Rational and Periodic Solutions for a (2+1)-Dimensional Breaking Soliton Equation Associated

with ZS-AKNS Hierarchy

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Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 430–434c© Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 3, March 15, 2010

Rational and Periodic Solutions for a (2+1)-Dimensional Breaking Soliton Equation

Associated with ZS-AKNS Hierarchy∗

HAO Hong-Hai (Ï÷°),1,† ZHANG Da-Jun (Ü��),2,‡ ZHANG Jian-Bing (ÜïW),2 and

YAO Yu-Qin (���)3

1Department of Plan and Finance, Hebei Branch, China Construction Bank, Shijiazhuang 050000, China

2Department of Mathematics, Shanghai University, Shanghai 200444, China

3Department of Mathematics, Tsinghua University, Beijing 100084, China

(Received February 16, 2009)

Abstract The double Wronskian solutions whose entries satisfy matrix equation for a (2+1)-dimensional breakingsoliton equation ((2+1)DBSE) associated with the ZS-AKNS hierarchy are derived through the Wronskian technique.Rational and periodic solutions for (2+1)DBSE are obtained by taking special cases in general double Wronskian solu-tions.

PACS numbers: 02.30.Ik, 05.45.Yv

Key words: (2+1)DBSE, double Wronskian, rational solution, periodic solution

1 Introduction

The breaking soliton equations (BSEs) are a kind of

nonlinear evolution equations which can be used to de-

scribe the (2+1)-dimensional interaction of a Riemann

wave propagating along the y-axis with a long-wave prop-

agating along the x-axis.[1] One of the most famous BSEs

is described as[2]

uxt − 4uxuxy − 2uyuxx + uxxxy = 0 . (1)

The algebraic properties and Lax integrability, the exis-

tence of localized coherent structures, solutions, soliton-

like solutions, the Hamiltonian structure, and the Painleve

property have been discussed in Refs. [3–7]. Reference [3]

has shown that the self-dual Yang–Mills equation belongs

to the class of BSEs.

The (2+1)-dimensional breaking soliton equa-

tion ((2+1)DBSE) associated with the ZS-AKNS

hierarchy,[8−9] which was proposed by Bogoyovlenskii,[1]

takes the form

qt = −qxy + 2q∂−1x (qr)y ,

rt = qxy − 2r∂−1x (qr)y , (2)

where q and r are potential functions, ∂ = ∂/∂x, ∂∂−1 =

∂−1∂ = 1. In 1993, Li et al. constructed many sym-

metries by infinitesimal dressing method,[10] recently, the

multisoliton solutions were obtained by Hirota’s bilinear

method.[11] In this letter, we turn our interest to the un-

derstanding of the properties belong to its solutions. We

would like to argue that such a discussion is more than

worthwhile as it allows one to uncover a lot of peculiar

phenomena.

Our discussion bases on the Wronski determinant

(Wronskian), which was first introduced to describing soli-

tary waves by Freeman and Nimmo.[12−13] Recently Chen

et al. extended the traditional condition equation to the

arbitrary matrix equation and they obtained the rational

and complexion solutions[14] in terms of double Wronskian

form for AKNS system.[15] In this paper, by making use

of Chen’s method, we discuss the rational solutions and

periodic solutions in terms of double Wronskian for the

(2+1)DBSE.

The paper is organized as follows. In Sec. 2, the hier-

archy of (2+1)DBSEs is derived and the Lax pair of (2)

is given. In Sec. 3, we give the double Wronskian solu-

tion of (2) whose entries satisfy general matrix equation.

The rational solutions and periodic solutions are obtained

by taking special cases in double Wronskian solutions. A

conclusion is given in the final section.

2 Lax Integrability of BSE

In this section, we derive the hierarchy of the (2+1)-

dimensional breaking soliton equations which concerns

with the ZS-AKNS soliton hierarchy closely. Consider the

ZS-AKNS spectral problem[9]

ψx = Mψ, M =

(−λ q

r λ

), (3)

where q, r are potential functions, the spectral parameter

λ = λ(y, t) is independent of x. The adjoint time evolution

ψt = 2λψy +Nψ, N =

(A B

C −A

). (4)

From (3) and (4), we know there have two main charac-

teristic features of BSEs, one is the spectral parameter λ

∗Supported by the National Natural Science Foundation of China under Grant No. 10671121†Corresponding author, E-mail: hao−haier@shu.edu.cn‡E-amil: djzhang@staff.shu.edu.cn

No. 3 Rational and Periodic Solutions for a (2+1)-Dimensional Breaking Soliton Equation Associated with ZS-AKNS Hierarchy 431

possesses so-called breaking behavior, in other word the

spectral value becomes a multivalued function about y

and t. The other feature is the time evolution not only

depends on N but also ψy. Consequently, the solutions of

these equations may be multivalued too and have much

lager types of properties than AKNS hierarchy.

