Generalized double Casoratian solutions to the four-potential isospectral Ablowitz–Ladik equation

Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

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Generalized double Casoratian solutions to the four-potential isospectral Ablowitz–Ladik equation Shouting Chen a,⇑, Jianbing Zhang b, Dengyuan Chen c a

School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Jiangsu 221008, China School of Mathematical Sciences, Jiangsu Normal University, Jiangsu 221116, China c Department of Mathematics, Shanghai University, Shanghai 200444, China b

a r t i c l e

i n f o

Article history: Received 18 February 2013 Received in revised form 22 April 2013 Accepted 22 April 2013 Available online 4 May 2013 Keywords: The four-potential isospectral Ablowitz– Ladik equation Wronskian technique Double Casoratian form

a b s t r a c t Generalized solutions in double Casoratian form of the four-potential isospectral Ablowitz– Ladik equation possessing bilinear form are derived through a matrix method for constructing double Casoratian entries. A novel class of explicit solutions, such as soliton, rational-like, Matveev, Complexiton and interaction solutions, are obtained by letting the general matrix be some special cases. Interestingly, a periodic solution is deduced from the Complexiton solution. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction As is well known, Wronskian technique is an important direct method of searching the soliton solutions to the nonlinear evolution equations [1,2]. Owing to its advantage of direct verification of solutions for soliton equations, Wronskian technique has received considerable attention to its applications and generalizations. As a generalization of Wronskian, the double Wronskian technique was proposed by Darboux in Ref. [3] and applied to constructing the double Wronskian solutions [4,5]. In terms of the technique, soliton solutions can be represented in (double) Wronski determinant form. Besides soliton solutions [6,7], many other exact solutions, such as rational solutions[8,9], Positon solutions [10,11], Complexitons[12], interaction solutions [13], and so forth, can be also expressed in (double) Wronskian form. In 1983, Nimmo and Freeman put forward the rational solutions in the Wronskian form for the KdV equation[8] under the basis of performing an appropriate limiting procedure on the soliton solutions proposed by Ablowitz and Satsuma[14]. In Refs. [9,15], Zhang derived the rational and mixed rational–soliton solutions to the Toda lattice and differential-difference KdV equation in the Casoratian (the discrete version of the Wronskian) form. As the generalization of the Wronskian technique, Chen et al. provided an arbitrary matrix equation satisfied by double Wronskian entries and derived a novel class of solutions of the second-order AKNS equation. By letting the spectral matrix be some special cases, the soliton solutions, rational solutions, Matveev solutions, Complexitons and interaction solutions were obtained [13]. This approach to construct double Wronskian solutions has been applied to some famous soliton equations, such as the general nonlinear Schrödinger equation with derivative [16], the Kadomtsev–Petviashvili equation [17], and so on. In this paper, we consider the generalized double Casoratian solutions of the first-order four-potential isospectral Ablowitz–Ladik (AL) equation. The paper is organized as follows. In Section 2, we derive the generalized double Casoratian ⇑ Corresponding author. Tel.: +86 13852487246. E-mail address: [email protected] (S.t. Chen). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.04.024

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solution of the equation under a matrix equation satisfied by double Casoratian entries. In Section 3, soliton solutions and rational-like solutions in double Casoratian form are deduced by letting the general matrix be some special cases. In Section 4, we derive the Matveev solutions. In Section 5, we obtain a Complexiton solution and a periodic solution. In Section 6, we construct the interaction solutions. Finally, a conclusion is given. 2. The generalized double Casoratian solutions Let E be the shift operator defined as Ek v ðnÞ ¼ v ðn þ kÞ; k 2 Z. Usually for convenience, we write confusion. The first-order four-potential isospectral AL equation reads[18]

Q n;t ¼ ð1  Q n Rn ÞSn ;

Rn;t ¼ ð1  Q n Rn ÞT n1 ;

Sn;t ¼ ð1  Sn T n ÞQ nþ1 ;

T n;t ¼ ð1  Sn T n ÞRn ;

v ðnÞ ¼ v n

without any

ð2:1Þ

where the functions of variable n fQ n ; Rn ; Sn ; T n g are four potentials. Its Lax pairs are [19,18]

EUn ¼ U n Un ;

Un ¼

Un;t ¼ V n Un ;

Vn ¼

z 2 þ Sn R n

zQ n þ z1 Sn

zT n þ z1 Rn

z2 þ T n Q n

 12 Q n T n1 þ 12 z2

!

Un ¼ !

Q nz 1 Q T 2 n n1

T n1 z

 ;

 12 z2

/1n /2n

 ;

ð2:2Þ

:

Through the dependent variable transformation[20]

Qn ¼

gn ; fn

Rn ¼

hn ; fn

Sn ¼

Gn ; Fn

Tn ¼

Hn ; Fn

ð2:3Þ

Eq. (2.1) can be transformed into the following bilinear form

Dt g n  fn ¼ F n1 Gn ;

ð2:4aÞ

Dt hn  fn ¼ F n Hn1 ;

ð2:4bÞ

Dt Gn  F n ¼ fn g nþ1 ;

ð2:4cÞ

Dt Hn  F n ¼ fnþ1 hn ;

ð2:4dÞ

fn2

 g n hn ¼ F n F n1 ;

ð2:4eÞ

F 2n

 Gn Hn ¼ fnþ1 fn ;

