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Rational and Periodic Solutions for a (2+1)-Dimensional Breaking Soliton Equation Associated
with ZS-AKNS Hierarchy
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2010 Commun. Theor. Phys. 53 430
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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−[email protected]‡E-amil: [email protected]
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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