Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 1063–1066c© International Academic Publishers Vol. 48, No. 6, December 15, 2007

Peakon Excitations and Fractal Dromions for General (2+1)-Dimensional Korteweg deVries System∗

MA Song-Hua,† LI Jiang-Bo, and FANG Jian-PingCollege of Mathematics and Physics, Lishui University, Lishui 323000, China

(Received December 5, 2006; Revised March 6, 2007)Abstract By means of an extended mapping approach and a linear variable separation approach, a new family ofexact solutions of the general (2+1)-dimensional Korteweg de Vries system (GKdV) are derived. Based on the derivedsolitary wave excitation, we obtain some special peakon excitations and fractal dromions in this short note.

PACS numbers: 05.45.Yv, 03.65.GeKey words: extended mapping approach, GKdV system, peakon excitation, fractal dromion

1 IntroductionSolitons and fractals are the two important aspects of

nonlinear science.[1] Because of the wide applications ofsoliton and fractal in many natural sciences such as chem-istry, biology, mathematics, communication, and partic-ularly in almost all branches of physics like fluid dy-namics, plasma physics, field theory, optics, and con-densed matter physics, etc.,[2] searching for exact andexplicit solutions of a nonlinear physical model, espe-cially for new exponentially localized structures like soli-ton solutions or for these excitations with novel prop-erties is a very significant work. Since the concept ofdromions was introduced by Boiti et al.,[3] the study ofsoliton-like solutions in higher dimensions has attractedmuch more attention. Now several significant (2+1)-and (3+1)-dimensional models, such as (2+1)-dimensionalKadomtsev–Petviashvili equation,[4] Davey–Stewartsonequation,[5] generalized Korteweg-de Vries equation,[6]

asymmetric NNV equation,[7] sine-Gordon equation,[8]

(3+1)-dimensional Korteweg-de Vries equation,[9] andJimbo-Miwa–Kadomtsev–Petviashvili equation[10] havebeen investigated and some special types of localizedsolutions for these models have also been obtained bymeans of different approaches, for instance, the bilinearmethod, the standard Painleve truncated expansion, themethod of “coalescence of eigenvalue” or “wavenumbers”,the homogenous balance method, the variable separationmethod,[11−20] and the mapping method,[21−25] etc. Fromthe above study of (2+1)- and (3+1)-dimensional models,one can see that there exist more abundant localized struc-tures than those in lower dimensions. This fact hints thatthere may exist new localized coherent structures that areunrevealed in some (2+1)-dimensional integrable models.

In this paper, by the extended mapping approach, wefound the new exact solutions of (2+1)-dimensional GKdVsystem

ut − uxxy − auuy − bux∂−1x uy = 0 , (1)

where a and b are arbitrary constants. Many researchershave investigated some interesting properties of Eq. (1).For instance, the GKdV system has been proved to beintegrable by Calogero,[26] and the Painleve property testfor the equation has been proved to be completely inte-grable by Clarkson et al.[27] only when a = 2b. Some vari-able separation solutions for the special model for a = b

have also been obtained by Lou and coworkers.[28] Zhengand Chen found some semifolded localized coherent struc-tures by the multilinear variable separation method.[29]However, to the best of our knowledge, the mapping exci-tations for the (2+1)-dimensional GKdV model were notreported in previous literature.

In the following discussion, we will investigate themapping solutions to the case a = b for Eq. (1). Forsimplicity, we introduce a transformation v = ∂−1

x uy andchange the GKdV system into a set of two coupled non-linear partial differential equations:

ut − uxxy − buvx − bvux = 0 , uy − vx = 0 . (2)

2 New Exact Solutions to the GKdV SystemAs is well known, to search for the solitary wave so-

lutions to a nonlinear physical model, we can apply dif-ferent approaches. One of the most efficient methods tofind soliton excitations of a physical model is the so-calledextended mapping approach. The basic ideal of the algo-rithm is as follows. For a given nonlinear partial differ-ential equation (NPDE) with the independent variablesx = (x0 = t, x1, x2, . . . , xm), and the dependent variableu, in the form

P (u, ut, uxi, uxixj

, . . .) = 0 , (3)

where P is in general a polynomial function of its argu-ments, and the subscripts denote the partial derivatives,the solution can be assumed to be in the form

u = A(x) +n∑

i=1

{Bi(x)φi[q(x)] + Ci(x)φ−i[q(x)]} (4)

with

φ′ = σ + φ2 , (5)

where A(x), Bi(x), Ci(x), and q(x) are functions of theindicated argument to be determined, σ is an arbitraryconstant, and the prime denotes φ differentiation with re-spect to q. To determine u explicitly, one may substitute(4) and (5) into the given NPDE and collect coefficientsof polynomials of φ, then eliminate each coefficient to de-rive a set of partial differential equations of A(x), Bi(x),Ci(x), and q(x), and solve the system of partial differential

