Coherent Structures and Chaos

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Commun. Theor. Phys. (Beijing, China) 40 (2003) pp. 25–32c International Academic Publishers Vol. 40, No. 1, July 15, 2003

Localized Coherent Structures with Chaotic and Fractal Behaviors in a

(2+1)-Dimensional Modified Dispersive Water-Wave System∗

ZHENG Chun-Long†

Department of Physics, Zhejiang Lishui Normal College, Lishui 323000, China‡

Shanghai Institute of Mathematics and Mechanics, Shanghai University, Shanghai 200072, ChinaDepartment of Physics, Zhejiang University, Hangzhou 310027, China

Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua 321004, China

(Received October 14, 2002; Revised December 24, 2002)

Abstract In this work, we reveal a novel phenomenon that the localized coherent structures of some (2+1)-dimensional physical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2+1)-dimensional modified dispersive water-wave system as a concrete example. Starting from a variable separation approach,a general variable separation solution of this system is derived. Besides the stable localized coherent soliton excitations like dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractal behaviors are derived by introducing some types of lower dimensional chaotic and fractal patterns.

PACS numbers: 03.40.Kf, 03.65.Ge, 05.45.Yv

Key words: variable separation approach, dispersive water-wave system, fractal, chaos

1 Introduction

In nonlinear science, solitons, chaos, and fractals are

three most important aspects,[1] which are widely ap-

plied in many natural sciences such as chemistry, biology,

mathematics, communication, and particularly in almost

all branches of physics like the fluid dynamics, plasma

physics, field theory, optics, condensed matter physics,

etc.[2] Conventionally, the three aspects are treated inde-

pendently since one often considers that solitons are basicexcitations of an integrable model while chaos and frac-

tals are elementary behavior of a nonintegrable system.

In other words, one does not analyze the possibility of

existence of the chaos and fractals in a soliton system.

However, the above consideration may not be complete,

especially in some higher dimensions. In recent study of

soliton systems, we have found that some characteristic

lower-dimensional arbitrary functions exist in exact exci-

tation of some two-dimensional integrable models. This

means that any lower-dimensional chaotic and/or frac-

tal solutions can be used to construct exact solution of

a higher-dimensional integrable model, which also implies

that any exotic behavior may propagate along this char-

acteristics.

To verify the above viewpoints, we take the (2+1)-

dimensional modified dispersive water-wave system[3] as a

concrete example,

uyt + uxxy − 2vxx − (u2)xy = 0 , (1)

vt − vxx − 2(uv)x = 0 , (2)

which was used to model nonlinear and dispersive long

gravity waves travelling in two horizontal directions on

shallow water of uniform depth, and can also be derived

from the celebrated Kadomtsev–Petviashvili (KP) equa-

tion by the symmetry constraint.[4]

It is worth while men-tioning that the system has been widely applied in many

branches of physics like plasma physics, fluid dynamics,

nonlinear optics, etc. So a good understanding of more

solutions of Eqs. (1) and (2) is very helpful, especially for

coastal and civil engineers to apply the nonlinear water

model in a harbor and coastal design. Meanwhile, find-

ing more types of solutions of system (1) and (2) is of

fundamental interest in fluid dynamics.

In Ref. [5], a special variable separation solution of a

similar system was derived via an extended homogeneous

balance approach. However, we find that the standardtruncated Painleve analysis and general variable separa-

tion approach are more efficient in searching for localized

excitation of nonlinear physical models. In this paper, we

investigate the modified dispersive water-wave system via

the standard truncated Painleve analysis and variable sep-

aration approach. Much attention is paid to its localized

∗The project supported by the Foundation of the State “973” Programme for Nonlinear Science of China and the Natural Science

Foundation of Zhejiang Province of China†E-mail: [email protected]‡Mailing address



26 ZHENG Chun-Long Vol. 40

coherent structures, especially the localized coherent soli-

ton structure with some novel properties like fractal and

chaotic properties. In Sec. 2, we outline the main proce-

dures of the variable separation approach and apply this

approach to the modified dispersive water-wave system.

In Sec. 3, we discuss the abundant structures of the local-

ized solution based on the result obtained by the variable

separation approach. A simple discussion and summary

is given in the last section.

