Multisoliton solutions to a nonisospectral (2+1 )-dimensional breaking soliton equation

Physics Letters A 372 (2008) 2017–2025 www.elsevier.com/locate/pla

Multisoliton solutions to a nonisospectral (2 + 1)-dimensional breaking soliton equation Yuqin Yao a,∗ , Dengyuan Chen b , Dajun Zhang b a Department of Mathematics, Tsinghua University, 100084, PR China b Department of Mathematics, Shanghai University, Shanghai 200444, PR China

Received 25 July 2007; accepted 29 October 2007 Available online 17 November 2007 Communicated by A.R. Bishop

Abstract A nonisospectral (2 + 1)-dimensional breaking soliton equation ((2 + 1)DBSE) is derived, which corresponds to the spectral parameter λ satisfying 2λλy − λt = λ2 . The bilinear form for the nonisospectral (2 + 1)DBSE is obtained and the multisoliton solutions are worked out by means of the Hirota method and Wronskian technique, respectively. The nonisospectral (2 + 1)-dimensional breaking nonlinear Schrödinger equation and its multisoliton solutions are also presented by reduction. © 2007 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 05.45.Yv Keywords: Nonisospectral breaking soliton equation; Hirota method; Wronskian technique; Reduction

1. Introduction Bilinear method including Hirota method and Wronskian technique is an efficient method for searching for soliton solutions of the nonlinear evolution equations. Hirota method was first proposed by Hirota in 1971 to obtain the N -soliton solutions of the KdV equation [1]. Soliton solutions can also be written in Wronskian form, which was first introduced by Satsuma [2] in 1979. Taking the advantage that special structure of a Wronskian contributes simple forms of its derivatives, Freeman and Nimmo [3] developed the Wronskian technique which admits direct verifications of solutions in Wronskian form to the bilinear equations. A determinant with double Wronskian structure is a generalization comparing with the standard Wronskian. Many soliton equations, such as the nonlinear Schrödinger equation [4,5], the 2-dimensional Toda lattice [6], the AKNS hierarchy [7] and some equations constrained from the KP hierarchy [8] admit solutions in double Wronskian form. The breaking soliton equations are a kind of nonlinear evolution equations which can be used to describe the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long-wave propagating along the x-axis. As for as the (2 + 1)DBSE associated with the AKNS hierarchy, which was proposed by O.I. Bogoyovlenskii [9,10]. In Ref. [11], many symmetries was constructed by infinitesimal “dressing” method. In Ref. [12], N -soliton solutions and double Wronskian solution were worked out by bilinear method. In recent years, there has been much interest in study of the variable coefficient generalizations of complete integrable nonlinear evolution equations [13–17]. In the present Letter, we aim to investigate the soliton solutions of the nonisospectral (2 + 1)DBSE associated with the AKNS hierarchy. We first deduce the nonisospectral (2 + 1)DBSE which corresponds to the spectral parameter λ satisfying 2λλy − λt = λ2 . Then it is transformed into the bilinear form by which N -soliton solutions in Hirota’s form and * Corresponding author. Tel.: +86 1013810633589.

E-mail address: [email protected] (Y. Yao). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.096

2018

Y. Yao et al. / Physics Letters A 372 (2008) 2017–2025

Wronskian form are obtained. Besides, the nonisospectral (2 + 1)-dimensional breaking nonlinear Schrödinger equation and its multisoliton solutions are also presented by reduction. The Letter is organized as follows. In Section 2 the nonisospectral (2 + 1)DBSE is derived and its Lax pair is given. In Section 3, we solve the nonisospectral (2 + 1)DBSE by Hirota method. In Section 4 solution in double Wronskian form is proven. In Section 5 the nonisospectral (2 + 1)-dimensional breaking nonlinear Schrödinger equation and its N -soliton solutions are given by reduction. 2. Lax integrability of the nonisospectral (2 + 1)DBSE Consider the following spectral problem   −λ q , λx = 0, ψx = Mψ, M = r λ with the time evolution ψt = 2yλψy + N ψ,

 N=

A C

 B . −A

(2.1)

(2.2)

The compatibility of (2.1) and (2.2) gives the zero-curvature equation Mt − Nx + [M, N ] − 2yλMy = 0.

