jie-fang zhang and chun-long zheng- abundant localized coherent structures of the (2+1)-dimensional...

8/3/2019 Jie-Fang Zhang and Chun-Long Zheng- Abundant Localized Coherent Structures of the (2+1)-dimensional Generalize

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CHINESE JOURNAL OF PHYSICS VOL. 41 , NO. 3 JUNE 2003

Abundant Localized Coherent Structures of the (2+1)-dimensional GeneralizedNozhnik-Novikov-Veselov System

Jie-Fang Zhang

Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua 321004, China andShanghai Institute of Mathematics and Mechanics,

Shanghai University, Shanghai 200072 ,China

Chun-Long ZhengInstitute of Nonlinear Physics, Zhejiang Normal University, Jinhua 321004, China and

Department of Physics, Zhejiang Lishui Normal College, Lishui 323000, China(Received September 9, 2002)

In a previous paper (Chin. Phys. 11, 651, (2002)), a rather general variable separation solution ofthe generalized Nizhnik-Novikov-Veselov(GNNV) system was obtained by using a special Backlundtransformation, which can be derived from the extended homogenous balance method. However we

did not discuss the related localized coherent structures of the model. In this article, the abundanceof the localized coherent structures of the system, particularly some localized excitations with fractalbehaviours, i.e. the fractal dromion and fractal lump excitations, were induced by the appropriateselection of the separated variables arbitrary functions.

PACS numbers: 03.40.Kf

I. INTRODUCTION

It is well-known that the (2+1)-dimensional Nizhnik-Novikov-Veselov(NNV) equation [1] isthe only known isotropic Lax extension of the well known (1+1)-dimensional KdV equation. Sometypes of soliton solutions have been studied by many authors. For Instance, Boiti, Leon, Mannaand Pempinelli [2] solved the NNV equation via the inverse scattering transformation. Tagami [3]

and Hu, and Li [4] obtained soliton-like solutions for the NNV equation by means of the B acklundtransformation. Hu [5] also gave a nonlinear superposition formula of the NNV equation. Ohta[6] obtained the Pfaffian solutions for the NNV equation. Radha and Lakshmanan [7] constructedonly the dromion solutions from its bilinear form after analyzing its integrability aspects. Lou[8] obtained some special new types of multisoliton solutions for the NNV equation by using thestandard truncated Painleve analysis.

In Ref. [9], a rather general variable separation solution of the generalized Nizhnik-Novikov-Veselov(GNNV) system

vt + avxxx + vyyy + cvx + dvy = 3a(uv)x + 3b(vw), (1)

vx = uy, vy = wx, (2)

where a,b,c and d are some arbitrary constants, was obtained by using a special Backlund trans-

formation, which can be derived from the extended homogenous balance method. Specifically:

v =2(A a1a2)pxqy

(1 + a1p + a2q + Apq)2, (3)

u =2(a1 + Aq)

2p2x(1 + a1p + a2q + Apq)2

2(a1 + Aq)pxx1 + a1p + a2q + Apq

+pt + apxxx + cpx

3apx, (4)

http://PSROC.phys.ntu.edu.tw/cjp 242 c 2003 THE PHYSICAL SOCIETYOF THE REPUBLIC OF CHINA


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VOL. 41 JIE-FANG ZHANG AND CHUN-LONG ZHENG 243

w =2(a2 + Ap)2q2y

(1 + a1p + a2q + Apq)2 2(a1 + Ap)qyy

1 + a1p + a2q + Apq+

qt + bqyyy + dqy3bqy

, (5)

where a1, a2 and A are some arbitrary constants, p = p(x, t) may be an arbitrary function of {x, t}, and q = q(y, t) may also be an arbitrary function of{y, t}. However we did not discuss the relatedlocalized coherent structures of the model. In this article, the abundance of the localized coherentstructures of the system, in particular, some localized excitations with fractal behaviors, i.e. thefractal dromion and fractal lump excitations, were induced by appropriately selecting the separatedvariables arbitrary functions.