The compatibility of (3) and (4), i.e. the zero-

curvature equation is

Mt −Nx + [M,N ] − 2λMy = 0 . (5)

From (5), we have(q

r

)

t

= L

(−B

C

)− 2λ

(−B

C

)

+ 2(2λλy − λt)σ

(xq

xr

)

+ 2λ

(q

r

)

y

− 2A0σ

(q

r

), (6)

where A0 is a constant and

L = σ∂ + 2

(q

−r

)∂−1

x (r, q) , σ =

(−1 0

0 1

). (7)

For the purpose of finding soliton hierarchy, we expand

(B,C)T as(−B

C

)=

n∑

j=1

(−bj

cj

)λn−j . (8)

To get the isospectral hierarchy of (2+1)DBSE, we take

2λλy − λt = 0 which is Riemann equation and A0 =

−(1/2)(2λ)n. Inserting (8) into (6), we obtain the isospec-

tral hierarchy by a direct calculation as(q

r

)

t

= Ln

(−q

r

)+ L

(−q

r

)

y

. (9)

When n = 1, (9) reduces to (2+1)DBSE (2) which is Lax

integrable with time evolution

ψt = 2λψy +Nψ, N =

(∂−1

x (qr)y −qy

ry −∂−1x (qr)y

). (10)

Similar as above, if taking 2λλy − λt = (1/2)(2λ)n which

is inhomogeneous Riemann equation and A0 = 0 in (6),

we obtain the nonisospectral hierarchy of (2+1)DBSE(q

r

)

t

= Ln

(−xq

xr

)+ L

(−q

r

)

y

. (11)

To sum-up, we have the following theorem

Theorem 1 Let γ(λ) and ω(λ) are n-th degree Polyno-

mial independent of x, if λ satisfies Riemann equation

λλy − λt =1

2ω(2λ) , (12)

then the hierarchy of (2+1)DBSE associated with ZS-

AKNS problem is(q

r

)

t

= γ(λ)

(−q

r

)+ ω(λ)

(−xq

xr

)

+ L

(−q

r

)

y

, (13)

with the boundary condition

N |(q,r)=(0,0) =

(−(1/2)(γ(2λ) + ω(2λ)x) 0

0 (1/2)(γ(2λ) + ω(2λ)x)

), (14)

where L is the recursive operator (7).

3 Rational and Periodic Solutions for(2+1)DBSE

In this section, we investigate two kinds of solutions for

(2) rational solution and periodic solution. First of all, we

give the double Wronskian solution for (2+1)DBSE. Base

on it, we discuss the rational and periodic solutions.

Equation (2) holds the bilinear form

(Dt +DxDy)g · f = 0, (Dt −DxDy)h · f = 0 ,

D2xf · f = 2gh , (15)

by introducing the dependent variable transformation

q =g

f, r = −

h

f, (16)

where D is the Hirota’s bilinear operator.[16]

The double Wronskian solution for (2+1)DBSE can be

described as the following theorem

Theorem 2 The (2+1)-DBSE (2) has the double Wron-

skian solution

f = |N̂ ; M̂ |, g = 2|N̂ + 1; M̂ − 1| ,

h = 2|N̂ − 1; M̂ + 1| , (17)

where ϕj and ψj satisfy the conditions

ϕx = −Γϕj , ψx = Γψj ,

ϕj,y = ϕj,xx, ψjy = −ψj,xx ,

ϕj,t = −2ϕj,xxx, ψjt = −2ψj,xxx , (18)

respectively, where Γ is an arbitrary constant matrix

whose rank is (N + M + 2). In this paper, we call (18)

Wronskian condition equations of (15).

The proof is given in Ref. [11]. Besides, in theorem 1,

we have adopted the compact notation of Wronskian as

Freeman and Nimmo did[12−13]

WN+1,M+1(ϕ;ψ) = det(ϕ, ∂xϕ, . . . , ∂Nx ϕ;ψ, ∂xψ, . . . , ∂

Mx ψ) = |N̂ ; M̂ | , (19)

where ϕ = (ϕ1, ϕ2, . . . , ϕN+M+2)T and ψ = (ψ1, ψ2, . . . , ψN+M+2)

T.