ð2:4fÞ

where D is the well-known Hirota bilinear operator defined by m n n 0 0 0 Dm t Dx f  g ¼ ð@ t  @ t Þ ð@ x  @ x0 Þ f ðt; xÞ  gðt ; x Þjt 0 ¼t;

x0 ¼x :

ð2:5Þ

Let us observe the matrix equations

EUn ¼ AUn ; E1 Wn ¼ AWn ; 1 1 Un;t ¼ EUn ; Wn;t ¼  E1 Wn ; 2 2

ð2:6aÞ ð2:6bÞ

where A ¼ ðaij Þ is an ðm þ p þ 2Þ  ðm þ p þ 2Þ arbitrary invertible, real matrix independent of n and t and

Un ¼ ð/1n ; /2n ; . . . ; /mþpþ2;n ÞT ;

Wn ¼ ðw1n ; w2n ; . . . ; wmþpþ2;n ÞT :

ð2:7Þ

The double Casorati determinant is a discrete version of a double Wronskian defined as [21]

b b Casmþ1;pþ1 ðUn ; Wn Þ ¼ jUn ; EUn ; . . . ; Em Un ; Wn ; EWn ; . . . ; Ep Wn j ¼ j m; p j:

ð2:8Þ

kj and jel; b kj[9] Besides, we introduce the definitions of jel; e

kj ¼ jEUn ; E2 Un ;    ; El Un ; EWn ; E2 Wn ; . . . ; Ek Wn j; jel; e

ð2:9aÞ

kj ¼ jEUn ; E2 Un ;    ; El Un ; Wn ; EWn ; . . . ; Ek Wn j: jel; b

ð2:9bÞ

In order to obtain the generalized double Casoratian solution, we first give the following lemma. Lemma 1. Suppose that M is an N  ðN  2Þ matrix and a; b; c; d are N-order column vectors, then

jM; a; bjjM; c; dj  jM; a; cjjM; b; dj þ jM; a; djjM; b; cj ¼ 0: Applying the double Casoratian technique, we have the following theorem.

ð2:10Þ

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

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Theorem 1. Eq. (2.4) has the following double Casoratian solution

b b fn ¼ j m; p j;

g n ¼ j md þ 1; pd  1j;

hn ¼ j md  1; pd þ 1j;

1 p j; F n ¼ jAj2 j mg þ 1; b

12

Gn ¼ jAj j mg þ 2; pd  1j;

ð2:11Þ

12

e pd Hn ¼ jAj j m; þ 1j;

which entries satisfy (2.6).

Proof. A direct calculation yields 1 b 1; pd F n1 ¼ jAj2 j m;  1j;

p j: g nþ1 ¼ j mg þ 2; e

ð2:12Þ

Noting that the derivatives of /jn and wjn with respect to t are

/jn;t ¼

1 E/ ; 2 jn

1 wjn;t ¼  E1 wjn 2

ðj ¼ 1; 2;    ; m þ p þ 2Þ:

ð2:13Þ

We have

1 1 b m þ 2; pd j m;  1j  j md þ 1; 1; pg  1j; 2 2 1 1 b 1; e p j  j m; p j;  1; m þ 1; b fn;t ¼ j md 2 2   1 1 Gn;t ¼ jAj2 j mg þ 1; m þ 3; pd  1j  j mg þ 2; 1; pg  1j ; 2  1 12  e m þ 2; b p j  j mg pj : þ 1; 1; e F n;t ¼ jAj j m; 2

g n;t ¼

ð2:14aÞ ð2:14bÞ ð2:14cÞ ð2:14dÞ

Substituting Eqs. (2.14) and (2.12) into (2.4a), (2.4c) and (2.4e) yield

  1 1 b m þ 2; pd b b j m; p j  j md pj  1j  j md þ 1; 1; pg  1j j m; þ 1; pd  1j j md  1; m þ 1; b 2 2 1 d g d b 1; e b 1; p  1jj m þ 2; p  1j; p jÞ  jAj j m; j m;   1 pj þ 1; m þ 3; pd  1j  j mg þ 2; 1; pg  1j j mg þ 1; b Gn;t F n  Gn F n;t  fn g nþ1 ¼ jAj1 j mg 2   1 e m þ 2; b b b p j  j mg p j  j m; p jj mg p j; þ 2; pd  1j j m; þ 1; 1; e þ 2; e  jAj1 j mg 2 2 1 2 d d d d g d b b b 1; p  1j: p j þ j m þ 1; p  1jj m  1; p þ 1j  jAj j m þ 1; b p jj m; fn  g n hn  F n F n1 ¼ j m;

g n;t fn  g n fn;t  F n1 Gn ¼

ð2:15aÞ

ð2:15bÞ ð2:15cÞ

According to Lemma 1, Eqs. (2.15a) and (2.15b) can be transformed into

 1 b m þ 2; pd b b b 1; pd  1jj m; þ 1; pd  1jj md  1; m þ 1; b þ 1; e  1j j m; p j  j md p j þ j md p jj m; 2 b 1; pd  1jj mg þ 2; pd  1j; ð2:16aÞ jAj1 j m; 1 1  g e m þ 2; b  fn g nþ1 ¼ jAj j m þ 1; m þ 3; pd p j  j mg pj  1jj mg þ 1; b þ 2; pd  1jj m; 2  b b þ j mg p jj mg p jj mg p j: ð2:16bÞ þ 2; e þ 1; 1; pd  1j  j m; þ 2; e

g n;t fn  g n fn;t  F n1 Gn ¼ Gn;t F n  Gn F n;t

From Eq. (2.6), we get

b b jAjj m; p j ¼ j mg þ 1; 1; pd  1j;

e m þ 2; 1; pd p j ¼ j m; jAjj md  1; m þ 1; b  1j;