II. SOME STABLE LOCALIZED COHERENT SOLITON STRUCTURES OF THE (2+1)-DIMENSIONAL GNNV SYSTEM

It is interesting that expression (3) is valid for many (2+1)-dimensional models such as theDS equation, the NNV system, the ANNV equation and the ADS model, the dispersive long waveequation, etc. [13, 2628]. Because of the arbitrariness of the functions p and q included in (3), thequantity v possesses quite rich structures. For instance, as mentioned in [13, 2628], if we select

the functions p and q appropriately, we can obtain many kinds of localized solutions, like the multi-solitoff solutions, multi-dromion and dromion lattice solutions, multiple ring soliton solutions, andso on. Although these types of localized solutions have been discussed for other models, we includesome special examples here for completeness.

II-1. Multi-solitoff solutions and multi-dromion solutions driven by straight-line soli-tons

If we restrict the functions p and q of (3) to bep = 1 +

Ni=1 exp(kix + it + x0i) 1 +

Ni=1 exp(i) ,

q =M

i=1 exp(Kiy + y0i)J

j=1 exp(jt) ,(6)

where x0i, y0i, ki, i, Ki and i are arbitrary constants, and M, N and J are arbitrary positive

integers, then we have the multi-solitoff solutions (we call a half straight line soliton solution asolitoff, which is caused by the resonance condition A = 0) and the first type of special multi-dromion solutions (A = 0), driven by multiple straight-line solitons. There is no dispersion relationamong ki, i, Ki and i.

Fig. 1 shows the structure of a two-solitoff solution for the quantity v shown by (3) with (6)and

M = 2, N = 1, k1 = K1 = 1, K2 = 2, a1 = a2 = 1, A = 0,x01 = y01 = 0, y02 = 9 , (7)

at time t = 0.Fig. 2 shows the structure of a single dromion solution for the quantity v shown by (3 ) with

(6) and

M = 1, N = 1, k1 = K1 = 3, a1 = 10, a2 = 3, A = 0, x01 = y01 = 0 , (8)

at time t = 0.

II-2. Multi-dromion solutions driven by curved-line ghost solitons

Recently, Lou has pointed out that for many (2+1)-dimensional models, a dromion may bedriven not only by straight line solitons [36] but also by curved line solitons [37]. Actually, (3) can


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244 ABUNDANT LOCALIZED COHERENT STRUCTURE OF THE . . . VOL. 41

Fig.1

50

510

1520

x

20

10

0

10

20

y

1

0v

FIG. 1: A special two-solitoff solution for the quantity v shown by (3) with (6) and (7) at time t = 0.

Fig.2

4

2

0

2

4

x

42

02

4

y

0

0.5

1

1.5

v

FIG. 2: A single dromion solution for the quantity v shown by (3) with (6) and (8) at time t = 0.

be rewritten as

v =QyPx(a1a2)

2[A1 cosh12(P + Q + C1) + A2 cosh

12(P Q + C2)]2

, (9)

where P and Q are related to p and q by p = b1 exp(P), q = b2 exp(Q) and

A1 =

A(a1b1 + a2b2 + Ab1b2), A2 =

(a1 + Ab2)(a2 + Ab1),

C2 = lna1 + Ab2a2 + Ab2

, C1 = lnA

a1b1 + a2b2 + Ab1b2,

with arbitrary constants b1 and b2.

