jie-fang zhang- generalized dromion structures of new (2 + 1)-dimensional nonlinear evolution...

8/3/2019 Jie-Fang Zhang- Generalized Dromion Structures of New (2 + 1)-Dimensional Nonlinear Evolution Equation

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Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 267270c International Academic Publishers Vol. 35, No. 3, March 15, 2001

Generalized Dromion Structures of New (2 + 1)-Dimensional Nonlinear Evolution

Equation

ZHANG Jie-Fang

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

(Received January 4, 2000)

Abstract We derive the generalized dromions of the new (2 + 1)-dimensional nonlinear evolution equation by thearbitrary function presented in the bilinearized linear equations. The rich soliton and dromion structures for this system

are released.

PACS numbers : 03.40.Kf, 02.90.+pKey words: (2 + 1) dimensions, nonlinear evolution equation, soliton, dromion

In the last decade there has been increasing great inter-

est in study of soliton generating nonlinear evolution equa-

tions in different branches of science not only in (1 + 1)-

dimensional systems, but also in (2 + 1)-dimensional or

higher-dimensional systems. Generally speaking, to find

exact analyzing coherent-type soliton structures of (2+1)-

dimensional systems is more complicated than that in

(1 + 1)-dimensional systems. Since the pioneering work of

Boiti et al.,[1] for some (2+1)-dimensional integrable mod-

els such as the DaveyStewartson (DS) equation,[2] theKadomtsevPetviashvilli (KP) equation,[3] the Nizhnik

NovikovVeselov (NNV) equation,[4] the breaking soliton

equation,[5] the long dispersive wave (LDW) equation,[6]

the scalar nonlinear Schrodinger (NLS) equation[7,8] and

for some (3 + 1)-dimensional integrable models such as

the KdV-type equation,[9] the JimboMiwaKadomtsev

Petviaashvili (JMKP) equation,[10] some types of expo-

nentially localized soliton solutions, called dromions, are

found by using different approaches. Usually, dromion so-

lutions are driven by two or more nonparallel straight lineghost solitons. Even more generalized dromion solutions

which are driven by curved and straight line soliton so-

lutions for the (1 + 1)-dimensional KdV equation,[11] the

(2+1)-dimensional potential breaking soliton equation,[12]

the (2+1)-dimensional LDW equation[13] and the (2+1)-

dimensional scalar NLS equation,[13] are also found. In

this article we investigate the generalized dromion struc-

tures of new (2+1)-dimensional nonlinear evolution equa-

tion discussed by Maccari[14] by suitably utilizing the ar-

bitrary functions presented in the system. The system is

of the form

i + = 0, i + = 0 ,

= (||2 + ||2) , (1)

where ( , , ), ( , , ) are complex and ( , , ) is

real. Equations (1) are derived from Nizhnik equations

through the reduction method. Uthayakunar et al.[15]

have established the integrability property of Eqs (1) by

using singularity structure analysis.

Equations (1) can be bilinearized by means of the fol-

lowing transformations

=g

f, =

h

f, = 2(ln f) , (2)

where g, h are complex functions and f is a real function.

Using Eq. (2) and the Hirotas bilinear operators,[16] equa-

tions (1) can be written as

(i D + D2)g f = 0 ,

(i D + D2)h f = 0 ,

(DD)f f = (|g|2 + |h|2) . (3)

We expand g, h and f in the form of a power series as

g = g(1) + 3g(3) + ,

h = h(1) + 3h(3) + ,

f = 1 + 2f(2) + 4f(4) + . (4)

Substituting Eqs (4) into Eqs (3) and comparing various

powers of , we get

o() : ig(1) + g(1) = 0 , ih

(1) + h

(1) = 0 , (5)

o(2) : f = 1

2(g(1)g(1) + h(1)h(1)) , (6)

o(3) : ig(3) + g(3) = (i D + D

2)g

(1) f(2) ,

ih(3) + h(3) = (i D + D

2)h

(1) f(2) , (7)

o(4) : f(4) + DDf

(2)f(2) = 1

2[ g(3)g(1)+g(1)g(3)

+ (h(3) h(1) + h(1) h(3))] , (8)


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268 ZHANG Jie-Fang Vol. 35

and so on.

Solving Eqs (5), we get

g(1) =Ni=1

Ai exp[ki + i + li()] , (9a)

h(1) =Ni=1

Bi exp[k

i +

i + l

i()] , (9b)

where li(), l

i() are arbitrary complex functions of and

ki, k

i, i,

i are complex constants,

iI = (k2iR k

2iI) , iR = 2kiRkiI ,

iI = (k2iR k

2iI) ,

iR = 2k

iRk

iI .

To construct the soliton solution, we put N = 1 so

that we have from Eq. (6),

f(2) =

1

2[A21 exp(2k1R + 2l1R() + 21R)

+ B21 exp(2k

1R + 2l

1R() + 2

1R)] . (10)

Solving the above equation, we obtain

f(2) = A21 exp(2k1R + 2G() + 21R + 21)

+ B21 exp(2k

1R + 2G() + 21R + 22) , (11)

where

exp[2G()] =

e2l1R()d, e21 =

1

4k1R,

exp[2G()] =

e2l

1R()d, e22 =

1

4k1R. (12)

Substituting g(1), h(1) and f(2) into Eqs (7) and (8), it canbe shown that g(j) = 0, h(j) = 0 for j 3 and f(j) = 0

for j 4. Now, using Eqs (9) and (11), we obtain the

solution of Eqs (1) as follows:

=g(1)

1 + f(2)=

A1 exp[k1 + 1 + l1()]

1 + A21 exp[2k1R + 2G() + 21R + 21] + B21 exp[2k

1R + 2G() + 21R + 22]