ð2:17aÞ

þ 1; e þ 2; pd  1j; jAjj md p j ¼ j mg

e 1; b jAjj md  1; pd þ 1j ¼ j m; p j:

ð2:17bÞ

Applying the above conclusions, (2.16) and (2.15c) become   1 b m þ 2; pd e b g n;t fn  g n fn;t  F n1 Gn ¼ jAj1 j m; þ 2; 1; pd  1jj mg þ 1; 1; pd  1j  j md þ 1; pd  1jj m;m  1j  j mg þ 2; pd  1jj m;1; pd  1j ;ð2:18aÞ 2   1 e m þ 2; b Gn;t F n  Gn F n;t  fn g nþ1 ¼ jAj1 j mg ð2:18bÞ p j  j mg p j  j mg p jj mg þ 1; m þ 3; pd  1jj mg þ 1; b þ 2; pd  1jj m; þ 2; e þ 1; 1; pd  1j ; 2  b b e 1; b b 1; pd f 2  g hn  F n F n1 ¼ jAj1 j m; þ 1; 1; pd  1j þ j md þ 1; pd  1jj m; þ 1; b  1j : ð2:18cÞ p jj mg p j  j mg p jj m; n

n

Directly calculation shows that (2.18a)–(2.18c) are equal to zero. Thus the proofs of Eqs. (2.4a), (2.4c) and (2.4e) are completed. The other equations of (2.4) can be verified similarly. h Finally, the generalized double Casoratian solution of Eq. (2.1) can be expressed as

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j md j md þ 1; pd  1j  1; pd þ 1j ; Rn ¼  ; b b b b j m; j m; pj pj e pd þ 2; pd  1j þ 1j j mg j m; Sn ¼ ; Tn ¼  : g j m þ 1; b j mg pj pj þ 1; b

Qn ¼

ð2:19aÞ ð2:19bÞ

3. Soliton solutions and rational-like solutions From (2.6), we get the general solution 1

Un ¼ An e2At C;

1

Wn ¼ An e2At D;

ð3:1Þ

T

T

1B 2

where C ¼ ðc1 ; c2 ;    ; cmþpþ2 Þ and D ¼ ðd1 ; d2 ;    ; dmþpþ2 Þ are real constant vectors. Let A ¼ e , (3.1) can be rewritten as 1

1B

1

1

Un ¼ e2nBþ2e2 t C;

1

1B

Wn ¼ e2nB2e2 t D:

ð3:2Þ

Expanding (3.2) leads to

" # 1 1 X s X tr X r l nsl Un ¼ e C¼ Bs C; 2r r! s¼0 l¼0 2s l!ðs  lÞ! r¼0 " # 1 r 1 X s 1B X t ð1Þr X r l nsl ð1Þsl s 12nB12e2 t B D: Wn ¼ e D¼ 2r r! s¼0 l¼0 2s l!ðs  lÞ! r¼0 1 1nBþ1e2B t 2 2

ð3:3aÞ ð3:3bÞ

If

0

k1

B B B¼B B @

0

C C C; C A

k2 .. 0

1

.

ki – kj ði – jÞ;

ð3:4Þ

kmþpþ2

we can obtain soliton solutions of Eq. (2.1) [20], where 1

1

1k j

/jn ¼ e2kj nþ2e2 t cj ;

1

1

1k j

wjn ¼ e2kj n2e2 t dj :

ð3:5Þ

If

0

0

0

1

C B1 0 C B C B¼B ; . . C B . . A @ . . 0 1 0 ðmþpþ2Þðmþpþ2Þ

ð3:6Þ

it is obvious that Bmþpþ2 ¼ 0. Thus (3.3) can be truncated as

" # mþpþ1 1 s X tr X X r l nsl Bs C; Un ¼ 2r r! s¼0 l¼0 2s l!ðs  lÞ! r¼0 " # mþpþ1 1 r s X t ð1Þr X X rl nsl ð1Þsl s Wn ¼ B D: 2r r! 2s l!ðs  lÞ! r¼0 s¼0 l¼0

ð3:7aÞ ð3:7bÞ

The components of Un and Wn are

" # j1 1 X X tr nþr n2 þ 2rn þ r 2 r l nj1l ; c þ c þ c þ    þ c 1 j j1 j2 j1 2 8 2r r! l!ðj  1  lÞ! r¼0 l¼0 2 " # j1 l j1l 1 r X X t ð1Þr n þ r n2  2rn þ r 2 rn ð1Þj1l wjn ¼ : dj þ dj1 þ dj2 þ    þ d1 j1 2 8 2r r! l!ðj  1  lÞ! r¼0 l¼0 2

/jn ¼

Taking c1 ¼ d1 ¼ 1; ck ¼ dk ¼ 0ðk ¼ 2; 3;    ; m þ p þ 2Þ, then (3.8) becomes

ð3:8aÞ ð3:8bÞ

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

# j1 1 X tr X r l nj1l /jn ¼ ; 2r r! l¼0 2j1 l!ðj  1  lÞ! r¼0 wjn ¼

j1 1 r X t ð1Þr X r l nj1l ð1Þj1l

2r r!

