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A class of coherent structures and interaction
behavior in Multidimensions
Jie-Fang Zhang 1,2,3
1 Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua 321004, P.R.China
2 Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11,3TU,UK
3 Shanghai Institute of Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China
Abstract
We solve the (2+1)-dimensional Davey-Stewartson (DS) equation, a multidimensional
analog of the nonlinear Schrodinger equation. A rather general solution for the variables
separation with two arbitrary functions is first obtained by applying a special Backlund
transformation and introducing the seed solutions. And then some new special types of
two-dimensional coherent structures are obtained. These structures exhibit interesting novel
features not found in one-dimensional solitons.
Key words: coherent structures, variable separation, Davey-Stewartson(DS) equation
PACC: 03.40.Kf, 03.65.Ge
1 Introduction
We consider the Davey-Stewartson(DS)[1] system of equations:
iqt +1
2(qxx + qyy) (x + |q|2)q = 0, xx yy + 2|q|2x = 0, (1)
which is the shallow-water limit of the Benney-Rokes equation[2], where q is the amplitude of a
surface wave packet while characterizes the mean motion generated by this surface wave(One
assumes a small-amplitude, nearly monochromatic,nearly one-dimensional wave train with dom-
inant surface tension[3]). Equations (1) provides a two-dimensional generalization of the cele-
brated nonlinear Schrodinger equation. Furthermore, it arises generically in both physics and
mathematics.Indeed, it has been shown that a very large class of nonlinear dispersive equa-
tions in 2+1 (two spatial and one temporal) dimensions reduce to the (2+1)-dimensional DS
equation in appropriate but generic asymptotic considerations[4]. Physical applications include
water waves, Plasma physics, and nonlinear optics[5]. Fokas and Santini[6]have solved an initial-
boundary value problem for DS system by using the inverse scattering transform (IST) method
Corresponding author. E-mail:[email protected] address.
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and obtained a kind of two-dimensional coherent structures and found that the coherent struc-
tures of Eqs.(1) exhibit interesting novel features not found in one-dimensional solitons. In this
paper, we consider further Eqs.(1). A rather general solution of the variable separation with
two arbitrary functions is first obtained by applying a special Backlund transformation and
introducing the seed solutions. And then some new special types of two-dimensional coherentstructures are constructed. These structures exhibit also interesting novel features not found in
one-dimensional solitons.
2 Exact variable separation solution of the (2+1)-dimensional
DS system
It is convenient to introduce characteristic coordinates = x + y, = x
y, and U1
12|q|2, U2 12 |q|2, then the second equation(2) can be integrated and Eqs.(1)and (2) reduce
to
iqt + (q + q) + (U1 + U2)q = 0, (2)
U1 =1
2
|q|2d + 01, (3)
U2 =1
2
|q|2d + 02, (4)
where 01(, t) = U1(, , t), 02(, t) = U2(, , t).To find soliton solutions of an equation, we can use different kinds of methods. One of
the powerful methods is the variable separation approach, which was recently presented and
successfully applied in some (2+1)-dimensional models[7-11]. Now we would use this method
to investigate the (2+1)-dimensional DS equation. To solve the system (2), we first take the
following Backlund transformation
q =g
f+ q0, U1 = (ln f) + U10, U2 = (ln f) + U20, (5)
which can be obtained from the standard Painleve truncated expansion, where f is a real, g is
complex, and (q0, U10, U20) is an arbitrary seed solution. Substituting (5) directly into system
(2)-(4) and integrating Eqs.(3)and (4) once to the argument and respectively, yields its
bilinear form:
(D2 + D2 + iDt)g f + q0(D2 + D2)f f + 2f g(U10 + U20)
+2f2q0(U10 + U20) + f2(q0 + q0) + if
2q0t = 0, (6)
DDf f + 2(gg + f gq0 + f gq0 + f2q0q0 + 2f2(1 U10)) = 0, (7)
DDf f + 2(gg + f gq0 + f gq0 + f2q0q0 + 2f2(1 U20)) = 0, (8)
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where 1 , 1 are integration symbols to the argument and , and D, D, Dt are defined as
Dm Dn D
kt f f =lim=,=,t=t (
)m(
)n(
t
t)k (9)
which is the usual bilinear operator introduced first by Hirota[12].
