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  • 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()]

    h3()sech[k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()] , (21)

    4 =

    Ni=1

    aii

    Mi=1

    bii

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

    h4()sech[k1Rx + 1Rt + G() + 1]exp i[k1Ix + 1It + l1I()] , (22)

    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.

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