r¼0

2j1 l!ðj  1  lÞ!

l¼0

2953

ð3:9aÞ # ð3:9bÞ

:

Thus we can calculate several rational-like solutions with double Casoratian form of Eq. (2.1) from (2.19)). For example, for m ¼ p ¼ 0, we obtain

et et ; Rn ¼  ; 2n þ t 2n þ t t e et Sn ¼  ; Tn ¼  : 2n þ t þ 1 2n þ t þ 1

Qn ¼ 

ð3:10aÞ ð3:10bÞ

For m ¼ 1; p ¼ 0, we have

2et ð4n2 þ 4nt þ 2n þ t 2 þ 2tÞet ; R ¼ ; n 4n2 þ 4nt þ 2n þ t 2 4n2 þ 4nt þ 2n þ t2 2 t 2 2e ð4n þ 4nt þ 6n þ t þ 4t þ 2Þet Sn ¼ 2 ; Tn ¼ : 2 4n þ 4nt þ 6n þ t þ 2t þ 2 4n2 þ 4nt þ 6n þ t 2 þ 2t þ 2

Qn ¼

ð3:11aÞ ð3:11bÞ

For m ¼ 0; p ¼ 1, we get

ð4n2 þ 4nt þ 2n þ t 2 Þet 2et ; Rn ¼  2 ; 2 2 4n þ 4nt þ 2n þ 2t þ t 4n þ 4nt þ 2n þ 2t þ t 2 ð4n2 þ 4nt þ 6n þ t2 þ 2t þ 2Þet 2et Sn ¼  ; Tn ¼  2 : 2 2 4n þ 4nt þ 6n þ t þ 4t þ 2 4n þ 4nt þ 6n þ t2 þ 4t þ 2

Qn ¼ 

ð3:12aÞ ð3:12bÞ

Choosing m ¼ p ¼ 1 yields 1 f ¼  768 ð16n4 þ 32n3 t þ 32n3 þ 24n2 t 2 þ 48n2 t þ 20n2 þ 8nt3 þ 24nt 2 þ 16nt þ 4n þ 6t2 þ 4t 3 þ t4 Þ; 1 ð8n3 þ 12n2 t þ 12n2 þ 6nt 2 þ 6nt þ 4n þ t 3 Þet ; g ¼  384 1 ð8n3 þ 12n2 t þ 12n2 þ 6nt 2 þ 18nt þ 4n þ 6t þ 6t 2 þ t 3 Þet ; h ¼  384 1

1 F ¼  768 jAj2 ð16n4 þ 32n3 t þ 64n3 þ 24n2 t 2 þ 96n2 t þ 92n2 þ 8nt 3 þ 48nt2 þ 88nt þ 56n þ 24t 2

3

ð3:13Þ

4

þ24t þ 8t þ t þ 12Þ; 1

1 jAj2 ð8n3 þ 12n2 t þ 24n2 þ 6nt 2 þ 18nt þ 22n þ 6t þ 3t2 þ t 3 þ 6Þet ; G ¼  384 1

1 H ¼  384 jAj2 ð8n3 þ 12n2 t þ 24n2 þ 6nt2 þ 30nt þ 22n þ 18t þ 9t 2 þ t3 þ 6Þet :

Substituting (3.13) into (2.3), we can derive corresponding rational-like solution of Eq. (2.1). Besides, we can also work out the rational-like solutions under the conditions of ðm; pÞ ¼ ð2; 0Þ and ðm; pÞ ¼ ð0; 2Þ, respectively. These solutions can be verified by direct substitution to Eq. (2.1). 4. Matveev solutions Set B to be a ðm þ p þ 2Þ  ðm þ p þ 2Þ Jordan matrix

0

Jðk1 Þ

B B B¼B B @

1

0

C C C; C A

Jðk2 Þ ..

.

0

ð4:1Þ

Jðks Þ

without loss of generality, we observe some Jordan block

1 k 0 C B1 k C B C JðkÞ ¼ B . . C B . . A @ . . 0

0

1

k

0 ¼ kIl þ Y l ;

0

0

C C C ; C A

B1 0 B Yl ¼ B .. .. B @ . .

ll

0

1

1

0

ð4:2Þ

ll

where Il denotes an l  l unite matrix. Obviously,

J s ðkÞ ¼ ðkIl þ Y l Þs ¼

  1 1 1 s Il þ Y l @ k þ Y 2l @ 2k þ    þ Y jl @ jk þ    þ Y sl @ sk k : 2! j! s!

ð4:3Þ

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S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

Hence, for an arbitrary positive integers, we have

0 B B B B B TðkÞ ¼ B B B B B @

s

J s ðkÞ ¼ TðkÞk ;

1

0 1

@k 1 2 @ 2 k 1 3 @ 6 k

@k

1

1 2 @ 2 k

.

@k .. .

1 .. .



1 3 @ 6 k

1 2 @ 2 k

.. 1 @ l1 ðl1Þ! k

..

.

@k

1

1 C C C C C C : C C C C A

ð4:4Þ

ll

Substituting JðkÞ for matrix B in (3.3) yields 1

1k

1

Un ðkÞ ¼ TðkÞe2nkþ2e2 t C;

1

1

1k

Wn ðkÞ ¼ TðkÞe2nk2e2 t D;

ð4:5Þ

which components are

  1 1 1 1 k /jn ðkÞ ¼ c1 @ j1 þ    þ cj1 @ k þ cj e2nkþ2e2 t ; k ðj  1Þ!   1 1 1 1 k @ j1 þ    þ dj1 @ k þ dj e2nk2e2 t wjn ðkÞ ¼ d1 k ðj  1Þ!