2 Variable Separation Approach and ItsApplication for the (2+1)-DimensionalModified Dispersive Water-Wave System

2.1 General Theory of the Variable Separation

Approach

It is well known that solving nonlinear physical mod-

els is more difficult than solving the linear ones. In

linear physics, the Fourier transformation and the vari-able separation approach (VAS) are the two most impor-

tant methods. The celebrated inverse scattering trans-

formation (IST) can be viewed as an extension of the

Fourier transformation method. However, it is difficult

to extend VAS to nonlinear physics. Recently, the two

kinds of “variable separating” procedures have been es-

tablished. The first is called symmetry constraints or non-

linearization of the Lax pairs,[6] because this type of pro-

cedure is used only for integrable models which possess

Lax pairs. Lou and Chen have also extended the method

to some nonintegrable models, so this method is equiv-

alently called the “formal variable separation approach”

(FVSA).[7] The independent variables of a reduced field in

FVSA have not been totally separated though the reduced

field satisfies some lower-dimensional equations. The sec-

ond kind of variable separation method is established first

for the (2+1)-dimensional DS (Davey–Stewartson) equa-

tions and asymmetric DS equation,[8] and then revised and

developed recently for various (2+1)-dimensional models

like the (2+1)-dimensional KdV equation, the generalized

AKNS system, the nonlinear Schrodinger equation, and

the generalized ANNV system, etc.[9−18] The main idea

is that by solving its bilinear equations or higher multi-linear equations of the original models and introducing a

prior ansatz, some special types of exact solution of the

(2+1)-dimensional nonlinear models can be obtained from

some (1+1)-dimensional variable separation fields. Here

we describe the basic procedures of the variable separation

approach briefly. For a general nonlinear physics system

P (v) ≡ P (x0 = t, x1, x2, . . . , xn, v , vxi , vxixj , . . .) = 0 , (3)

where v = v(v1, v2, . . . , vq)T (T indicates the transposi-

tion of a matrix), P (v) = (P 1(v), P 2(v), . . . , P q(v))T, and

P i(v) are polynomials of vi and their derivatives.

Firstly, we make a Backlund transformation

vi =

αi

j=0

vijf j−αi , i = 1, 2, . . . , q , (4)

where viαiare arbitrary known seed solutions of Eq. (3).

In usual cases, αi should be chosen as small as possi-

ble since substituting Eq. (4) into Eq. (3) can yield rela-tively simple multi-linear equations in this situation, and

it is determined by the leading term analysis (suppose

f ∼ 0). Inserting Eq. (4) with αi into Eq. (3) and van-

ishing the leading and sub-leading terms, we can derive

{vij , j = 0, 1, 2, . . . , αi − 1}. If the original model is inte-

grable, this procedure will result in its bilinear or higher

multi-linear equations.

Secondly, after obtaining the multi-linear equations of

the original system, we select an appropriate variable sep-

aration hypothesis. For integrable models, it can be cho-

sen as the modifying Horita’s multi-soliton forms. Forinstance, we often take f as such an ansatz for some cele-

brated physical models[19−21]

f = a0 + a1 p(x, t) + a2q (y, t) + a3 p(x, t)q (y, t) , (5)

where the variable separation functions p(x, t) ≡ p and

q (y, t) ≡ q are only the functions of (x, t) and (y, t), re-

spectively, while a0, a1, a2, and a3 are arbitrary constants.

When p and q are set as exponential functions, equa-

tion (5) is just the Hirota’s two-soliton form.

Finally, we determine the variable separation equa-

tions which the variable separation functions p and q

should satisfy after substituting the ansatz (5) into themulti-linear equations. It is worth while mentioning that

this procedure is a tedious work since different physical

models should be processed in different ways. In the

next subsection, we apply this approach to the (2+1)-

dimensional modified dispersive water-wave system. The

variable separation approach shows its efficiency in search-

ing for localized excitation of nonlinear physical models.