3.1 Rational Solutions for (2+1)DBSE

The Wronskian condition equations (18) have general solutions

432 HAO Hong-Hai, ZHANG Da-Jun, ZHANG Jian-Bing, and YAO Yu-Qin Vol. 53

ϕ = e−Γx+Γ2y+2Γ3tC, ψ = eΓx−Γ2y−2Γ3tD , (20)

where C = (C1, C2, . . . , CN+M+2)T, D = (D1, D2, . . . ,

DN+M+2)T are constant vectors. Equation (20) admits

the Taylor expansions

ϕ =

∞∑

s=0

[ s3]∑

l=0

[ s−3l

2]∑

n=0

(−1)s−3l−2n 2lxs−3l−2nyntl

n!l!(s− 3l− 2n)!ΓsC,

ψ =

∞∑

s=0

[ s3]∑

l=0

[ s−3l2

]∑

n=0

(−1)l+n 2lxs−3l−2nyntl

n!l!(s− 3l− 2n)!ΓsD . (21)

3.1.1 Soliton Solutions for (2+1)DBSE

Suppose the Γ is diagonal matrix, i.e. Γ =

diag(k1, k2, . . . , kN+M+2), obviously we have

ϕj = Cje−kjx+k2

j y+2k3

j t, ψj = Djekjx−k2

j y−2k3

j t

(j = 1, 2, . . . , N +M + 2) . (22)

The Wronskians composed by (22) are the normal soliton

solutions for (2).

3.1.2 Rational Solutions Related to Γ

If we consider Γ as

Γ =

0 0

1. . .

. . .

0 1 0

(N+M+2)×(N+M+2)

. (23)

Noticing that ΓN+M+2 = 0, the series (21) are truncated

as

ϕ =

N+M+1∑

s=0

[ s3]∑

l=0

[ s−3l

2]∑

n=0

(−1)s−3l−2n 2lxs−3l−2nyntl

n!l!(s− 3l− 2n)!ΓsC,

ψ =N+M+1∑

s=0

[ s3]∑

l=0

[ s−3l2

]∑

n=0

(−1)l+n 2lxs−3l−2nyntl

n!l!(s− 3l − 2n)!ΓsD . (24)

As a special result, let C1 = 1, Cj = 0, j = 2, 3, . . . , N +

M + 2, then Eqs. (24) reduce to

ϕj =

[ j−1

3]∑

l=0

[ j−3l−1

2]∑

n=0

(−1)j−3l−2n−1 2lxj−3l−2n−1yntl

n!l!(j − 3l − 2n− 1)!,

ψj =

[ j−1

3]∑

l=0

[ j−3l−1

2]∑

n=0

(−1)l+n 2lxj−3l−2n−1yntl

n!l!(j − 3l − 2n− 1)!. (25)

The double Wronskians composed by (25) will generate

the rational solutions for (2) through the transformation

(16). For examples, when N + M = 0; N = 1,M = 0;

N = 0,M = 1; N = M = 1; N = 2,M = 0;

N = 0,M = 2, we obtain the rational solution for (2),

respectively

q = r = −1

x, (26a)

q =1

x2 − y, r =

x2 + y

x2 − y, (26b)

q = −x2 − y

x2 + y, r = −

1

x2 + y, (26c)

q = 2x3 − 3t− 3xy

x4 + 3y2 + 6tx, r = 2

x3 − 3t+ 3xy

x4 + 3y2 + 6tx, (26d)

q =3

2(3t− x3 + 3xy), r = 2

x4 + 3y2 + 6tx

3t− x3 + 3xy, (26e)

q = 2x4 + 3y2 + 6tx

3t− x3 − 3xy, r =

3

2(3t− x3 − 3xy). (26f)

We depict those rational solutions in Figs. 1 and 2.

Fig. 1 The shape and motion of q (a) and r (b) in (26c),when t = 1.

Fig. 2 The shape and motion of q (on the left) in (26d)and r (on the right) in (26f), when t = 1.

No. 3 Rational and Periodic Solutions for a (2+1)-Dimensional Breaking Soliton Equation Associated with ZS-AKNS Hierarchy 433

3.2 Periodic Solutions for (2+1)DBSE

In 1996, Porubov proposed Weierstrass elliptic func-

tion expansion method, Liu et al. proposed Jacobi el-

liptic sine function expansion methods obtained some

exact periodic solutions of some nonlinear evolution

equations.[17−18] different from them, we deduce the peri-

odic solution from the view of double Wronskian solution.