To discuss further, we take the seed solution (q0, U10, U20) as
q0 = 0, U10 = u0(, t), U20 = v0(, t) (10)
then(6),(7)and (8) can be simplified to
(D2 + D2 + iDt)g f + 2f g(u0 + v0) = 0, (11)
DDf f + 2gg = 0. (12)
To find some interesting solutions of equations (11) and (12), we can use the variable separation
ansatz
f = a1u + a2v + a3uv, g = u1v1 exp(ir + is), (13)
where a1, a2, a3 are arbitrary constants and u u(, t), v v(, t), u1 u1(, t), v1 v1(, t), r r(, t), s s(, t) are all real functions of the indicated variables. SubstitutingEq.(13) into Eqs.(11) and (12) and separating the real and imaginary parts of the resulting
equations,we have
u21v21 2a1a2uv = 0, (14)
(a1u + a2v + a3uv)(v1u1 + u1v1 u1v1(2rt + 2st 2u0 2v0 + r2
+ s
2
))
+v1(a1 + a3v)(u1u 2u1u) + u1(a2 + a3u)(v1v 2v1v) = 0, (15)
(a1u + a2v + a3uv)(v1(2ru1 + 2u1t + u1r) + u1(2sv1 + 2v1t + v1s))
2u1v1(vt + 2sv)(a2 + a3u) 2u1v1(ut + 2ru)(a1 + a3v) = 0. (16)
Because the functions u0, u , u1 and r are only functions of{, t} and the functions v0, v, v1and s are only functions of {, t}, the equation system (14)-(16) can be solved by the followingvariable separated equations:
u1 = 12a1a2c10 u, (17)v1 = 2
c0v, (
21 =
22 = 1), (18)
ut + ru = c1(a2 + a3u)2 + c2(a2 + a3u) + a1a2c3, (19)
vt + sv = c3(a1 + a3v)2 c2(a1 + a3v) a1a2c1, (20)
4(2st + s2 2v0)v2 2vv + v2 + c4v2 = 0, (21)
4(2rt + r2 2u0)u2 2uu + u2 c4u2 = 0. (22)
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In Eqs.(17)-(22), c0, c1, c2, c3, c4 are all arbitrary functions of t. From Eqs.(17) and (18),
we know that the real conditions of u and v require
a1a2c10 u 0, (23)
c0v 0. (24)Although it is not an easy task to obtain general solutions of Eqs.(19)-(22) for any fixed u0
and v0, we can treat the problem in an alternative way. Because u0 and v0 are arbitrary seed
solutions, we can view u and v as arbitrary functions of {, t} and {, t} with the conditions(23), (24) respectively. The functions r and s can be expressed by u and v simply by integrations
from Eqs.(19) and (20). Then the seed solutions u0 and v0 can be fixed by Eqs.(21), (22). Finally,
substituting Eq.(13) with (17)-(22) into Eq.(8), we get a quite general solution of the (2+1)-
dimensional system (5)-(7)
q =12
a1a2uv exp(ir + is)
a1u + a2v + a3uv , (25)
U1 = u0 + (a1u + a3vu
a1u + a2v + a3uv (a1u + a3vu)
2
(a1u + a2v + a3uv)2), (26)
U2 = v0 + (a2v + a3uv
a1u + a2v + a3uv (a2v + a3uv)
2
(a1u + a2v + a3uv)2), (27)
with two arbitrary u and v under the conditions (24) and (25) and u0, v0 are determined by
Eqs.(22) and (23). Especially, for the module square of the field reads
=
|q
|2 =
a1a2uv(a1u + a2v + a3uv)2
(28)
=a1a2UV
2(A1 cosh1
2(U + V + C1) + A2 cosh
1
2(U V + C2))2
, (29)
where
u = b1 + eU, v = b2 + e
V , (30)
and
A1 =
a3(a1b1 + a2b2 + a3b1b2), A2 =
(a1 + a3b2)(a2 + a3b1), (31)
C1 = lna3
a1b1 + a2b2 + a3b1b2, C2 = ln
a1 + a3b2
a2 + a3b1, (32)
for b1 and b2 being arbitrary constants. U and V are also arbitrary functions of{, t} and {, t}respectively under the conditions
a1a2UV 0. (33)
Because u(, t) and v(, t) are arbitrary the functions of indicated arguments, Eq.(28) (or
Eq.(29))reveals the quite abundant soliton structures. From Eqs.(28) and (29), it is easy to
know that for arbitrary u and v with the boundary conditions
u
|
B1, u
|+
B2, v
|
B3, v
|+
B4, (34)
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where B1, B2, B3 and B4 are arbitrary constants which may be infinities.
