Þtþai x : ¼ oajs i ðj sÞ! uj2 ðai Þ s¼1 !) cs1 cos½4ð3a2i bi b3i Þt þ bi x þ cs2 sin½4ð3a2i bi b3i Þt þ bi x ; cs1 sin½4ð3a2i bi b3i Þt þ bi x þ cs2 cos½4ð3a2i bi b3i Þt þ bi x ðj ¼ 1; 2; 3; . . . ; li Þ; ! ( j X wj1 ðai Þ 1 2 3 js ¼ oai e½4ðai 3ai bi Þtþai x : wj ðai Þ ¼ ðj sÞ! wj2 ðai Þ s¼1 d s1 cos½4ð3a2i bi b3i Þt þ bi x d s2 sin½4ð3a2i bi b3i Þt þ bi x d s1 sin½4ð3a2i bi b3i Þt þ bi x þ d s2 cos½4ð3a2i bi b3i Þt þ bi x
ð31aÞ
!) ;
ðj ¼ 1; 2; 3; . . . ; li Þ;
ð31bÞ
where cs1 ; cs2 ; d s1 ; d s2 are real constants. Therefore, we obtain complexitons of Eq. (3) in double Wronskian form, where u ¼ ðu1 ða1 ÞT ; . . . ; ul1 ða1 ÞT ; u1 ða2 ÞT ; . . . ; ul2 ða2 ÞT ; . . . ; u1 ðalm ÞT ; . . . ; ulm ðalm ÞT ÞT ; ðl1 þ l2 þ þ lm ¼ N þ M þ 2Þ; w ¼ ðw1 ða1 ÞT ; . . . ; wl1 ða1 ÞT ; w1 ða2 ÞT ; . . . ; wl2 ða2 ÞT ; . . . ; w1 ðalm ÞT ; . . . ; wlm ðalm ÞT ÞT ; ðl1 þ l2 þ þ lm ¼ N þ M þ 2Þ: For example, letting ðN ; MÞ ¼ ð0; 0Þ and a1 ¼ 0; c11 ¼ 1; c12 ¼ 0; d 11 ¼ d 12 ¼ q¼r¼v¼
2b1 ; sin½2ð4b31 t þ b1 xÞ þ h
pffiffi 2 , 2
then we have
tan h ¼ 1:
The singularity of the solution can be reflected in Fig. 1. In Fig. 1b, white points represent positive infinity, while dark points represent negative infinity. The grey stripes and dark ones denote the alternate changes of positive and negative amplitudes as it is described in Fig. 1a. Because the solution is periodic, so the width of each stripe is unchanged and it moves with constant speed 4b21 .
934
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
a
b
t=0
density graphics 6
30 4 20 2
10 -4
-6
-2
4
2
6
-10
0 -2
-20
-4
-30 -6 -6
-4
-2
0
2
4
6
Fig. 1. (a) t = 0. (b) Density image of v for x 2 ½2p; 2p and t 2 ½6; 6 and plot range of ½25; 25, b1 ¼ 1; h ¼ p4.
On the other hand, for oai Ani ¼ r2 obi Ani ;the partial derivative with respect to ai can be replaced by the partial derivative with respect to bi in (29)–(31).
6. Interaction solutions In what follows, we give a few double Wronskian interaction solutions of Eq. (1). Letting ðN ; MÞ ¼ ð1; 0Þ; ð0; 1Þ and choosing u ¼ ð1; x; e4k
3
tkx T
w ¼ ð1; x; e4k
Þ ;
3
tþkx T
Þ ;
then we get 3
2k 2 2ð2kx 1Þe2ð4k tþkxÞ þ 2 ; r ¼ ; 3 3 2kx þ 1 e2ð4k tþkxÞ 2kx þ 1 e2ð4k tþkxÞ 3 2ð2kx þ 1Þe2ð4k tþkxÞ 2 2k 2 ; r¼ : q¼ 3 3 2ð4k tþkxÞ 2kx 1 þ e2ð4k tþkxÞ 2kx 1 þ e
q¼
ð32Þ ð33Þ
When ðN; MÞ ¼ ð1; 1Þ, taking u ¼ ð1; x; e4k w ¼ ð1; x; e
3
3
tkx
4k tþkx
; ð12k 2 t xÞe4k 2
; ð12k t þ xÞe
3
3
tkx T
Þ ;
4k tþkx T
Þ ;
we have 3
q ¼ 4k
ðkx þ 1Þe2ð4k tþkxÞ þ 12k 3 t þ kx 1 ; 3 e2ð4k tþkxÞ þ e2ð4k tþkxÞ 2 4k 2 xð12k 2 t þ xÞ 3
ð34aÞ
3
r ¼ 4k
e2ð4k
3
ðkx 1Þe2ð4k tþkxÞ þ 12k 3 t þ kx þ 1 : tþkxÞ þ e2ð4k 3 tþkxÞ 2 4k 2 xð12k 2 t þ xÞ
ð34bÞ
If we choose u ¼ ð1; x; en cos g; en sin gÞT ;
w ¼ ð1; x; en cos g; en sin gÞT ;
then, q¼
ða2 b2 Þ sin 2g 2ab cos 2g þ 2b½ða2 þ b2 Þx þ ae2n ; ða2 þ b2 Þx sin 2g þ bðcos 2g cosh 2nÞ
ð35aÞ
r¼
ða2 b2 Þ sin 2g þ 2ab cos 2g þ 2b½ða2 þ b2 Þx ae2n ; ða2 þ b2 Þx sin 2g þ bðcos 2g cosh 2nÞ
ð35bÞ
F.-m. Yin et al. / Chaos, Solitons and Fractals 39 (2009) 926–935
935
where n ¼ 4ða3 3ab2 Þt þ ax; g ¼ 4ð3a2 b b3 Þt þ bx: Of course, we can get more general interaction solutions among three or more kinds of solutions such as soliton solutions, rational solutions, Matveev solutions and complexitons.
7. Conclusions In this paper, by means of the Wronskian technique, we have verified Eq. (3) which possesses generalized double Wronskian solutions. These solutions are rational, Matveev, complexitons and interaction solutions. Further, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.
Acknowledgements The authors are very grateful to Professor M.Wadati for his help. The authors express their sincere thanks to Dr. Sun YP for his valuable advice. This project is supported by National Science Foundation of China (10371070) and the Special Found for Major Specialities of Shanghai Education Committee.
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