(2.3)

From (2.3), we have

  −B + (λt − 2λλy )x + A0 , A = ∂x−1 (r, q) C             −B q −B xq q q + 2(2λλy − λt )σ − 2λ + 2yλ y − 2A0 σ , =L C C xr r r t ry

(2.4a) (2.4b)

where A0 is a constant and  −∂x + 2q∂x−1 r L= −2r∂x−1 r

   2q∂x−1 q −1 0 (2.4c) , σ= . 0 1 ∂x − 2r∂x−1 q  Expanding (−B, C)T as (−B, C)T = nj=1 (−bj , cj )T λn−j and taking n = 2, 2λλy − λt = λ2 , A0 = 0 in Eq. (2.4), we have  qt = −qx + q∂x−1 qr − 12 x(qxx − 2q 2 r) + y[−qxy + 2q∂x−1 (qr)y ], (2.5) rt = rx − r∂x−1 qr + 12 x(rxx − 2qr 2 ) + y[rxy − 2r∂x−1 (qr)y ]. The corresponding A, B and C in N are 1 1 A = −xλ2 + xqr + ∂x−1 qr + y∂x−1 (qr)y , 2 2 1 1 B = xqλ − xqx − q − yqy , 2 2 1 1 C = xrλ + xrx + r + yry . 2 2

(2.6a) (2.6b) (2.6c)

3. Bilinear form and multisoliton solutions Introducing the dependent variable transformation q=

g , f

h r =− , f

(2.5) can be transformed into the bilinear form   x Dt + Dx2 + yDx Dy g · f = −gx f, 2   x 2 Dt − Dx − yDx Dy h · f = hx f, 2 Dx2 f · f − 2gh = 0,

(3.1)

(3.2a) (3.2b) (3.2c)

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where D is the well-known Hirota bilinear operator [18] defined by Dtm Dxn Dyl f · g = (∂t − ∂t  )m (∂x − ∂x  )n (∂y − ∂y  )l f (t, x, y)g(t  , x  , y  )|t  =t,x  =x,y  =y . We expanding f, g, h into power series of a small parameter  as f = 1 + f (2)  2 + f (4)  4 + · · · , +g

(3.3a)

 + ···,

(3.3b)

h = h  + h  + ···.

(3.3c)

g=g

(1)

(3) 3 (3) 3

(1)

Substituting (3.3) into (3.2) and equating coefficients of  yield x (1) (1) + gxx + ygxy + gx(1) = 0, 2   x (3) x 2 (3) (3) (3) gt + gxx + ygxy + gx = − Dt + Dx + yDx Dy g (1) · f (2) − gx(1) f (2) , 2 2 (1)

(3.4a)

gt

(3.4b)

···, x (1) (1) − yh(1) ht − h(1) xy − hx = 0, 2 xx   x x 2 (3) (3) (3) (1) (2) − yh − h = − D − − yD D · f (2) + h(1) , D ht − h(3) t x y h xy x x f 2 xx 2 x

(3.5a) (3.5b)

···, and (2) = g (1) h(1) , fxx (4) 2fxx

= −Dx2 f (2)

·f

(2)

(3.6a)

  + 2 g (1) h(3) + g (3) h(1) ,

(3.6b)

···. Take g (1) = ω1 (t)eξ1 ,

h(1) = υ1 (t)eη1 ,

ξ1 = k1 (t)x + p1 (t)y,

η1 = l1 (t)x + q1 (t)y.

(3.7)

From (3.4)–(3.6), we have 1 k1,t (t) = − k12 (t), 2

p1,t (t) = −k1 (t)p1 (t),

1 l1,t (t) = l12 (t), 2

q1,t (t) = l1 (t)q1 (t),

ω1,t (t) = −k1 (t)ω1 (t), υ1,t (t) = l1 (t)υ1 (t), (t)υ (t) ω 1 1 f (2) = eξ1 +η1 , f (j ) = 0, j = 4, 6, . . . , (k1 (t) + l1 (t))2

(3.8a) (3.8b)

and g (j ) = 0,

h(j ) = 0,

j = 3, 5, . . . .