We known that,in addition to the continuous localized excitations in (1+1)-dimensional non-
linear systems, some type of significant weak solutions like the peakon[13] and compacton[14]and
multi-valued localized solution like loop soliton[15]. The so-called peakon solution (u = c exp( |x ct |))which is called a weak solution of the celebrated (1+1)-dimensional Camassa-Holmequation
ut + 2kux uxxt + 3uux = 2uxuxx + uuxxx, (35)
was firstly given by Camassa and Holm[13]. While the so called (1+1)-dimensional compacton
solutions which describes the typical (1+1)-dimensional soliton solutions with finite wavelength
when the nonlinear dispersion effects are included was firstly given by Rosenau and Hyman[14].
It has found that peakon and compacton may have many interesting properties and possible
physical applications[16]. Moreover, in natural world, there exist very complicated folded phe-
nomena such as the folded protein[17]folded brain and skin surface and many many other kinds
of folded biology system[18]. The bubbles on (or under) a fluid surface may be thought to be the
simplest folded waves. Further, various kinds of ocean waves are really folded waves also. These
phenomena have been applied in some physical fields like the string interaction with external
field, quantum field theory and particle physics[19]. Recently, the higher dimensional peakon
solution and compacton solution and foldon solution, which are new types of soliton if the inter-
action between the folded solitary waves is completely elastic, have also been investigated and
obtained in some (2+1)-dimensional models[20-22]. Here we focus our attention on giving the
three kinds of interesting coherent structures from the expression(28) for the (2+1)-dimensional
DS system.
3 Compacton solutions and their interaction behavior
Because of the entrance of arbitrary functions in expression(26), we can easily find some types of
multiple compacton solutions by selecting the arbitrary functions appropriately. For instance,if
we fixed the functions u and v as
u =a0
a1+
M
i=1
0 + it 0i 2kibi sin(ki( + it
0i)) + bi 0i
2ki< + it
0i +
2ki,
2bi + it > 0i +2ki
(36)
and
v =N
j=1
0 + jt 0j 2li ,cj sin(lj( + jt 0j)) + cj 0j 2lj < + jt 0j + 2lj ,2ci + jt > 0j +
2lj
(37)
in the expressions (36) and (37), bi, ki, cj , i, j , lj , 0i and 0j are all arbitrary constants, then
the solution (28) with (36)and (37) becomes a multi-compactons solution.
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From(36) and (37), we can see that the piecewise functions u and v of the compacton solutions
are once differential
u =M
i=1
0 + it 0i 2kibiki cos(ki( + it
0i)) + bi 0i
2ki
< + it
0i +
2ki
,
0 + it > 0i +2ki
(38)
and
v =N
j=1
0 + jt 0j 2lj ,cj lj cos(lj( + jt 0j)) + cj 0j 2lj < + jt 0j + 2lj .0 + jt > 0j +
2lj
(39)
If selecting M = 2, N = 2, a0 = 20, a1 = a2 = 1, a3 =1
25, b1 = b2 = 1, c1 = 1, c2 = 1.5, k1 =
0.6, k2 = 0.4, 1 = 1, 2 = 3, 1 = 2 = 0, l1 = 1, l2 = 0.7, 01 = 02 = 0, 01 = 0, 02 = 5we can obtain a four-compactons excitation for the (2+1)-dimensional DS system. Fig.1(a)-(f)
shows the evolution behavior of interaction among four compactons. We see that the interaction
among four compactons exhibits a new phenomenon. Their interaction is non-elastic and do not
completely exchange their shapes each other.
4 Peakon solutions and their interaction behavior
Similarly, considering the arbitrariness of the functions u and v in expression (28), we can
construct the peakons of the (2+1)-dimensional DS system by selecting appropriate functions.