∗The project supported by the Natural Science Foundation of Zhejiang Province under Grant No. Y604106 and the Natural Science

Foundation of Zhejiang Lishui University under Grant No. KZ05010†Corresponding author, E-mail: msh6209@yahoo.com.cn

1064 MA Song-Hua, LI Jiang-Bo, and FANG Jian-Ping Vol. 48

equations to obtain A(x), Bi(x), Ci(x), and q(x). Finally,equation (5) possesses the general solutions

φ =

−√−σ tanh(

√−σq) , σ < 0 ,

−√−σ coth(

√−σq) , σ < 0 ,

√σ tan(

√σq) , σ > 0 ,

−√

σ cot(√

σq) , σ > 0 ,

−1/q , σ = 0 .

(6)

Substituting A(x), Bi(x), Ci(x), q(x) and Eq. (6) intoEq. (4), one can obtain the exact solutions to the givenNPDE.

Now we apply the extended mapping approach toEq. (2). By the balancing procedure, the ansatz (4) be-comes

u = f + gφ + hφ2 +Q

φ+

R

φ2,

v = F + Gφ + Hφ2 +A

φ+

B

φ2, (7)

where f , g, h, Q, R, F , G, H, A, B, and q are functionsof (x, y, t) to be determined. Substituting Eqs. (7) and (5)into Eq. (2) and collecting coefficients of polynomials ofφ, then setting each coefficient to zero, we have

f =−qxxxqy + 3qxyqxx − 3qxqxxy − 8q3

xqyσ + 12q2xσ

∫(−qxqxy + qxxqy)dx + qxqt

bqyqx,

g = −6qxx

b, h = −6

q2x

b, Q = 6

qxxσ

b, R = −6

q2xσ2

b, F =

12σ

b

∫(qxqxy − qxxqy)dx ,

G = −6qxy

b, H = −6

qxqy

b, A = 6

qxyσ

b, B = −6

qxqyσ2

b(8)

with the function q in a special variable separation formq = χ(x) + ϕ(y − ct) , (9)

where c is an arbitrary constant. Based on the solutions of Eq. (5), one thus obtains an explicit solution of equation(2).

Case 1 For σ = −1, we can derive the following solitary wave solutions of Eq. (2):

u1 = 6χxx tanh(χ + ϕ)− χ2

x

b tanh(χ + ϕ)2− χxxx + 4χ3

x + χxc− 6χxx tanh(χ + ϕ)χx + 6χ3x tanh(χ + ϕ)2

bχx, (10)

v1 = −6ϕyχx sech(χ + ϕ)4

b tanh(χ + ϕ)2, (11)

u2 = 6χxx coth(χ + ϕ)− χ2

x

b coth(χ + ϕ)2− χxxx + 4χ3

x + χxc− 6χxx coth(χ + ϕ)χx + 6χ3x coth(χ + ϕ)2

bχx, (12)

v2 = −6ϕyχx csch(χ + ϕ)4

b coth(χ + ϕ)2(13)

with two arbitrary functions being χ(x) and ϕ(y − ct).Case 2 For σ = 1, we can obtain the following periodic wave solutions of Eq. (2):

u3 = 6χxx tan(χ + ϕ)− χ2

x

b tan(χ + ϕ)2− χxxx − 4χ3

x + χxc + 6χxx tan(χ + ϕ)χx + 6χ3x tan(χ + ϕ)2

bχx, (14)

v3 = −6ϕyχx sec(χ + ϕ)4

b tan(χ + ϕ)2, (15)

u4 = −6χxx cot(χ + ϕ) + χ2

x

b cot(χ + ϕ)2− χxxx − 4χ3

x + χxc− 6χxx cot(χ + ϕ)χx + 6χ3x cot(χ + ϕ)2

bχx, (16)

v4 = −6ϕyχx csc(χ + ϕ)4

b cot(χ + ϕ)2(17)

with two arbitrary functions being χ(x) and ϕ(y − ct).Case 3 For σ = 0, we can derive the following vari-

able separation solution of Eq. (2):

u5 = −χxxx + χxc

bχx+ 6

χxx

b(χ + ϕ)− 6

χ2x

b(χ + ϕ)2, (18)

v5 = −6ϕyχx

b(χ + ϕ)2(19)

with two arbitrary functions being χ(x) and ϕ(y − ct).