(3.8c)

Further, from (3.8a), we can work out k1 (t) =

2 , t + 2c1

p1 (t) = ω1 (t) =

1 , (t + 2c1 )2

l1 (t) =

2 , 2c1 − t

q1 (t) = υ1 (t) =

1 . (2c1 − t)2

So the one-soliton solution to the nonisospectral (2 + 1)DBSE (2.5) is  1 (t)υ1 (t) ] 1 ω1 (t)υ1 (t) ω1 (t) 12 [ξ1 −η1 −ln (kω(t)+l 2 (t)) 1 1 Sech ξ1 + η1 + ln e , q= 2 2 (k1 (t) + l1 (t))2  1 (t)υ1 (t) ] υ1 (t) 12 [η1 −ξ1 −ln (kω(t)+l 1 ω1 (t)υ1 (t) 2 1 1 (t)) r= Sech ξ1 + η1 + ln e . 2 2 (k1 (t) + l1 (t))2 From Fig. 1, we can easily find that (3.10a) provide a line-soliton traveling with time-space-varying amplitude

(3.9)

(3.10a) (3.10b) ω1 (t) 2

×

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Fig. 1. The shape and motion of the one-soliton q for c1 = 0.5, t = 0.7. 1

e2

[ξ1 −η1 −ln

ω1 (t)υ1 (t) ] (k1 (t)+l1 (t))2

x(t, y) =

and top trace

t 2 + 4c12 4c1 (t 2 − 4c12 )

y−

t 2 − 4c12 ln 64c12 . 8c1

Similar to the one-soliton, if we take g (1) = ω1 (t)eξ1 + ω2 (t)eξ2 ,

h(1) = υ1 (t)eη1 + υ2 (t)eη2 ,

ξj = kj (t)x + pj (t)y,

ηj = lj (t)x + qj (t)y,

j = 1, 2,

(3.11)

then 1 kj,t (t) = − kj2 (t), 2

pj,t (t) = −kj (t)pj (t),

1 lj,t (t) = lj2 (t), 2

qj,t (t) = lj (t)qj (t),

ωj,t (t) = −kj (t)ωj (t), υj,t (t) = lj (t)υj (t), j = 1, 2, ω1 (t)υ1 (t) ω1 (t)υ2 (t) ω2 (t)υ1 (t) ω2 (t)υ2 (t) f (2) = eξ1 +η1 + eξ1 +η2 + eξ2 +η1 + eξ2 +η2 , (k1 (t) + l1 (t))2 (k1 (t) + l2 (t))2 (k2 (t) + l1 (t))2 (k2 (t) + l2 (t))2

(3.12) (3.13)

f (4) =

(k1 (t) − k2 (t))2 (l1 (t) − l2 (t))2 ω1 (t)ω2 (t)υ1 (t)υ2 (t) eξ1 +ξ2 +η1 +η2 , (k1 (t) + l1 (t))2 (k2 (t) + l1 (t))2 (k1 (t) + l2 (t))2 (k2 (t) + l2 (t))2

g (3) =

(k1 (t) − k2 (t))2 ω1 (t)ω2 (t)υ1 (t) ξ1 +ξ2 +η1 (k1 (t) − k2 (t))2 ω1 (t)ω2 (t)υ2 (t) ξ1 +ξ2 +η2 e + e , (k1 (t) + l1 (t))2 (k2 (t) + l1 (t))2 (k1 (t) + l2 (t))2 (k2 (t) + l2 (t))2

(3.15)

h(3) =

(l1 (t) − l2 (t))2 ω1 (t)υ1 (t)υ2 (t) ξ1 +η1 +η2 (l1 (t) − l2 (t))2 ω2 (t)υ1 (t)υ2 (t) ξ2 +η1 +η2 e + e , (k1 (t) + l1 (t))2 (k1 (t) + l2 (t))2 (k2 (t) + l1 (t))2 (k2 (t) + l2 (t))2

(3.16)

g (l) = 0, h(l) = 0, l = 5, 7, . . . .

f (l) = 0,

l = 6, 8, . . . ,

(3.14)

(3.17)

Therefore the two-soliton solution is obtained from (3.1) here f = 1 + f (2) + f (4) ,

g = g (1) + g (3) ,

h = h(1) + h(3) .