For instance, when u and v are taken the following simple form
u =a0a1
+Mi=1
di exp(mi it + 0i), mi it + 0i 0,di exp(mi + it 0i), mi it + 0i > 0,
(40)
v =N
j=1
ej exp(nj jt + y0j), nj jt + y0j 0,ej exp(nj + jt y0j) + 2, nj jt + y0j > 0,
(41)
where di, mi, ej , nj , i, j , 0 and 0j are all arbitrary constants, then the solution (28) with (40)
and(41)becomes a multi-peakon solution.
If we selecting M = 2, N = 2, a0 = 200, a1 = a2 = 1, a3 =1
200, d1 = d2 = 1, m1 = 0.5, m2 =
1, 1 = 1, 2 = 2e1 = e2 = 1, n1 = 1, n2 = 1, 1 = 1, = 2, 01 = 4, 02 = 4, 01 = 4, 02 =4, we can obtain a four-peakons excitation for the (2+1)-dimensional DS system. Fig.2(a)-(f)shows the evolution behavior of interaction among four peakons at different times. We can find
that the interaction of four-peakons is not completely elastic but four peakons may completely
exchange their shapes.
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5 Dromion solutions and their interaction behavior
In order to compare with the results given in Ref.[6], we also discuss the corresponding dromion
solution for the (2+1)-dimensional DS system.If selecting
u =Mi=1
tanh(ki + it), v =Ni=1
exp(lj + jt), (42)
we can obtain another kind of the multi-dromions solution. Figure 3(a)-(f)shows the evolution
behavior of interaction among four dromions when M = 4, N = 1, a0 = 10, a1 = a2 = a3 =
1, k1 = 1, k2 = 0.5, k3 = 1, k4 = 1, 1 = 0.5, 2 = 1, 3 = 3, 4 = 5, l1 = 1 and 1 = 1 at different
times. From figure 3(a)-(f), we can find that the interaction of four dromions is not completely
elastic and four dromions do not completely exchange their shapes each other.
6 Foldon solutions and and their interaction behavior
In order to construct these kinds of interesting folded solitary waves and/or foldons for the
module square of the field q, we should introduce some suitable multi-valued functions. For
example,
v =Mi=1
Vi( + cit), = +Mi=1
Yi( + it), (43)
where 1 < 2 < < Mare all arbitrary constants and Vi, Yi, j are localized functionswith properties Vi() = 0, Yi() = consts. From Eq.(43), one knows that may be a
multi-valued function in some suitable regions of by selecting the functions Yi appropriately.Therefore, the function v, which is obviously an interaction solution of M localized excitations
since the property | , may be a multi-valued function of in these areas though it isa single valued functions of .Actually, most of the known multi-loop solutions are the special
situations of Eq.(43). Similarly, we also treat the function u(, t) in this way
u =N
j=1
Uj( + jt), = +N
j=1
Xj( + jt), (44)
where 1 < 2 < < Nare all arbitrary constants andUj , Xj , j are all localized functions
with properties Uj() = 0, Xj() = consts. Now we further discuss the properties ofthe interaction among the folded solitary waves. If we select u and v to be some appropriate
multi-valued functions, then we can see that the interactions among the folded solitary waves
are completely elastic. For example, when set
u = sec h2(), = 2 tanh(), (45)
v =4
5sec h2() +
1
2sec h2( t/4),
= 32
(tanh() + tanh( t/4)), (46)
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and a1 = a2 = a3 = 1, then we can derived some interesting foldons depicted in Figs.(2a), (2b)
and (2c) at different times (a) t = 18, (b)t = 8, (c)t = 11.8,(d) t = 14,(e)t = 18 respectively.From figures (4a) and (4f), one can find the interaction between the two foldons is completely
elastic since the velocity of one of the foldons has set to be zero and there are still phase shifts for
the two foldons. To see more carefully, one can easily find that the position located by the largestatic foldon is altered from about = 1.5 to = 1.5 and its shape is completely preservedafter interaction.
7 Summary and discussion
In summary, with help of Backlund transformation and the variable separation procedures,
the (2+1)-dimensional DS system has also solved as Fokas and Santini by use of the inverse
scattering transform. By selecting arbitrary functions appropriately, three kinds of new coherent
structures(peakon, compacton and foldon) and a kind of known coherent structures (dromion)
have been constructed. The interactions among peakons and compactons exhibit interesting
novel features not found in one-dimensional solitons. The interaction among four foldons is
completely elastic. The interaction among four compactons or four dromions is non-elastic and
do not completely exchanged their shapes each other. While the interaction of four travelling
peakons or four dromions is not non-elastic but four peakons may completely exchange their
shapes.