3 Some New Localized Coherent Structuresin the GKdV SystemDue to the arbitrariness of the functions χ(x) and

ϕ(y − ct) included in the above cases, the physical quan-tities u and v may possess rich structures. For example,

when χ = ax and ϕ = y−ct, all the solutions of the abovecases become simple travelling wave excitations. More-over, based on the derived solutions, we may obtain richlocalized structures such as peakons and fractal dromions.In the following discussion, we merely analyze some spe-cial localized excitations of solution v2 in Eq. (13) in Case1, namely

V = v2 = −6ϕyχx csch(χ + ϕ)4

b coth(χ + ϕ)2. (20)

3.1 Peakon Excitations

According to the solution (20), we first discuss itspeakon excitations. For instance, if we choose χ and ϕas

No. 6 Peakon Excitations and Fractal Dromions for General (2+1)-Dimensional Korteweg de Vries System 1065

χ = 1 + exp(−|x + 1|) , ϕ = 1 + tanh(y − ct) , (21)χ = 1 + exp(−|x + 1|) , ϕ = 1 + sech(y − ct) , (22)

we can obtain a type of peakon excitation for the physicalquantity V of Eq. (20) presented in Figs. 1(a) and 1(b)with fixed parameters b = −0.1, c = 1, and t = 0. If wechoose χ and ϕ asχ = 1 + tanh(−|x + 1|) , ϕ = 1 + 0.1 tanh(y − ct) , (23)χ = 1 + tanh(−|x + 1|) , ϕ = 1 + 0.1 sech(y − ct) , (24)we can obtain a type of peakon excitation for the phys-

ical quantity V of Eq. (20) presented in Figs. 2(a) and2(b) with fixed parameters b = −0.2, c = 1, and t = 0.Furthermore, if we choose χ and ϕ asχ = 1 + 2 exp(−|x + 3|) , ϕ = 1 + exp(−|y − ct|) , (25)χ = 1 + 2 exp(−|x + 3|) + 0.8 exp(−|x− 3|) ,

ϕ = 1 + exp(−|y − ct|) , (26)we can obtain another type of peakon excitation for thephysical quantity V of Eq. (20) presented in Figs. 3(a) and3(b) with fixed parameters b = −6, c = −3, and t = 1.

Fig. 1 A plot of a special type of peakon structure for the physical quantity V given by the solution (20) with thechoices (21) and (22) and b = −0.1, c = 1, t = 0.

Fig. 2 A plot of a special type of peakon structure for the physical quantity V given by the solution (20) with thechoices (23) and (24) and b = −0.2, c = 1, t = 0.

Fig. 3 A plot of a special type of peakon structure for the physical quantity V given by the solution (20) with thechoices (25) and (26) and b = −6, c = −3, t = 1.

3.2 Fractal Dromion

In (2+1) dimensions, one of the most important nonlinear solutions is the dromion excitation, which is localized inall directions exponentially. Recently, it is found that many lower-dimensional piecewise smooth functions with fractalproperties can be used to construct exact localized solutions of higher-dimensional soliton systems which also possessfractal structures.[30] This situation also occurs in the (2+1)-dimensional GKdV system. If we appropriately select thearbitrary functions χ and ϕ, we find that some special types of fractal dromions for the field V will be revealed. Forexample, if we take

χ = 1 + exp [−x(x + 2sin(ln x2))] , ϕ = 1 + exp [−(y − ct)((y − ct) + 2sin(ln(y − ct)2))] , (27)

1066 MA Song-Hua, LI Jiang-Bo, and FANG Jian-Ping Vol. 48

we can derive a fractal dromion. Figure 4(a) shows a plot of this special types of fractal dromion structure for the Vgiven by Eq. (20) with the choice (27) at b = −1, c = 1, t = 0. Figure 4(b) shows the density of the fractal structure ofthe dromion at the region (x ∈ [−0.002, 0.002], y ∈ [−0.002, 0.002]). To observe the self-similar structure of the fractaldromion more clearly, one may enlarge a small region near the center of Fig. 4(b). For instance, if we reduce the regionof Fig. 4(b) to (x ∈ [−0.0002, 0.0002], y ∈ [−0.0002, 0.0002]), (x ∈ [−0.000 02, 0.000 02], y ∈ [−0.000 02, 0.000 02]) andso on, we find totally similar structure to that presented in Fig. 4(b).

Fig. 4 (a) A fractal dromion structure for the V given by Eq. (20) with the choice (27) and b = −1, c = 1, t = 0. (b)Density plot of the fractal structure at the region (x ∈ [−0.002, 0.002], y ∈ [−0.002, 0.002]).

4 Summary and DiscussionIn summary, via an extended mapping approach and

a special variable separation form q = χ(x) + ϕ(y − ct),we find some new exact solutions of the general (2+1)-dimensional Korteweg de Vries system. Based on thederived solitary wave solution (13), we obtain some spe-cial peakon excitations. Then, we use the sine functionand exponent function to find the fractal dromion of the(2+1)-dimensional GKdV system. Additionally, using thepiecewise function, Zheng recently obtained some peakon

excitations in the new (2+1)-dimensional long dispersivewave system.[19] Along with the above line, we use thepiecewise function to get the new peakon excitations ofGKdV system, which are different from the ones of theprevious work.

AcknowledgmentsThe authors would like to thank Prof. Chun-Long

Zheng for his fruitful and helpful suggestions.

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