The limits of this solution are easily obtained: ⎧ ω2 (t)eξ2 ⎪ , ⎪ ω2 (t)υ2 (t) ⎪ ⎨ 1+ (k (t)+l (t))2 eξ2 +η2 2 2 q→ ω2 (t)eξ2 ⎪ ⎪ , 2 2 ⎪ (k (t)+l (t)) (l ξ2 +η2 ⎩ 2 1 1 (t)−l2 (t)) ω2 (t)υ2 (t) (k1 (t)−k2 (t))2

q→

⎧ ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩

+

(k1 (t)+l2 (t))2 (k2 (t)+l2 (t))2

ω1 (t)eξ1 ω1 (t)υ1 (t) eξ1 +η1 (k1 (t)+l1 (t))2

e

ξ1 → +∞, η1 → +∞, ξ2 → −∞, η2 → −∞,

,

ω1 (t)eξ1 (k1 (t)+l1 (t))2 (l1 (t)−l2 (t))2 ω1 (t)υ1 (t) + eξ1 +η1 (k1 (t)−k2 (t))2 (k1 (t)+l1 (t))2 (k2 (t)+l1 (t))2

ξ1 → −∞, η1 → −∞,

,

ξ2 → +∞, η2 → +∞.

(3.18)

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Fig. 2. The shape and motion of the two-soliton q for c1 = 2, c2 = 1, t = 0.

The ξi → −∞, ηi → −∞ (i = 1, 2) limits are of the standard form of the one-soliton (3.10). While for ξi → +∞, ηi → +∞ (i = 1 −k2 )(l1 −l2 ) 1, 2) there are rightward phase shift ln (k (k1 +l2 )(k2 +l1 ) which can be showed in Fig. 2. We can continue to work out the three-, four-solitons and so on. Generally, we have  2n  2n       A1 (μ) exp μj ξj + ln ωj (t) + μj μi θ j i , fn = (3.19) μ=0,1

gn =



 A2 (μ) exp

hn =

2n  j =1

μ=0,1



j =1

 A3 (μ) exp

2n 

1j
 2n    μj ξj + ln ωj (t) + μj μi θ j i , 

 2n    μj ξj + ln ωj (t) + μj μi θ j i , 

j =1

μ=0,1

(3.20)

1j
(3.21)

1j
where ξj = kj (t)x + pj (t)y, 1 kj,t (t) = − kj2 (t), 2

ηj = lj (t)x + qj (t)y, pj,t (t) = −kj (t)pj (t),

ξn+j = ηj ,

ωn+j (t) = υj (t),

1 lj,t (t) = lj2 (t), 2

ωj,t (t) = −kj (t)ωj (t),

υj,t (t) = lj (t)υj (t),

eθj,n+i =

 2 eθj i = kj (t) − ki (t) ,

 2 eθ(n+j )(n+i) = lj (t) − li (t) ,

j = 1, 2, . . . , n,

(3.22)

qj,t (t) = lj (t)qj (t),

1 , (kj (t) + lj (t))2

j = 1, 2, . . . , n,

j < i = 2, 3, . . . , n,

(3.23) (3.24)

A1 (μ), A2 (μ), and A3 (μ) take over all possible combinations of μj = 0, 1 (j = 1, 2, . . . , 2n) and satisfy the following conditions n 

μj =

j =1

n  j =1

μn+j ,

n  j =1

μj = 1 +

n 

μn+j ,

j =1

1+

n  j =1

μj =

n 

μn+j ,

(3.25)

j =1

respectively. 4. N -soliton solutions in double Wronskian form In the present section, we show that the bilinear equation (3.2) have the solution in double Wronskian form. We start from the following two propositions [3]. Proposition 4.1. |Q, a, b||Q, c, d| − |Q, a, c||Q, b, d| + |Q, a, d||Q, b, c| = 0, where Q is an (N + M + 2) × (N + M) matrix, a, b, c and d represent (N + M + 2) column vectors.