2.2 Variable Separation Approach for the (2+1)-

Dimensional Modified Dispersive

Water-Wave System

According to the above procedures, we first take the

following Backlund transformation for u and v in Eqs. (1)

and (2),

u =

α1

j=0

ujf j−α1 , v =

α2

j=0

vjf j−α2 , (6)

where uα1 and vα2 are arbitrary seed solutions of the mod-

ified dispersive water-wave system. By using the leading

term analysis, we obtain

α1 = 1 , α2 = 2 . (7)



No. 1 Lo calized Coherent Structures with Chaotic and Fractal Behaviors in · · · 27

Substituting Eqs. (6) and (7) directly into Eqs. (1) and

(2) and considering the fact that the functions u1 and v2

are seed solutions of the model, yield3

i=0

P 1if i−4 = 0 , (8)

3i=0

P 2if i−4 = 0 , (9)

where P 1i, P 2i are the functions of {uj , vj , f } and their

derivatives. Due to the complexity of the expression of

P 1i and P 2i, we omit their concrete forms. Vanishing the

leading and sub-leading terms of Eqs. (8) and (9), the

functions {u0, v0, v1} are determined, which read

u0 = f x , v0 = −f xf y , v1 = f xy . (10)

Inserting the result (10) with Eq. (7) into Eq. (6) and

rewriting its form, the Backlund transformation becomes

u = (ln f )x + u1 , v = (ln f )xy + v2 . (11)

For convenience of discussions, we choose the seed solu-

tions u1 and v2 as

u1 = p0(x, t) , v2 = 0 , (12)

where p0(x, t) is an arbitrary function of indicated argu-

ments.

Now substituting Eqs. (11) and (12) yields two multi-

linear equations in f which degenerate into the same tri-

linear form

(f xyt − f xxxy − 2( p0xf xy + p0f xxy))f 2 + [(f xxy − f yt + 2 p0xf y + 4 p0f xy)f x + (f xx − f t)f xy

+ (f xxx − f xt + 2 p0f xx)f y ]f + 2(f t − f xx − 2 p0f x)f xy = 0 . (13)Inserting the ansatz (5) into Eq. (13) and performing some tedious calculations, we finally obtain the following two

variable separated equations

pt = pxx + 2 px p0 + (a3a0 − a1a2)(c1 − c2 p + c0 p2) , (14)

q t = c0(a0 + a2q )2 + c1(a1 + a3q )2 + c2(a0 + a2q )(a1 + a3q ) , (15)

where c0 ≡ c0(t), c1 ≡ c1(t), and c2 ≡ c2(t) are arbitrary functions of time t.

Although it is not easy to obtain general solutions of Eqs. (14) and (15) for any fixed p0, we can treat the problem

in an alternative way. Since p0 is an arbitrary seed solution, we can view p as an arbitrary function of {x, t}, then the

seed solutions p0 can be fixed by Eq. (14), which reads

p0 =1

2 px[ pt − pxx − (a3a0 − a1a2)(c1 − c2 p + c0 p

2)] . (16)

As to the Riccati equation (15), its general solution has the form

q (y, t) =A1(t)

A2(t) + F (y)+ A3(t) , (17)

where F (y) is an arbitrary function of y, while A1, A2, and A3 are arbitrary functions of time t, which are linked with

c0, c1, and c2 by

c0 =1

A1Γ(a23A1A3t − (a1 + a3A3)2A2t − a3(a1 + a3A3)A1t) , (18)

c1 =1

A1Γ(a22A1A3t − (a0 + a2A3)2A2t − a2(a0 + a2A3)A1t) , (19)

c2 =

1

A1Γ (−2a2a3A1A3t − 2(a0 + a2A3)(a1 + a3A3)A2t + (a0a3 + a1a2 + 2a2a3A3)A1t) , (20)

where Γ = (a1a2 − a0a3)2. Using the relations (18) ∼ (20), equation (15) becomes

q t =−1

A1[A2tq 2 − (A1t + 2A3A2t)q + A2

3A2t + A3A1t −A1A3t] . (21)

One can verify easily that equation (17) is a general exci-

tation of Eq. (21).

Finally, we obtain a general excitation of the modified

dispersive water-wave system

u =(a1 + a3q ) px

(a0 + a1 p + a2q + a3 pq )+ p0 , (22)

v =(a3a0 − a1a2) pxq y

(a0 + a1 p + a2q + a3 pq )2, (23)




where p is an arbitrary function of {x, t}, q is determined

by Eq. (17) and p0 is expressed by Eq. (16).