ð4:6aÞ ðj ¼ 1; 2;    ; lÞ:

ð4:6bÞ

Particularly, let c1 ¼ d1 ¼ 1; cj ¼ dj ¼ 0 ðj ¼ 2; 3; . . . ; lÞ, (4.6) becomes 1 1 1nkþ1e2k t 2 2 @ j1 ; k e ðj  1Þ! 1 1 1 1 k wjn ðkÞ ¼ @ j1 e2nk2e2 t : ðj  1Þ! k

/jn ðkÞ ¼

ð4:7aÞ ð4:7bÞ

Thus, the Matveev solutions to Eq. (2.1) are worked out, where

Un ¼ ð/1n ðk1 Þ; . . . ; /l1 n ðk1 Þ; . . . ; /1n ðks Þ; . . . ; /ls n ðks ÞÞT ;

ð4:8aÞ

Wn ¼ ðw1n ðk1 Þ; . . . ; wl1 n ðk1 Þ; . . . ; w1n ðks Þ; . . . ; wls n ðks ÞÞT

ð4:8bÞ

ðl1 þ l2 þ    þ ls ¼ m þ p þ 2Þ:

ð4:8cÞ

Setting

Un ¼ ð/1n ðkÞ; /2n ðkÞÞT ;

Wn ¼ ðw1n ðkÞ; w2n ðkÞÞT

ð4:9Þ

in (4.8), where /jn ðkÞ and wjn ðkÞ are defined by (4.7), we can derive the Matveev solution of Eq. (2.1) 1

Qn ¼  Rn ¼  Sn ¼  Tn ¼ 

e2k

1k

2n þ te 1 e2k

2n þ te ek

1k 2

1k 2

enkþe2 t ;

ð4:10aÞ

1k

enke2 t ;

ð4:10bÞ 1k

1k 2

2n þ 1 þ te ek

2n þ 1 þ te

enkþe2 t ;

ð4:10cÞ

1k

1k 2

enke2 t :

ð4:10dÞ

Now, let us take

Un ¼ ð/1n ðkÞ; /2n ðkÞ; /3n ðkÞÞT ;

Wn ¼ ðw1n ðkÞ; w2n ðkÞ; w2n ðkÞÞT :

ð4:11Þ

When choosing ðm; pÞ ¼ ð1; 0Þ, we get

Qn ¼ Rn ¼ Sn ¼

2ek

1k

enkþe2 t ;

1k 2

4n2 þ 2n þ 4nte þ t 2 ek 1 ð4n2 þ 2nÞek þ ð4nt þ 2tÞe2k þ t2 4n2

1k 2

þ 2n þ 4nte þ 3 2e2k

t 2 ek

ð4:12aÞ 1k

enke2 t ; 1k

1k 2

4n2 þ 6n þ 2 þ ð2t þ 4ntÞe þ

t2 ek

enkþe2 t ;

3

Tn ¼

1

ð4n2 þ 6n þ 2Þe2k þ ð4t þ 4ntÞek þ t2 e2k 4n2

ð4:12bÞ

1k 2

þ 6n þ 2 þ ð2t þ 4ntÞe þ

t 2 ek

ð4:12cÞ 1k

enke2 t :

ð4:12dÞ

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

2955

While ðm; pÞ ¼ ð0; 1Þ, we obtain 3

Qn ¼ 

ð4n2 þ 2nÞek þ 4nte2k þ t 2 e2k þ 2n þ

t2 ek

þ 2n þ

t2 ek

4n2

þ ð4nt þ 2tÞe

1k

2ek

Rn ¼ 

4n2

enkþe2 t ;

1k 2

1k

þ ð4nt þ 2tÞe

enke2 t ;

1k 2

3

ð4:13bÞ

5

ð4n2 þ 6n þ 2Þe2k þ ð2t þ 4ntÞe2k þ t2 e2k

Sn ¼ 

ð4:13aÞ

1k 2

4n2 þ 6n þ 2 þ ð4t þ 4ntÞe þ

t 2 ek

3

Tn ¼ 

2e2k 4n2

1k

enkþe2 t ; 1k

1k 2

t 2 ek

þ 6n þ 2 þ ð4t þ 4ntÞe þ

enke2 t :

ð4:13cÞ ð4:13dÞ

It is easy to verify that Eq. (2.1) admits the above Matveev solutions. 5. Complexiton solutions In order to construct complexitons from (3.2), let us begin to consider that B is a real Jordan matrix as follows:

0

J1

1

0

B B B¼B B @

C C C; C A

J2 ..

.