Acknowledgement: The author is in debt to thank the useful discussions with Professor
Lou Senyue. Appreciation is also given to Prof. Roger Grimshaw, Dr. Rod Halburd and Dr.Zhiming Lu for their kind help. This work is supported by the Pao Yu-Kong and Pao Zhao-Long
Scholarship for Chinese Students Studying Abroad and the the Natural Science Foundation of
China Granted No.10272072.
References
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[7] Lou S Y 2000 Phys. Lett. A 277 94;2000 Phys. Lett. A 277 94; 2001 Physica Scripta64 1;
Lou S Y, Chen C L and Tang X Y 2002 J. Math. Phys 43 4078;Lou S Y and Ruan H Y
2001 J. Phys. A: Math. Gen.34 305
[8] Ruan H Y and Chen Y X 2001 Acta Phys.Sin.50 586(in Chinese)
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C L 2002 Chin. Phys. Lett. 19 769
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[12] Hirota R 1971 Phys.Rev.Lett.27 1192
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174 ;
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[22] Tang X Y and Lou S Y 2002 arXiv. nlin. SI/0210009. 4 1; 2003 J. Math. Phys. in press
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Fig.1a
50
510
2
0 2
46
8
0
0.0005
0.001
Fig.1b
42
0
2 46
2
02
46
8
0
0.0005
0.001
Fig.1c
42
02
4
2
02
46
8
0
0.0005
0.001
0.0015
Fig.1d
86
42
0
2 4
2
02
46
8
0
0.0005
0.001
Fig.1e
108
64
20
24
2
02
46
8
0
0.0005
Figure 1: Evolution plot of a four-compactons solution determined by (28) with (36) and (37) at (a)
t = 3, (b) t = 0.9, (c) t = 0.02, (d) t = 1, (e) t = 1.8.
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Fig.2a
108
64
2
024
108
64
20 2
4
0
1e05
2e05
Fig.2b
86
42
02
86
42
02
01e052e053e05
Fig.2c
54
32
10
12
54
32
10
1 2
0
4e05
8e05
Fig.2d
6
4
2
0
2
6
4
2
0
2
0
1e05
2e05
Fig.2e
86
42
02
46
8
86
42
02
46
8
0
1e05
2e05
Figure 2: Evolution plot of a four-peakons solution determined by (28) with (40) and (41) at (a) t = 5,(b) t = 3, (c) t = 2, (d) t = 1.4, (e) t = 0.5.
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Fig.3a
010
203040
50
4
8
12
16
0
0.001
0.002
0.003
0.004
Fig.3b
50
510
1520
4
0
4
8
0
0.001
0.002
0.003
0.004
Fig.3c
4321
0123
64
20
24
6
0
0.002
0.004
0.006
0.008
Fig.3d
86
42
02
8
4
0
4
0
0.001
0.002
0.003
0.004
0.005
Fig.3e
2520
1510
50
8
4
0
4
0
0.001
0.002
0.003
0.004
Figure 3: Evolution plot of a four-dromions solution determined by (28) with (44) at (a) t = 10, (b)t = 3, (c) t = 0, (d) t = 1, (e)t = 4.
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Fig.4a
2
1
0
1
2
3.5
32.5
21.51
0
0.0005
0.001
0.0015
0.002
Fig.4b
2
1
0
1
2
0
0.51
1.52
0
0.0004
0.0008
0.0012
0.0016
Fig.4c
2
1
0
1
2
1
1.21.4
1.61.8
0
0.0005
0.001
0.0015
Fig.4d
2
1
0
1
2
1.21.62
2.4
0
0.0005
0.001
0.0015
0.002
Fig.4e
2
1
0
1
2
1
1.52
2.53
3.5
0
0.0005
0.001
0.0015
0.002
Figure 4: Evolution plots of two foldons for the module square of the field q given by Eq.(28) with
conditions (45), (46) and a1 = a2 = 1, a3 =1
20at different times (a) t = 18, (b) t = 8, (c) t = 11.8, (d)
t = 14, (e) t = 18