Hence the general multi-dromion solutions of the model expressed by (3) (or equivalently(9)) are driven by two sets of straight-line solitons and some curved-line solitons. The first set ofstraight-line solitons appears in the factor Qy. One can take

Qy =Ni=1

Qi(y yi0) , (10)


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Fig.3

32

10

1

2

3

x 321

01

23

y

0

0.4v

FIG. 3: A special four-dromion solution shown by (3) with (9) and (12) at time t = 0.

where Qi = Qi(y yi0) denotes a straight-line soliton which is finite at the line y = yi0 anddecays rapidly away from the line. Similarly, the second set of straight-line solitons appears in thefactor Px. Finally, the curved-line solitons are determined by the factors A1 cosh(P + Q + C1) andA2 cosh(PQ + C2) of (9) and the curves are determined by

P + Q + C1 = min(P + Q + C1), PQ + C2 = min(P Q + C2) , (11)while the number of curved-line solitons is determined by the branches of the equations in (11). Thedromions are located at the cross points and/or the closed points of the straight and curved lines.

Figure 3 is a plot of a multi-dromion solution driven by the curved line solitons via takingP = (x v1t)3 + xv1t15 , Q = (yv2t)

5

20 + (y v2t)3 + yv2t100 ,a1 = a2 = 1, A = 2, b1 = 3, b2 = 40, t = 0.

(12)

II-3. Multi-lump solutionsIt is also known that in high dimensions, like the KP and DSII equations, a special type oflocalized structure (called lump solutions) may also be formed by rational functions. Actually, themulti-lump solutions of (2+1)-dimensional integrable models can be found by taking the appropriatearbitrary functions.

For the GNNV system, if we select the functions p and q of (3) to be rational functionssatisfying the conditions

p > 0, q > 0, x, y, t (13)and a1 > 0, a2 > 0, A > 0, then we can obtain the nonsingular localized lump solutions. In Figure4, we plot a special lump solution (3) with

p = 1 +1

1 + (x v1t 30)2

+1

1 + (x v2t)2

+1

1 + (x v3t + 30)2

, (14)

q =1

1 + (y v4t 15)2 +1

1 + (y v5t + 15)2 , a1 = 10, a2 = 10, A = 1 , (15)

at time t = 0.


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Fig.4

40

20

0

20

40

x 3020

100

1020

30

y

0.1

00.1v

FIG. 4: A special lump solution for (3) with the selection (14), (15) at time t = 0.

Fig.5

2010

010

20

x

15

10

5

05

10

15

y

1

0

1

v

FIG. 5: A oscillating dromion solution for (3) with the selection (16) at time t = 0.

II-4. Oscillating dromion solutions

If some periodic functions in the space variables are included in the functions p and q , wemay obtain some types of multi-dromion solutions with oscillating tails. The oscillating dromionsolution in Fig. 5 is related to

p = 1 + exp ((x v1t) cos(2(x v1t)) + 5/4), q = exp(y v2t), (16)a1 = a2 = 10, A = 1, t = 0.

II-5. Ring soliton solutions

In high dimensions, in addition to the point-like localized coherent excitations, there may besome other types of physically significant localized excitation. For instance, in the (2+1)-dimensionalcases, there may be some types of ring soliton solutions which are not identically equal to zero onsome closed curves and decay exponentially away from the closed curves [2628, 30]. In Figures 6and 7, we plot the interaction property of a travelling two saddle type of ring soliton solution withthe selection

p = 1 + exp (x cos(2x) + 5/4), q = exp(y), a1 = a2 = 10, A = 1. (17)


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Fig.6a

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.6b

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.6c

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.6d

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.6e

4020

020

40

x

20

10

0

10

20

y

1

01

v

FIG. 6: The evolution of two saddle type ring soliton solutions given by (3) with the selection (17) at thetimes: (a) t = 1, (b) t = 0.4, (c) t = 0, (d) t = 0.4, (e) t = 1.

In Fig. 6, we plot the evolution of the two ring soliton solution for quantity v expressed by (3)with (17) at times (a) t = 1, (b) t = 0.4, (c) t = 0, (d) t = 0.4, and (e) t = 1, respectively.From Figs. 6(a)6(e), we can see that the interaction of the two ring soliton solutions is elastic. Tosee more clearly the completely elastic interaction properties between the two travelling ring solitonsolutions, two counter plots related to Figure 6(a) and Figure 6(e) are plotted in Figure 7.