0

ð5:1Þ

Jh

where

0

Bi BI B 2 Ji ¼ B B @

0 Bi .. .. . .

0

I2

1 C C C; C A

Bi ¼





ai bi ; bi ai

ð5:2Þ

Bi

and ai ; bi ði ¼ 1; 2; . . . ; hÞ are real constants. We first observe the simplest case of B (dropping the subscript) when

B¼









a b 0 1 ¼ aI2 þ br2 ; r2 ¼ : b a 1 0

ð5:3Þ

Substituting (5.3) into (3.3) yields 1

1

1

1aI

Un ¼ e2naI2  e2nbr2  e2te2

Wn ¼ e

12naI2

12nb 2

e

r

e

1 2 e2br2

C;

ð5:4aÞ

1aI 1br 12te2 2 e2 2 

D:

Expanding the above Un ; Wn and noticing



ð5:4bÞ

r ¼ I2 , we get 2 2



    1 1 1 1 2a 1 1 1 Un ¼ e2na cos nb I2 þ sin nb r2  e2te ½cos ð2bÞI2 þsin ð2Þr2  C; 2 2       1 1 1 1 2a 1 1 1 Wn ¼ e2na cos nb I2  sin nb r2  e2te ½cos ð2bÞI2 þsin ð2Þr2  D: 2 2

ð5:5aÞ ð5:5bÞ

Next, setting B to be a Jordan block J i as follows:

B ¼ J i ¼ B0 þ Y 0 ; 0

Bi

B B B0 ¼ Ili  Bi ¼ B B @

0

C C C C A

Bi ..

0 0 0 BI B 2 Y 0 ¼ Y li  I 2 ¼ B B @ 0

. Bi 0

.

I2

1

0

;

ð5:6bÞ

:

ð5:6cÞ

2li 2li

C C C C A

0 ..

ð5:6aÞ

1

2li 2li

2956

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

where  means direct product. Taking advantage of the equality B0 Y 0 ¼ Y 0 B0 , we have

  1 1 Bs ¼ ðB0 þ Y 0 Þs ¼ I2li þ Y 0 @ ai þ    þ Y 0j @ jai þ    þ Y 0s @ sai B0s : j! s!

ð5:7Þ

Making use of the following formula

@ ai Bhi ¼ @ ai ðai I2 þ bi r2 Þh ¼ hðai I2 þ bi r2 Þh1

ðh ¼ 1; 2; 3;   Þ;

ð5:8Þ

we can transform (5.7) into

0

Bs ¼ Tð@ ai ÞB0s ;

B B B B Tð@ ai Þ ¼ B B B B @

I2

0

I2 @ ai

I2

1 I @2 2 2 ai

I 2 @ ai

I2

.. .

..

..

1 I @ l1 ðl1Þ! 2 ai

.



..

.

1 I @2 2 2 ai

.

I 2 @ ai

I2

1 C C C C C C C C A

ð5:9Þ

:

2l2l

Substituting (5.9) into (3.3) yields

  1 0 1 1 0 1 B 1 1 Bi /jn ðai Þ ¼ Tð@ ai Þe2nB þ2e2 t C ¼ Tð@ ai Þ Ili  e2nBi þ2e2 t C;

ð5:10Þ

which can be rewritten as

/jn ðai Þ ¼

1 1 1 1 1nB þ1e2Bi t 1 1 Bi 1 1 Bi 2 i 2 @ j1 c1 þ    þ @ ai e2nBi þ2e2 t cj1 þ e2nBi þ2e2 t cj ; ai e ðj  1Þ!

ð5:11Þ

where

 T /jn ðai Þ ¼ /jn;1 ðai Þ; /jn;2 ðai Þ ;

cj ¼ ðcj1 ; cj2 ÞT :

ð5:12Þ

Similarly, we work out

wjn ðai Þ ¼

1 1 1 1 1 1 Bi 1 1 Bi 1 1 Bi @ j1 e2nBi 2e2 t d1 þ    þ @ ai e2nBi 2e2 t dj1 þ e2nBi 2e2 t dj ; ðj  1Þ! ai

ð5:13Þ

where

 T wjn ðai Þ ¼ wjn;1 ðai Þ; wjn;2 ðai Þ ;

dj ¼ ðcj1 ; dj2 ÞT :

ð5:14Þ

By virtue of (5.5), (5.11) and (5.13) can be reduced to /jn ðai Þ ¼

wjn ðai Þ ¼

/jn;1 ðai Þ /jn;2 ðai Þ

wjn;1 ðai Þ wjn;2 ðai Þ

!

2 0       13 1 1 cs1 cos 12 nbi þ 2t e2ai sin 12 bi  cs2 sin 12 nbi þ 2t e2ai sin 12 bi 1a 1 C7 js 6 12nai þ12e2 i cos ð12bi Þt B   ¼ @ 4e @     A5; 1 1 ðj  sÞ! ai c sin 1 nb þ t e2ai sin 1 b þ c cos 1 nb þ t e2ai sin 1 b s¼1 j X

s1

!

i

2

s2

2 i

2

i

2

2 0     13 1 1 ds1 cos 12 nbi þ 2t e2ai sinð12 bi Þ þ ds2 sin 12 nbi þ 2t e2ai sinð12 bi Þ 1a 1 1 1 1 i 6 B C7     A5: ¼ @ js 4e2nai 2e2 cosð2bi Þt  @ 1 1 ðj  sÞ! ai d sin 1 nb þ t e2ai sinð1 b Þ þ d cos 1 nb þ t e2ai sinð1 b Þ s¼1 j X

s1

2

i

2

2 i

s2

2

i

ð5:15aÞ

2 i

2

2

ð5:15bÞ

2 i

Then we derive the complexiton solutions of Eq. (2.4) in double Casorati determinant form (2.11), where Un and Wn satisfy