, (13)

=h(1)

1 + f(2)=

B1 exp[k

1 +

1 + l

1()]

1 + A21 exp[2k1R + 2G() + 21R + 21] + B21 exp[2k

1R + 2G() + 21R + 22]

. (14)

If we choose

k1 = k

1, 1 =

1, l1() = l

1() , (15)

then solution of Eqs (1) becomes

= A1M()sech(k1R + 1R + G() + 1 + )exp i[k1I + 1I + l1I()] , (16)

= B1M()sech(k1R + 1R + G() + 1 + ) exp i[k1I + 1I + l1I()] , (17)

where

exp(2) =1

A21 + B21

, M() =

k1R

A21 + B21

exp l1R()

exp G(). (18)

In Eq. (18) because l1R() is an arbitrary function of , we can take M() as an arbitrary function of . Tounderstand the meaning of solutions (16) and (17), we discuss some special cases.

Case 1 Single Dromion Driven by One Line Soliton and One Curve Soliton

If we set

=

A1=

B1,

then when M() is fixed as a single line soliton which is parallel to the -axis and we combine the line soliton M() and

curve soliton sech [k1Rx + 1Rt + G() + 1R(0)+ 1] together properly (multiplying them together simply in the case

of solutions (17) or (18)), the original straight line and curved line solitons disappear (become ghost) and only a single

peak localized in all directions, which is called a dromion, survives. The dromion is located at the interaction of the

line and curve solitons. Because G() is an arbitrary function of , the single dromion still possesses rich structures.Here are five concrete simple examples,

1 = sechn( 0)sech[k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()]

h1()sech[k1Rx + 1Rt + g() + 1] exp i[k1Ix + 1It + l1I()] , (19)


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No. 3 Generalized Dromion Structures of New (2 + 1)-Dimensional Nonlinear 269

2 = sechn[cosh( 0 1)] sech[k1Rx + 1Rt + G(y) + 1] exp i[k1Ix + 1It + l1I()]

h2()sech[k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()] , (20)

3 =1

[( 0)2n + 1]sech [k1Rx + 1Rt + G() + 1] exp i[k1Ix + 1It + l1I()]


4 =

Ni=1

aii

Mi=1

bii

sech [k1Rx + 1Rt + G() + 1] exp i[k1Ix + 1It + l1I()]


5 = [A + sin B( + 0) + C]hj sech [k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()] ,

j = 1, 2, , 4 . (23)

The first type of generalized dromion (19) decays exponentially in all directions. The second type of generalized

dromion (20) decays much more quickly than the first in the direction. The third type of generalized dromion (21)

decays much slower than the first in the direction. The fourth type of generalized dromion (22) does not decay quickly

in the direction. While the fifth type of generalized dromion (23) has an oscillatory structure in the direction.

Case 2 Multi-dromion Bounded States

If G() is selected to be N parallel line solitons (parallel to x-axis), then we get N-dromion bound state

NB =N

j=1

fj()sech[k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()] (24)

driven by N straight line ghost solitons and curved line ghost soliton. In Eq. (24), fj() can be selected quite freely,

say h1(), h2(), h3(), h4() in Eqs (19) (23). Because all N parallel line solitons are static in -direction, the

N-dromion can only move with the same speed in the x direction as the curved line ghost soliton moves and they

cannot pass through each other. In other words, the behaviors of this dromion looks like a bounded state.

Case 3 Single Soliton

From Eq. (9), by choosing

ki = kiR, i = k2iR, li() = liR , (25)

we can obtain basic single soliton solution when taking N = 1,

=exp[k1Rx + ik21Rt + l1R()]

1 + exp[2k1Rx + 2l1R + 21]=

l1Rk1R sech [k1Rx + l1R + 1] exp ik21Rt . (26)

Case 4 Basic Dromion

To generate a (1, 1) basic dromion solution, if we may put N = 2 in Eq. (9) and take

f(1) = exp(k1Rx + ik21Rt) + exp(l1R) , (27)

we have

g(2) = M11 exp(1 +

1) + L11x exp(21) + K11 exp(1 + 1) + K11 exp(

1 + 1) , (28)

where

M11 = 1

2k1R , L11 =

1

2l1R , K11 =

1

k1RllR , 1 = k1Rx + ik21Rt , 1 = l1R , (29)

thus a (1, 1) dromion solution is obtained as follows:

=exp(1 + 1)

1 + M11y exp(1 +

1) + L11x exp21 + K11 exp(1 + 1) + K11 exp(

1 + 1). (30)


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270 ZHANG Jie-Fang Vol. 35

This expression describes an exponentially localized solution with one bound state in the x-direction and one bound

state in the -direction. The above analysis can be further generalized to (M, N) dromions

u =

Mi=1

Nj=1

exp(i + j)

1 +Mi=1

Nj=1

[Mij exp(i + j ) + Lijx exp(i + j) + Kij exp(i + j) + Kij exp(

i + j)]

, (31)

where

i = kiRx + ik2i t, j = ljR, Mij =

1

ki + kj,

Lij = 1

li + lj, Kij =

1

kiRljR, (32)

which represents an exponentially localized solution with

M bound states in the x-direction and N bound states in

the -direction.

In summary, we can not only generate the basic local-

ized solutions but also generate the generalized localized

solutions of Eq. (1) by harnessing the different arbitrary

functions of suitably. To look for the dromion of other

(2 + 1)-dimensional nonlinear evolution equations, further

studies are worthy to be done.

References

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4746.

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[9] S.Y. LOU, J. Phys. A29 (1996) 5989.

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[14] A. Maccari, J. Math. Phys. 38 (1997) 4151.

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