(4.1)

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Y. Yao et al. / Physics Letters A 372 (2008) 2017–2025

Proposition 4.2. N+M+2 

|α1 , . . . , αj −1 , bαj , αj +1 , . . . , αN+M+2 | =

N+M+2 

j =1

 bj |α1 , . . . , αN+M+2 |,

(4.2)

j =1

where αj are (N + M + 2) column vectors and bαj denotes (b1 α1j , b2 α2j , . . . , bN+M+2 αN+M+2,j )T . First we recall some results about the isospectral (2 + 1)-DBSE qt = −qxy + 2q∂x−1 (qr)y ,

(4.3a)

rt = rxy − 2r∂x−1 (qr)y ,

(4.3b)

which has the bilinear form (Dt + Dx Dy )g · f = 0,

(4.4a)

(Dt − Dx Dy )h · f = 0,

(4.4b)

Dx2 f

· f − 2gh = 0,

(4.4c)

through the transformation (3.1). In Ref. [12], we have proved the following theorem: Theorem 4.1. The following double Wronskian   ˆ f = φ, φ (1) , . . . , φ (N ) ; ψ, ψ (1) , . . . , ψ (M)  = |Nˆ ; M|,     + 1; M − 1|, g = 2φ, φ (1) , . . . , φ (N+1) ; ψ, ψ (1) , . . . , ψ (M−1)  = 2|N     − 1; M + 1|, h = 2φ, φ (1) , . . . , φ (N−1) ; ψ, ψ (1) , . . . , ψ (M+1)  = 2|N

(4.5) (4.6) (4.7)

solve the bilinear isospectral (2 + 1)DBSE (4.4) where φ = (φ1 , φ2 , . . . , φN+M+2 )T , ψ = (ψ1 , ψ2 , . . . , ψN+M+2 )T and φj , ψj (j = 1, 2, . . . , N + M + 2) enjoy the following conditions    φjy = φj xx , φj x = kj φj , φj t = −2φj xxx , (4.8) ψjy = −ψj xx , ψj x = −kj ψj , ψj t = −2ψj xxx . In case of the nonisospectral (2 + 1)DBSE (2.5), we have Theorem 4.2. The double Wronskian (4.5)–(4.7) with the conditions    φjy = φj xx , φj x = kj φj , φj t = −2yφj xxx − xφj xx + N φj x , ψjy = −ψj xx , ψj x = −kj ψj , ψj t = −2yψj xxx + xψj xx − Mψj x

(4.9)

solve the bilinear nonisospectral (2 + 1)DBSE (3.2). In order to prove Theorem 4.2, let us first focus our attention on Eqs. (3.2) and (4.4). Further, with the Theorem 4.1 in hand, we find we only need to prove the following result: Lemma 4.1. The double Wronskian (4.5)–(4.7) with  φj t = −xφj xx + N φj x , ψj t = xψj xx − Mψj x , solve the bilinear equation   x Dt + Dx2 g · f = −gx f, 2   x 2 Dt − Dx h · f = hx f. 2

(4.10)

(4.11a) (4.11b)

Prove. We first prove (4.11a). Note that the lth-order derivatives of φj , ψj with respect to x are (l)

(l+2)

φj t = −xφj (l) ψj t

(l+2) = xψj

(l+1)

+ (N − l)φj

+ (−M

,

(l+1) + l)ψj .

(4.12a) (4.12b)

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We have   ˆ + |Nˆ ; M − 1, N + 1; M| − 1, M + 1|, fx = |N     ˆ + |N ˆ + 2|N fxx = |N − 1, N + 2; M| − 2, N, N + 1; M| − 1, N + 1; M − 1, M + 1|   ˆ M + |N; − 1, M + 2| + |Nˆ ; M − 2, M, M + 1|,       ˆ − |N − 2, N, N + 1; M| ˆ − |Nˆ ; M − 1, M + 2| + |Nˆ ; M − 2, M, M + 1| , ft = −x |N − 1, N + 2; M|      − 1| + |N + 1; M − 2, M| , gx = 2 |Nˆ , N + 2; M      ˆ N + 3; M − 1| + |N − 1, N + 1, N + 2; M − 1| + 2|Nˆ , N + 2; M − 2, M| gxx = 2 |N,      + |N + 1; M − 2, M + 1| + |N + 1; M − 3, M − 1, M| ,       − 1| − |N − 1, N + 1, N + 2; M − 1| − |N + 1; M − 2, M + 1| gt = −2x |Nˆ , N + 3; M       + |N + 1; M − 3, M − 1, M| − 2|Nˆ , N + 2; M − 1| − 2|N + 1; M − 2, M|. Substituting f, g and their derivatives with respect to x, t into (4.11a) gives        ˆ M| ˆ −|Nˆ , N + 3; M x |N; − 1| + 3|N − 1, N + 1, N + 2; M − 1| + 3|N + 1; M − 2, M + 1|         ˆ − |N + 1; M − 3, M − 1, M| + 2|Nˆ , N + 2; M − 2, M| + |N + 1; M − 1| 3|N − 1, N + 2; M|