T

Un ¼ /T1n ða1 Þ; . . . ; /Tl1 n ða1 Þ; /T1n ða2 Þ; . . . ; /Tl2 n ða2 Þ; . . . ; /T1n ðah Þ; . . . ; /Tlh n ðah Þ ;

 T Wn ¼ wT1n ða1 Þ; . . . ; wTl1 n ða1 Þ; wT1n ða2 Þ; . . . ; wTl2 n ða2 Þ; . . . ; wT1n ðah Þ; . . . ; wTlh n ðah Þ ;

ð5:16aÞ ð5:16bÞ

l1 þ l2 þ    þ lh ¼ m þ p þ 2: m Otherwise, according to the equality of @ ai Bm i ¼ r2 @ bi Bi , we can get the similar conclusions for the complexitons by substituting @ ai for @ bi in (5.15). 1 1 For example, taking m ¼ p ¼ 0; c11 ¼ d11 ¼ 1; c12 ¼ d12 ¼ 0, n ¼ 12 na þ 12 e2a cosð12 bÞt, g ¼ 12 nb þ 12 e2a sinð12 bÞt (dropping the subscript) and

Un ¼ ðen cosg; en singÞT ;

Wn ¼ ðen cosg; en singÞT ;

we work out the complexiton solution of Eq. (2.1) as follows:

ð5:17Þ

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959

sin

1 b 2

1

a ðnþ1Þaþe2 cos ð12bÞt ; 1  e 2 sin nb þ e sin 2 b t  1a sin 12 b 1 1   eðnþ2Þae2 cos ð2bÞt ; Rn ¼  1a 1 sin nb þ e2 sinð2 bÞt  1a sin 12 b 1   eðnþ1Þaþe2 cos ð2bÞt ; Sn ¼   1 a 1 1 2 sin n þ 2 b þ e sin 2 b t  1a sin 12 b 1   eðnþ1Þae2 cos ð2bÞt : Tn ¼   1 sin n þ 12 b þ e2a sin 12 b t



Qn ¼ 

2957

ð5:18aÞ

1a 2

ð5:18bÞ

ð5:18cÞ

ð5:18dÞ

When sinð12 bÞ ¼ 1, i.e., cosð12 bÞ ¼ 0, (5.18) is the periodic solution about t to Eq. (2.1) 1

Sn ¼ 

1

eðnþ2Þa  ; 1 sin nb þ e2a t

Rn ¼ 

eðnþ2Þa  ; 1 sin nb þ e2a t

ð5:19aÞ

eðnþ1Þa  ; 1 cos nb þ e2a t

Tn ¼ 

eðnþ1Þa  : 1 cos nb þ e2a t

ð5:19bÞ

Qn ¼ 

6. Interaction solutions In order to obtain more exact solutions in the double Casoratian form of Eq. (2.1), we set B as the paradiagonal matrix constituting of the matrices (3.6) ðBr Þ, (4.1) ðBm Þ and (5.1) ðBc Þ

0 B B¼@

Br

0 Bm

0

1 C A;

ð6:1Þ

Bc

then Un and Wn defined as (2.6) are described by

Un ¼ ðUTnr ; UTnm ; UTnc ÞT ;

Wn ¼ ðWTnr ; WTnm ; WTnc ÞT ;

ð6:2Þ

where

EUnz ¼ Az Unz ; E1 Wnz ¼ Az Wnz ; 1 1 Unz;t ¼ EUnz ; Wnz;t ¼  E1 Wnz ; 2 2

ð6:3aÞ z ¼ r; m; c:

ð6:3bÞ

Obviously, the double Casorati determinant (2.11) constructed by (6.2) is a solution to Eq. (2.4) and it is called the interaction solution. Choosing

Un ¼ ð/1;nr ; /2;nr ; Unm ÞT ;

Wn ¼ ðw1;nr ; w2;nr ; Wnm ÞT

ð6:4Þ

and ðm; pÞ ¼ ð1; 0Þ, we derive 1

Qn ¼

ekþt  2e2kþt þ et 1k

1

ð2n þ t þ 1Þ þ ð2n þ tÞe2k þ etnke2

1k

1k

Rn ¼

ð2n þ t þ 1Þenke2 t þ ð2n þ tÞenke2 1k 2

ð2n þ t þ 1Þ þ ð2n þ tÞe þ e 3

Sn ¼

t12k

ð6:5bÞ

;

1

1k

1

ð2n þ t þ 1Þek  ð2n þ t þ 2Þe2k þ etnke2 1k

e2kt þ ð2n þ t þ 1Þenke2

t12k

ð6:5cÞ

; t 1k

 ð2n þ t þ 2Þenke2 1k 2

1k tnke2 t

ð2n þ t þ 1Þek  ð2n þ t þ 2Þe þ e When ðm; pÞ ¼ ð0; 1Þ, we get

þ et

1k tnke2 t

e2kþt  2ekþt þ e2kþt 1

Tn ¼

ð6:5aÞ

; t

t

:

ð6:5dÞ

2958

S.t. Chen et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 2949–2959 1k

1k

Qn ¼

ð2n þ t þ 1Þenkþe2 t þ ð2n þ tÞenkþe2 12k

ð2n þ t þ 1Þ  ð2n þ tÞe

tþ12k

þ et

1k nkþe2 tt

ð6:6aÞ

;

e

1

ekt  2e2kt þ et

Rn ¼

1k

Sn ¼

1k

1

ð2n þ t þ 1Þ  ð2n þ tÞe2k  enkþe2 ð2n þ t þ 1Þenkþe2

tþ12k

1k

1

 ð2n þ t þ 2Þenkþe2 t þ et2k 1k

1

ð2n þ t þ 1Þek þ ð2n þ t þ 2Þe2k  enkþe2 3

Tn ¼

ð6:6bÞ

; tt

;