3 Some Chaotic and Fractal Coherent SolitonStructures of the (2+1)-DimensionalModified Dispersive Water-Wave System

It is interesting that expression (23) is valid for many

(2+1)-dimensional models like the DS equation, NNV sys-

tem, ANNV equation, ADS model, and the generalized

AKNS system etc.,[9,10,13,18] Because of the arbitrariness

of the functions p and q included in Eq. (23), the physical

quantity or field v possesses quite rich structures. For in-

stance, as mentioned in Refs. [9], [10], [13], and [18], when

we select functions p and q appropriately, we can obtain

many kinds of localized solutions like the multi-solitoff so-

lutions, multi-dromion and dromion lattice solutions, mul-

tiple ring soliton solutions, multiple peakon solutions, and

so on. Since these types of localized solutions have been

discussed widely for other models, we omit all the stable

localized coherent soliton structures here.

Now an important matter is whether we can find some

new types of solitons which possess chaotic and/or fractal

behaviors, say, the chaotic and/or fractal localized exci-

tations for the soliton system. The answer is apparently

positive since p(x, t) and q (y, t) are arbitrary functions.

There are various chaotic and fractal dromion and lump

excitations because any types of (1+1)- and/or (0+1)-

dimensional chaos and fractal models can be used to con-

struct localized excitations of high-dimensional models.

Some interesting possible chaotic and fractal patterns arecited here. For simplification in the following discussions,

we set a0 = a1 = a2 = 1 and a3 = 2 in expression (23).

3.1 Chaotic Localized Excitations

(i) Chaotic dromions

In (2+1) dimensions, one of the most important non-

linear solutions is the dromion excitation which is localized

in all directions. Now we set p and q as

p = exp(x) , q = 1 + (60 + f (t))exp(y) , (24)

where f (t) is arbitrary function of time t. From excitation

(23) with equation (24), one knows that the amplitude of

the dromion is determined by the function f (t). If we

select the function f (t) as a solution of a chaotic dynam-

ical system such as Lorenz system,[22] Rossler system,[23]

chemical dynamical system,[24] etc., then we can obtain a

type of chaotic dromion solution. In Fig. 1(a), we plot the

shape of the dromion for the physical quantity v shown

by expression (23) at a fix time (for f (t) = 0) with condi-

tion (24). The amplitude V of the dromion related to (a)

is changed chaotically with f (t) as depicted in Fig. 1(b),

where f (t) is a solution of the following chemical dynam-

ical chaos system

f t = f (A1 − k1f − g − h) + k2g2 + A3 ,

gt = g(f − k2g −A5) + A2 ,

ht = h(A4 − f − k3h) + A3 , (25)

where A1 = 30, A2 = A3 = 0.01, A4 = 16.5, A5 = 10,k1 = 0.25, k2 = 0.001, and k3 = 0.5.

Fig. 1 (a) A plot of single dromion structure for the

physical quantity v given by solution (23) with condi-tions (24) and f (t) = 0; (b) Evolution of the amplitudeV of chaotic dromion related to (a) with f (t) being asolution of the chemical dynamical chaos system (25) atdifferent times.

This fact hints that the functions p or q may take a

more general form. For example, if we set p and q as

[f 3(t) > 0, f 7(t) > 0],

p =f 1(t)

f 2(t) + exp(f 3(t)(x + f 4(t))),

q =f 5(t)

f 6(t) + exp(f 7(t)(x + f 8(t)))(26)

with f i(t), i = 1, 2, . . . , 8 being chaotic solutions, then

solution (23) becomes a chaotic dromion which may be

chaotical in different ways. The amplitude of the dromion

(23) with Eq. (26) will be chaotic if f 1(t), f 2(t), f 5(t)

and/or f 6(t) are chaotic while f 4(t) and/or f 8(t) are

chaotic the position of the dromion located becomes

chaotic. The shape (width) of the dromion may be chaotic

if the functions f 3(t) and/or f 7(t) are chaotic. Since a de-

tailed physical discussion has been given in Ref. [25], we

omit the related plots.