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Fig.7a

10

5

0

5

10

y

30 20 10 0 10 20 30x

Fig.7b

10

5

0

5

10

y

30 20 10 0 10 20 30x

FIG. 7: (a) The contour plot related to Fig. 6 (a); (b) The contour plot of Fig. 6 (e). The values of thecontours in these figures from the outside to the inside are: |v| = 0.01, |v| = 0.1 and |v| = 0.4, respectively.

II-6. Multi-breather like soliton solutions

In (1+1)-dimensional cases, the breather solutions are another important type of nonlinearexcitations. Because of the arbitrariness that appeared in the functions p and q of (3), the breathersolutions to the (2+1)-dimensional models may also have quite rich structures. On the one hand,any (1+1)-dimensional breather solution of the (1+1)-dimensional integrable models (like the sine-Gordon model and the nonlinear Schrodinger equation) can be used to construct a breather solutionof the higher dimensional models, say the GNNV system. In Figure 8, the well known breathersolution

p = 4 arctan

1 2 sin(t)

cosh

1 2x

, (18)

of the sine-Gordon model, pxx ptt = sinp, is taken as the function p of (3), while q is taken asq = exp(y) (19)

with the parameters = 1/2, a1 = a2 = 10, A = 1. On the other hand, one can put any periodicfunctions of t into the localized excitations, as shown in the above examples 15, to construct moreinteresting new breather-like solutions. Fig. 9 shows the evolution behavior of a breather-like ringsoliton solution given by (5) with

p = exp 110 (x 20 sin(t))2 + 5(1.1 + sin(t)) ,

q = expy2

10 5 + sin(t)

, a1 = a2 = 1, A = 0.(20)

From Figs. 9a9e, we can see that the breather-like ring soliton solution can breath in somedifferent ways, in particular, it can breath not only in its amplitude but also in its shape (like theradius of the loop), and the position.

II-7. Multiple instanton solution.

If some types of decaying functions of the time t are included in the solution (3), then we canfind some types of instanton solutions. In Figure 10, the behavior of a special ring type of instantonsolution (3) with

p = exp

x2sech2t10 + 5(1.1 + sin(t))

,

q = expy2sech2t

10 5

, a1 = a2 = 1, a3 = 0(21)


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Fig.8a

86

42

02

46

8

x

86

42

02

46

8

y

0.04

0

0.04

v

Fig.8b

86

42

02

46

8

x

86

42

02

46

8

y

0.02

0

0.02

v

FIG. 8: The plots of the point-like breather solution (3) with (18) and (19) and the parameters = 1/2, a1 =a2 = 10, a3 = 1, at the times: (a) t = 4 and (b) t =

4, respectively.

is exhibited. From Fig. 10a and Fig. 10b, we can see that the amplitude of the ring type of instantonsolution (3) with (21) decays rapidly from |v| 1 to |v| 109 as the time increases from t = 0 tot = 10.

III. SOME SPECIAL FRACTAL LOCALIZED COHERENT SOLITON STRUCTURES OFTHE (2+1)-DIMENSIONAL GNNV SYSTEM

III-1. Regular fractal dromions and lumps with self-similar structures.

It is known that in (2+1)-dimensions one of the most important basic excitations are the socalled dromions, which are exponentially localized in all directions. In [28], the authors found thatmany lower-dimensional piecewise smooth functions with fractal structure can be used to constructexact localized solutions of higher-dimensional soliton systems which also possess fractal structures.This situation also occurs in the (2+1)-dimensional GNNV system. With appropriately selected

arbitrary functions p and q, we were surprised to find that some special types of fractal dromionsfor the potential v (3) can be revealed. For example, if we take

p = 1 + exp(x(x + sin(ln(x2)) + cos(ln(x2)))), (22)

q = 1 + exp(y(y + sin(ln(y2)) + cos(ln(y2)))), (23)with a1 = a2 = 1, A = 2, then we can obtain a simple fractal dromion. Figure 11(a) shows a plotof this special type of fractal dromion structure for the potential v given by (3) with the conditions(22-23). Figure 11(b) is a density plot of the fractal structure of the dromion in the region {x=[-0.016, 0.016], y=[-0.016, 0.016]}. To observe the self-similar structure of the fractal dromion moreclearly, one may enlarge a small region near the center of Figure 11(b). For instance, if we reducethe region in Figure 11(b) to