      ˆ + 2|N − |N − 2, N, N + 1; M| − 1, N + 1; M − 1, M + 1| − |Nˆ ; M − 1, M + 2| + 3|Nˆ ; M − 2, M, M + 1|         ˆ + |Nˆ ; M − 2 |Nˆ , N + 2; M − 1| + |N + 1; M − 2, M| |N − 1, N + 1; M| − 1, M + 1| .

Utilizing the following identities:     ˆ − |N; ˆ M| ˆ ˆ N + 2; M ki |Nˆ , N + 2; M − 1| = 0, |N, − 1| ki |Nˆ ; M|       ˆ − |Nˆ ; M| ˆ |N + 1; M − 2, M| ki |Nˆ ; M| ki |N + 1; M − 2, M| = 0,         ˆ − |N ˆ |N + 1; M − 1| ki |N − 1, N + 1; M| − 1, N + 1; M| ki |N + 1; M − 1| = 0,         |N + 1; M − 1| ki |Nˆ ; M − 1, M + 1| − |Nˆ ; M − 1, M + 1| ki |N + 1; M − 1| = 0,

(4.13)

(4.14) (4.15) (4.16) (4.17)

(4.13) becomes            ˆ M| ˆ |N ˆ 4x |N; − 1, N + 1, N + 2; M − 1| + |N + 1; M − 2, M + 1| + |N + 1; M − 1| |N − 1, N + 2; M|         ˆ − |N + |Nˆ ; M − 2, M, M + 1| − |Nˆ , N + 2; M − 1||N − 1, N + 1; M| + 1; M − 2, M||Nˆ ; M − 1, M + 1| .

(4.18)

Using Proposition 4.1 we can know that Eq. (4.18) is equal to zero. Thus, we have proven (4.11a). Similarly, we can prove (4.11b). 2 5. Reduction Setting r = q ∗ and replacing t by −it in Eq. (2.5), where ∗ denotes the complex conjugate, i = nonisospectral (2 + 1)-dimensional breaking nonlinear Schrödinger equation:    x iqt − qx + q∂x−1 |q|2 − qxx − 2q|q|2 − y qxy − 2q∂x−1 |q|2y = 0, 2 with the bilinear form   x iDt + Dx2 + yDx Dy g · f = −gx f, 2 Dx2 f · f + 2gg ∗ = 0,

gn =

 μ=0,1

−1, we obtain the following (5.1)

(5.2a) (5.2b)

and the solutions can be denoted as  2n  2n        A1 (μ) exp μj ξj + ln ωj (t) + μj μi θj i , fn = μ=0,1

√

 A2 (μ) exp

j =1 2n  j =1

(5.3)

1j
 2n     μj ξj + ln ωj (t) + μj μi θj i , 

1j
(5.4)

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Y. Yao et al. / Physics Letters A 372 (2008) 2017–2025

(a)

(b)

(c) Fig. 3. A stationary soliton of Eq. (5.1) for (a) c1 = 0.15, t = 0, (b) c1 = 2, t = 0, (c) c1 = 1 + 2i, t = 0.