(ii) Chaotic line solitons

It is interesting to mention that the localized exci-

tations are not only chaotic with time t, but also with

space, say, in direction x or/and y. If one of p and

q is selected as a localized function while other one is

chaotic solutions of some (1+1)-dimensional (or (0+1)-

dimensional) nonintegrable models, then excitation (23)

becomes a chaotic line soliton which may be chaotical in

x or y direction. For example, we set p or q as the solution

of (ζ = x + ωt,η = y + νt)

pζ = p(A1 − k1 p − g − h) + k2g2 + A3 ,

gζ = g( p− k2g −A5) + A2 ,

hζ = h(A4 − p− k3h) + A3 , (27)

or

q η = q (A1 − k1q − g − h) + k2g2 + A3 ,

gη = g(q − k2g −A5) + A2 ,

hη = h(A4 − q − k3h) + A3 , (28)

where ω and ν are all arbitrary constants.

Fig. 2 (a) A plot of chaotic line soliton structure forthe physical quantity v given by expression (23) with theconditions (27), (29) and (30); (b) The typical plot of chaotic solution p in the chemical dynamical chaos sys-tem (25).Figure 2(a) is a plot of chaotic line soliton solution ex-

pressed by Eq. (23) with the following selections: p is a

chaotic solution of chemical chaos system (27) and

q (y, t) = tanh(y + ct) , (29)

while the parameters are selected as

A1 = 30 , A2 = A3 = 0.01 , A4 = 16.5 , A5 = 10 ,

k1 = 0.25 , k2 = 0.001 , k3 = 0.5 , t = 0 . (30)

The typical plot of chaotic solution p in the above chemical

chaos system is presented in Fig. 2(b).

(iii) Non-localized chaotic patternsCertainly, if both p and q are selected as chaotic so-

lutions of some lower-dimensional nonintegrable models,

then excitation (23) becomes chaotical both in x and y di-

rections, which yields a non-localized chaotic-chaotic pat-

terns. Figure 3(a) is a plot of a special chaotic-chaotic

pattern expressed by Eq. (23) with p and q being given by

Eqs. (27) and (28) under conditions Eqs (30) and

a0 = 200 , a1 = a2 = 1 , a3 = 0 . (31)

Fig. 3 (a) A plot of the special chaotic-chaotic patternexpressed by Eq. (23) and p and q being the chaotic solu-tions of Eqs. (27) and (29) under the conditions Eqs. (30)

and (31); (b) A plot for the typical chaotic solution of chemical dynamical model (27) with Eq. (30).

Figure 3(b) is a corresponding plot for the typical

chaotic solution of the chemical dynamical model (27)

with Eq. (30).

3.2 Fractal Localized Excitations

(i) Stochastic fractal dromions and lumps

Now, we discuss the localized coherent excitations with

fractal property. It is well known that there are some




lower-dimensional stochastic fractal functions, which may

be used to construct high-dimensional stochastic fractal

dromion and lump excitations. For instance, one of the

most well-known stochastic fractal functions is the Weier-

strass function

Γ ≡ Γ(ξ ) =

N

k=0

λ(s−2)k

sin(λk

ξ ) , N →∞ , (32)

where {λ, s} are constants and the independent variable

ξ may be a suitable function of {x + at} and/or {y + bt},

say ξ = x + at and ξ = y + bt in the functions of p and

q , respectively, for the following selection (33) or (34).

If the Weierstrass function is included in the dromion or

lump excitations, then we can derive the stochastic fractal

dromions and lumps.

Fig. 4 A plot of typical stochastic fractal dromion so-lution determined by Eq. (23) with selections (32) and

(33).

Figures 4 and 5 respectively show plots of typical

stochastic fractal dromin and lump solutions which are

determined by Eq. (23) with the conditions Eq. (32) and

p = 2 + 0.5Γ(x + at) tanh[4(x + at)− 20] ,

q = 0.1 tanh(y + bt) + 0.12 tanh[(2(y + bt)− 15] , (33)

or

p = Γ(x + at) + (x + at)

2

+ 10

3

,q = Γ(y + bt) + (y + bt)2 + 103 (34)

with λ = s = 1.5 at t = 0. In Fig. 5, the vertical axis

denotes the quantity V which is only a re-scaling of the

quantity v: v = V × 10−7.

From Fig. 4, one can easily find that the amplitudes of

the multi-dromion are changed stochastically. Similarly,

the shapes of the multi-lump in Fig. 5 are stochastically

altered, too.

Fig. 5 A plot of typical stochastic fractal lump solutiondetermined by Eq. (23) with selections (32) and (34).