{x=[-0.0032, 0.0032], y=[-0.0032, 0.0032]

},{

x=[-0.000138, 0.000138],y=[-0.000138, 0.000138]}, {x=[-0.000006, 0.000006], y=[-0.000006, 0.000006]} and so on, we find astructure totally similar to that presented in Figure 11(b).

It is also known that in high dimensions, such as the KP equations, the NNV equationsand the ANNV equations, a special type of localized structure, which is called the lump solution(algebraically localized in all directions), is formed by rational functions. These localized coherentsoliton structures are another type of significant localized excitation. If the functions p and q of


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Fig.9a

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.9b

4020

020

40

x

20

10

0

10

20

y

0.5

0

0.5v

Fig.9c

4020

020

40

x

20

10

0

10

20

y

0v

Fig.9d

4020

020

40

x

20

10

0

10

20

y

1

0

1

v

Fig.9e

4020

020

40

x

20

10

0

10

20

y

1

01

v

Fig.9f

4020

020

40

x

20

10

0

10

20

y

1

01

v

FIG. 9: The evolution of a special ring shaped breather solution (3) with the selection (20) at the times:(a) t = 1, (b) t = 0.5, (c) t = 0.2, (d) t = 0, (e) t = 0.2, (f) t = 1, respectively.

the potential v (3) are selected appropriately, we can find some types of lump solutions with fractalbehaviors. Figure 12(a) shows a fractal lump structure for the potential v, where the p and q in

solution (3) are selected as follows:

p = 1 +|x|

1 + x4(sin(ln(x2)) + cos(ln(x2)))2, (24)

q = 1 +|y|

1 + y4(sin(ln(y2)) + cos ln(y2)))2, (25)


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Fig.10a

2010

010

20

x

20

10

0

10

20

y

1

0

1

v

Fig.10b

15000050000

50000150000

x

150000100000

500000

50000100000

150000

y

0v

FIG. 10: (a) The evolution plots of the ring type of instanton solution (3) with (21) at the times (a) t = 0and (b) t = 10.

Fig.11(a)

42

02

4

x

43

21

01

23

y

0.1

0

0.1v

Fig.11(b)

.015

0.01

.005

0

.005

0.01

.015

y

0.015 0.01 0.005 0 0.005 0.01 0.015x

FIG. 11: (a) A plot of the fractal dromion structure for the potential v given by the solution (3) with the

conditions (22-23) and a1 = a2 = 1, A = 2. (b) is a density plot of the fractal structure of the dromion inthe region {x = [0.016, 0.016], y = [0.016, 0.016]}.

with a1 = a2 = 1, A = 2. From Figure 12(a), we can see that the solution is localized in alldirections. Near the center there are infinitely many peaks which are distributed in a fractal manner.In order to investigate the fractal structure of the lump, we must look at the structure more carefully.Figure 12(b) presents a density plot of the structure of the fractal lump at the region {x=[-0.016,0.016], y=[-0.016, 0.016]}. A more detailed study shows us the interesting self-similar structure ofthe lump. For example, if we reduce the region of Figure 12(b) to {x=[-0.0032, 0.0032], y=[-0.0032,0.0032]}, {x=[-0.00066, 0.00066], y=[-0.00066, 0.00066] }, {x=[-0.000028, 0.000028], y=[-0.000028,0.000028]} and so on, we find structure totally similar to that plotted in Figure 12( b).