where ξj = kj (t)x + pj (t)y,

ξn+j = ξj∗ ,

pj,t (t) = ikj (t)pj (t),

ωj,t (t) = ikj (t)ωj (t),

 2  eθj i = kj (t) − ki (t) ,

e

 θ(n+j )(n+i)

i kj,t (t) = kj2 (t), 2 1 = , j = 1, 2, . . . , n, (kj (t) + kj∗ (t))2

ωn+j (t) = ωj∗ (t), 

eθj,n+i

 2 = kj∗ (t) − ki∗ (t) ,

j < i = 2, 3, . . . , n,

(5.5) (5.6) (5.7)

A1 (μ), A2 (μ), take over all possible combinations of μj = 0, 1 (j = 1, 2, . . . , 2n) and satisfy the following conditions n  j =1

μj =

n  j =1

μn+j ,

n  j =1

μj = 1 +

n 

μn+j ,

(5.8)

j =1

respectively. For example, when n = 1 we can obtain the one-soliton of the Eq. (5.1).   Re[ω1 ]2 + Im[ω1 ]2 Re[ω1 ]2 + Im[ω1 ]2 1 Re[ω1 ]2 + Im[ω1 ]2 2 |q|2 = Sech ] + ln ln Re[ξ , 1 4 2 4 Re[k1 ]2 4 Re[k1 ]2

(5.9)

where Re[·] and Im[·] mean the real part and imaginary part, respectively, which can be described in Fig. 3. In the following we deduce the double Wronskian solution of Eq. (5.2). Let AN×M and BN×M be two N × M matrices, which have the following forms ⎛ ⎛ ⎞ ⎞ ϕ1 ∂ϕ1 · · · ∂ M−1 ϕ1 ψ1 ∂ψ1 · · · ∂ M−1 ψ1 ∂ϕ2 · · · ∂ M−1 ϕ2 ⎟ ∂ψ2 · · · ∂ M−1 ψ2 ⎟ ⎜ϕ ⎜ψ BN×M = ⎝ 2 AN×M = ⎝ 2 ⎠, ⎠, ··· ··· ··· ··· ··· ··· ··· ··· ϕN ∂ϕN · · · ∂ M−1 ϕN ψN ∂ψN · · · ∂ M−1 ψN where ϕj and ψj satisfy the following conditions    φjy = φj xx , φj x = kj φj , φj t = 2iyφj xxx + ixφj xx − iNφj x , ψjy = −ψj xx , ψj x = −kj ψj , ψj t = 2iyψj xxx − ixψj xx + iNψj x ,

(5.10)

Y. Yao et al. / Physics Letters A 372 (2008) 2017–2025

respectively. We define   A B(N+1)×(N+1)  ˆ ˆ f˜ =  (N+1)×(N+1) ∗  ≡ |N ; N|, A∗(N+1)×(N+1) B(N+1)×(N+1)    A B(N+1)×N    ≡ 2|N + 1; N − 1|. g˜ = 2  (N+1)×(N+2) ∗ −B(N+1)×(N+2) −A∗(N+1)×N  According to the determinantal properties, we have   ∗  ∗   A(N+1)×(N+2)  B(N+1)×N  = 2(−1)N+1  −B∗(N+1)×(N+2) g˜ ∗ = 2   A  −B(N+1)×(N+2) −A(N+1)×N (N+1)×(N+2)     4N+1  −A(N+1)×N −B(N+1)×(N+2)    = −2|N − 1; N + 1|. = 2(−1)  B∗ A∗(N+1)×(N+2)  (N+1)×N

2025

(5.11) (5.12)  −A(N+1)×N  ∗  B(N+1)×N (5.13)

Letting M = N in Eqs. (4.5)–(4.7) gives ˆ Nˆ |, f = |N;

  g = 2|N + 1; N − 1|,

  h = 2|N − 1; N + 1|.

Therefore, in the similar way, we have Theorem 5.1. The nonisospectral (2 + 1)-dimensional breaking nonlinear Schrödinger equation (5.2) has the double Wronskian solution ˆ Nˆ |, f˜ = |N;

  g˜ = 2|N + 1; N − 1|,

  g˜ ∗ = −2|N − 1; N + 1|,

(5.14)

where ϕj and ψj enjoy the conditions (5.10). Acknowledgements The authors are very grateful to the Referees for their invaluable comments. This project is supported by the National Science Foundation of China (10671121) the National Basic Research Program of China (973 program) (2007 CB814800) and the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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