If Γ has the form as

Γ =

(α −β

β α

)= αI2 + βσ2 ,

I2 =

(1 0

0 1

), σ2 =

(0 −1

1 0

). (27)

by a direct inserting into (21), we have

ϕ = e−αx+(α2−β2)y+2(α3−3αβ2)t

× (cos[−βx+ 2αβy + 2(3α2β − β3)t]I2

+ sin[−βx+ 2αβy + 2(3α2β − β3)t]σ2)C,

ψ = eαx−(α2−β2)y−2(α3−3αβ2)t

× (cos[−βx+ 2αβy + 2(3α2β − β3)t]I2

− sin[−βx+ 2αβy + 2(3α2β − β3)t]σ2)D. (28)

Furthermore, consider Γ is Jordan block matrix

Γ =

J1 0

J2

. . .

0 Jl

, (29)

and

Jk =

Γk 0

I2 Γk

. . .. . .

0 I2 Γk

, Γk =

(αk −βk

βk αk

). (30)

Then Jsk could be expressed as

Jsk =

I2 0

I2∂αkI2

12I2∂

2αk

I2∂αk

···. . .

. . .. . .

1(lk−1)!I2∂

lk−1αk

. . . 12I2∂

2αk

I2∂αkI2

Γ′s = T∂αkΓ′s . (31)

where Γ′ = diag(Γk,Γk, . . . ,Γk)2lk . Replace Γs in (21) with (31), by a direct but complicated calculation, we obtain

the following expressions of ϕ and ψ

ϕj(αk) =

j∑

s=1

1

(j − s)!∂j−s

αkeξk

(cs1 cos θk − cs2 sin θk

cs1 sin θk + cs2 cos θk

),

ψj(αk) =

j∑

s=1

1

(j − s)!∂j−s

αke−ξk

(cs1 cos θk + cs2 sin θk

−cs1 sin θk + cs2 cos θk

), (32)

where ξk = −αkx+ (α2k − β2

k)y + 2(α3k − 2αkβ

2k)t and θk = −βkx+ 2αkβky + 2(2α2

kβk − β3k)t. Similarly, we can find

ϕj(βk) =

j∑

s=1

(−σ2)j−s

(j − s)!∂j−s

βkeξk

(cs1 cos θk − cs2 sin θk

cs1 sin θk + cs2 cos θk

),

ψj(βk) =

j∑

s=1

(−σ2)j−s

(j − s)!∂j−s

βke−ξk

(cs1 cos θk + cs2 sin θk

−cs1 sin θk + cs2 cos θk

). (33)

It is easy to show that (32) and (33) are equivalent, so we get the complexiton solutions[14,19] for (2) where the

Wronskian entries satisfy

ϕ = (ϕT1 (α1), . . . , ϕ

Tl1

(α1);ϕT1 (α2), . . . , ϕ

Tl2

(α2); . . . ;ϕT1 (αh), . . . , ϕT

lh(αh))T ,

ψ = (ψT1 (α1), . . . , ψ

Tl1

(α1);ψT1 (α2), . . . , ψ

Tl2

(α2); . . . ;ψT1 (αh), . . . , ψT

lh(αh))T ,

where l1 + l2 + · · · + lh = N +M + 2. Specially, if N = M = 0, l1 = 1, lj = 0 , j = 2, 3, . . . , h, we can obtain the

periodic solution

ϕ = eξ1

(c11 cos θ1 − c12 sin θ1

c11 sin θ1 + c12 cos θ1

), ψ = e−ξ1

(c11 cos θ1 + c12 sin θ1

−c11 sin θ1 + c12 cos θ1

). (34)

Hence

f = |ϕ;ψ| = −(c211 + c212) sin 2θ1, g = 2|ϕ; ∂xϕ| = −2β1(c211 + c212)e

2ξ1 ,

h = 2|ϕ; ∂xϕ| = 2β1(c211 + c212)e

−2ξ1 , (35)

Noticing the transformation (16), we get the periodic solution as follows

q = 2β1e2ξ1 csc 2θ1, r = 2β1e

−2ξ1 csc 2θ1 . (36)

434 HAO Hong-Hai, ZHANG Da-Jun, ZHANG Jian-Bing, and YAO Yu-Qin Vol. 53

We describe the periodic solution in Figs. 3 and 4.

Fig. 3 The shape and motion of q (a) and r (b) in (36), when t = 1.

Fig. 4 The shape and motion of q (a) and r (b) in (36), when x = −0.5, t = 0.

4 Conclusion

In summary, we have given the hierarchy of (2+1)DBSEs. The double Wronskian solutions for (2) which satisfy

condition equations are obtained. Based on the solution, we deduce the rational solutions and the periodic solutions

through special choosing of the condition matrix Γ.

Acknowledgements

The authors are very grateful to Professor Deng-yuan Chen for his instructive suggestion.

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