III-2. Stochastic fractal dromions and lumps.

In addition to the self-similar regular fractal dromions and lumps, the lower-dimensionalstochastic fractal functions may also be used to construct higher-dimensional stochastic fractaldromion and lump excitations. For instance, one of the most well-known stochastic fractal functions


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Fig.12(a)

4

2

0

2

4

x

4

2

0

2

4

y

0v

Fig.12(b)

.015

0.01

.005

0

.005

0.01

.015

y

0. 015 0. 01 0. 005 0 0.00 5 0. 01 0.015x

FIG. 12: (a) A fractal lump structure for the potential v with the conditions (24-25) and a1 = a2 =1, A = 2 . (b) A density plot of the fractal lump related to (a) for the region {x = [0.0016, 0.0016], y =[0.0016, 0.0016]}.

Fig.13

6040

200

2040

60

x

6040

200

2040

60

y

2

0

2

4

ab

FIG. 13: A plot of a typical stochastic fractal lump solution determined by (3) with the selections (26-27)and a1 = a2 = 1, A = 2.

is the Weierstrass function,

w w() =Nk=0

(3/2)k/2 sin((3/2)k), N, (26)

where the independent variable may be a suitable function of {x + at} and/or {y + bt}, say = x + at and = y + bt in the functions u and v, respectively, for the following selection (27).If the Weierstrass function is included in the dromion or lump excitations, then we can derive thestochastic fractal dromions and lumps. Figure 13 shows a plot of a typical stochastic fractal lump

solution, which is determined by (3) with (26) and

p = w(x + at) + (x + at)2 + 103, q = w(y + bt) + (y + bt)2 + 103, (27)

and a1 = a2 = 1, A = 2 at t = 0. In Figure 13 the vertical axis denotes the quantity v which isonly a re-scaling of the potential v: v = v 107.


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IV. SUMMARY AND DISCUSSION

In summary, with the help of the Backlund transformation and the variable separation ap-proach, a (2+1)-dimensional GNNV system is solved. The abundant localized coherent soliton

structures of the solution (3), such as multi-dromion, multi-ring, multi-lump solutions, breathersand instantons etc., can be constructed by selecting the appropriate arbitrary functions.In addition to these stable localized coherent soliton structures, we find some new localized

excitationsthe fractal dromion and lump solutions for the (2+1)-dimensional GNNV system shownin Figures 11(b) and 12(b). As is known, fractals not only belong to the realms of mathematics andcomputer graphics, but also exist nearly everywhere in nature, such as in tree branching, leaves,coastlines, fluid turbulence, crystal growth patterns, human veins, fern shapes, galaxy clustering,cloud structures and in numerous other examples. By selecting different types of lower-dimensionalfractal models, one may obtain various beautiful higher dimensional fractal patterns. These beautifulpictures may be useful in architecture, costume design, and so on. In the future, perhaps the mostfamous artists will also be the most famous physicists and/or mathematicians, because paintings willbe produced not by brushes, but by mathematical expressions. Generally, fractals are the oppositeto solitons in nonlinear science, since solitons are trepresentatives of an integrable system, whilefractals represent non-integrable systems. However, in this paper, we found some fractal structuresfor dromion and lump solutions for the (2+1)-dimensional GNNV model. Naturally, as pointed outin [28], the question of what on earth the integrability definition is, weighs on people mind, as doesthe question of how to find and make use of this novel phenomena in reality.

Even though some issues baffle us, we believe that the variable separation approach is usefuland powerful and can be used in other (2+1)-dimensional nonlinear physical models. Additionalapplications of this method to other (2+1)-dimensional physical models, and the properties of themultiple localized coherent excitations, especially excitations with fractal behaviors, are worthy offurther study.

Acknowledgments

This project was supported by the Foundation of 151 Talent Engineering of ZhejiangProvince and the Natural Science Foundation of Zhejiang Province of China under Grant No.101032.

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