(ii) Regular fractal lumps and dromions

In addition to the stochastic fractal dromions and frac-

tal lumps, there may exist some regular fractal localized

excitations. We do know that it is a difficult task to

find some appropriate functions which can be used to de-

pict regular fractal patterns possessing self-similar struc-

tures. However, in recent study,[13−17] we find that many

lower-dimensional piecewise smooth functions with frac-

tal structure can be used to construct exact solutions of

higher-dimensional soliton system which also possess frac-

tal structures. This situation also occurs in the (2+1)-

dimensional modified dispersive water-wave system. If se-

lecting the functions p and q appropriately, we can find

some types of lump or dromin solutions with fractal be-

haviors. For instance, if p and q in expression (23) are

simply selected as

p = 1 + exp[

(x− c1t)2(1 + sin(ln(x− c1t)2))] , (35)

q = 1 + exp[

(y − c2t)2(1 + sin(ln(y − c2t)2))], (36)

then we can derive a fractal dromion solution which is

localized in all directions. Figure 6(a) shows a plot of

the special type of fractal dromions structure for the po-

tential v given by Eq. (23) with the conditions Eqs. (35)

and (36) at time t = 0. Figure 6(b) is a density plot

of the fractal structure of the dromion in the region

{x ∈ [−0.12, 0.12], y ∈ [−0.12, 0.12]}.

Similarly, when we take

p = 1 +|x − c1t|

1 + (x− c1t)4cos2(ln(x− c1t)2) , (37)

q = 1 +|y − c2t|

1 + (y − c2t)4cos2(ln(y − c2t)2) , (38)

then we can obtain a simple fractal lump pattern. Since

all the situations have been widely investigated in other

physical models such as in Refs. [13]–[17] and [25]–[27], we

omit these related plots and do not discuss further in this




paper.

Fig. 6 (a) A plot of fractal dromions structure for thepotential v given by solution (23) with the conditions (35)and (36) at time t = 0; (b) A density plot of the frac-tal structure of the dromion related to (a) in the region{x ∈ [−0.12, 0.12], y ∈ [−0.12, 0.12]}.

4 Summary and Discussion

In summary, with the help of the variable separa-tion approach, the (2+1)-dimensional modified dispersive

water-wave system is solved. Abundant localized coherent

soliton structures of solution (23), such as multi-dromion,

multi-ring, multi-lump solutions, peakons, breathers, and

instantons, etc. can be easily constructed by selecting ap-

propriate arbitrary functions.

Besides these usual localized coherent soliton struc-

tures, we find some new localized excitations — the

chaotic and fractal soliton solutions for the (2+1)-

dimensional modified dispersive water-wave system. As

is known, the chaos and fractals not only belong to the

realms of mathematics or computer graphics, but also ex-

ist nearly everywhere in nature, such as in fluid turbu-

lence, crystal growth patterns, human veins, fern shapes,

galaxy clustering, cloud structures, and in numerous other

examples. Conventionally, chaos and fractals are the

opposite circumstances to solitons in nonlinear science

since solitons are the representatives of integrable sys-

tem while chaos and fractals represent the behalf of non-

integrable systems. However, in this paper, we find some

chaotic and fractal structures for localized solutions of the

(2+1)-dimensional integrable modified dispersive water-

wave model. Why the localized excitations possess some

kinds of chaotic and/or fractal behaviors? If one consid-

ers the boundary and/or initial conditions of the chaotic

and/or fractal solutions obtained here, one can find that

the initial and/or boundary conditions possess the chaotic

and/or fractal property. In other words, the chaotic

and/or fractal behaviors of the localized excitations of the

integrable models come from the nonintegrable boundary

and/or initial conditions. Because of the wide applications

of the soliton, chaos and fractal theory in many physics

fields like the fluid dynamics, plasma physics, field theory,

optics, etc., it is a significant work to learn more about

the localized chaotic and/or fractal excitations of the “in-

tegrable” physical model with “nonintegrable” boundaryand/or initial conditions such as chaotic and/or fractal

boundary and/or initial conditions.

Acknowledgments

The author is in debt to helpful discussions with

Prof. J.F. Zhang, Prof. L.Q. Chen, and Dr. X.Y. Tang.

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