ð6:6cÞ

tt

1

e2kt  2ekt þ e2kt 1

1k

ð2n þ t þ 1Þek þ ð2n þ t þ 2Þe2k  enkþe2

:

ð6:6dÞ

tt

Taking

Un ¼ ð/1;nr ; /2;nr ; /1;nm ; /2;nm ÞT ;

Wn ¼ ðw1;nr ; w2;nr ; w1;nm ; w2;nm ÞT

ð6:7Þ

and ðm; pÞ ¼ ð2; 0Þ, we derive 1k 1k 1k 1 1 1 1 3 1 1 1 3 1 1 fn ¼  ð2n þ tÞenkþe2 tþ2k þ ð2n þ t þ 1Þenkþe2 tþk  ð2n þ t þ 2Þenkþe2 tþ2k  e2kþt t  ðn  t þ 1Þekþt þ ð4n  t þ 2Þe2kþt  net ; 4 2 4 4 2 4 2 1k 1k 1k 1k 1k 1 3 1 5 3 1 g n ¼ enkþe2 tþ2kþt  enkþe2 tþ2kþt þ enkþe2 tþ2kþt  enkþe2 tþkþt þ enkþe2 tþ2kþt ; 4 2 4 1k 1k 1 1 1 1 1 1 1 1 1 hn ¼ enkþe2 tþ2kt þ enke2 t2kþt  ð2nt þ t2 þ tÞek  ð4n2  2nt þ 4n  2t2  t þ 1Þe2k  ð4n2 þ 2nt þ 4n þ t þ 1Þe2k 4 4 4 4 4 1 þ ð8n2 þ 2nt þ 8n  t 2 þ tÞ;ð6:8cÞ 4

 1k 1k 1k 1 1 1 1 1 1 5 3 3 F n ¼ jAj2  ð2n þ t þ 1Þenkþe2 tþ2k þ ð2n þ t þ 2Þenkþe2 tþ2k  ð2n þ t þ 3Þenkþe2 tþ2k  e2kþt t  ð2n  2t þ 3Þe2kþt 4 2 4 4 4  1 1 1 þ ð4n  t þ 4Þekþt  ð2n þ 1Þe2kþt ; 4 4   1k 1k 1k 1k 1k 1 1 3 1 7 5 3 Gn ¼ jAj2 enkþe2 tþ2kþt  enkþe2 tþ3kþt þ enkþe2 tþ2kþt  enkþe2 tþ2kþt þ enkþe2 tþ2kþt ; 4 2 4  1 1 nke12k t1kþt 1 1 3k 12 1 nkþe2k tþ3kt 2 2 2 e þ e  ð2nt þ t þ 2tÞe2  ð2n2  nt þ 4n  t2  t þ 2Þek Hn ¼ jAj 4 4 4 2  1 1 1k 2 2 2 þ ð8n þ 2nt þ 16n  t þ 2t þ 6Þe2  ð2n þ nt þ 4n þ t þ 2Þ : 4 2

ð6:8aÞ ð6:8bÞ

ð6:8dÞ ð6:8eÞ

ð6:8fÞ

Substituting (6.8) into (2.3), we obtain the interaction solution of Eq. (2.1) between the rational-like and the Matveev solutions. Besides, we can also work out the interaction solution under the conditions of ðm; pÞ ¼ ð1; 1Þ and ðm; pÞ ¼ ð0; 2Þ. Similarly, we can also deduce the interaction solutions between the rational-like cases and complexiton cases or the Matveev cases and complexiton cases. 7. Conclusions In this paper, we have expressed a matrix equation satisfied by double Casoratian entries for the first-order four-potential isospectral AL equation. Through the Casoratian technique we get the generalized double Casoratian solutions. By expanding the general solutions of Un and Wn as the series of the arbitrary matrix B, we have constructed the soliton solutions and rational-like solutions through letting B be a triangular matrix and Jordan matrix with zero on the main diagonal. Furthermore, taking B be some other special matrices, such as Jordan matrices and combination of different Jordan form matrices with respect to rational cases, Matveev cases and Complexiton cases, we have derived the Matveev, Complexiton and interaction  solutions, respectively. It is interesting that under the condition sin 12 b ¼ 1, the complexiton solution (5.18) is reduced to a periodic solution about t. Acknowledgments The authors are very grateful to Professor D.J. Zhang and Professor R.G. Zhou for their ardent guidance and help. This project is supported by the National Natural Science Foundation of China (Grant Nos. 11071157, 11101350, 11271168), the National Natural Science Foundation of the colleges and universities of Jiangsu Province (Grant No. 11KJB110016), School Foundation of Xuzhou Institute of Technology (Grant No. XKY 2011204) and PAPD of Jiangsu Higher Education Institutions. References [1] Freeman NC, Nimmo JJC. Soliton solutions of the KdV and KP equations: the Wronskian technique. Phys Lett A 1983;95:1–3. [2] Nimmo JJC. Soliton solutions of three differential-difference equations in Wronskian form. Phys Lett A 1983;99:281–6.

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