Localized coherent structures and integrability in a generalized (2 + 1)-dimensional nonlinear Schrödinger equation

Chaos, Solrrom

& Fnzcrals Vol. 8, No. 1. pp. 17-K 1997 Copyright fQ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0%x-0779197 $17 00 + o.on

PII: SO960-0779(%)00090-2

Localized Coherent Structures and Integrability in a Generalized (2 + l)-Dimensional Nonlinear Schriidinger Equation R. RADHA

and M. LAKSHMANAN

Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli-620024,

India

(Accepted 4 July I996) Abstract-A generalized (2 + 1)-dimensional nonlinear Schriidinger equation introduced recently by Fokas is investigated and is shown to admit the Painleve property. The Hirota bilinearization directly follows from the singularity analysis. Localized dromion solutions, which arise essentially due to the interaction of two nonparallel ghost solitons and localized breather solutions (time oscillating solutions), are constructed using the Hirata method. This method can be rigorously pursued to generate multidromions and multibreathers. Copyright @ 1997 Elsevier Science Ltd

1. INTRODUCTION

Recently, Fokas [l] has introduced a novel (2 + l)-dimensional integrable equation, which is probably the simplest and symmetric generalization of nonlinear Schrodinger (NLS) equation in two spatial dimensions. It has the following form:

- (a - p,(,” lq/;drl’ + udt, 1))1 = 0, (1) --3: where E and 17are characteristic

coordinates

defined by

rj=x-y. c=x+y, (2) In the above equation, the scalar function q(E, n, t) is complex representing some physical quantity and ut(n, t) and uZ(& t) are arbitrary real-valued functions of the arguments indicated and they constitute the so-called ‘boundary flows’ of the system (1). In eqn (l), )3.= +l and (Y and p are arbitrary real constants. Physical realization of eqn (1) includes some of the important integrable models of the generalization of NLS equation in (2 + 1) dimensions. In fact, it is really one of three systems containing the following interesting cases. Case (i).

When LY= /3 = -i and u1 # 0, it yields

’ ]q(x, y’, t)Ifdy’ + u,(x, t) = 0, il = +l. (3) 0 -cc 1 As Fokas [l] has already pointed out, this equation is perhaps the simplest complex scalar equation in two dimensions, which can be solved by the inverse scattering transform (ET) method. When y = x and u1 = 0, eqn (3) reduces to the NLS equation. It is interesting to compare (3) with the equation of Boiti et al. [2], which is a (2 + 1)-dimensional generalized version of the Korteweg-deVries equation of the form

iq, + ax - %

17

!8

R. RADHA

and M. LAKSHMANAN

ut + U@ = 3(u&

u; = u9.

Pa)

Both eqns (3) and (3a) are (2 + 1)-dimensional generalizations of well-known integrable (1 + 1)-dimensional soliton equations, containing only one boundary and admit exponentially localized coherent structures [3, 41 for nonvanishing boundaries (see Section 4 below). Case (ii).

& = 0, /3 = 1 yields the celebrated Davey-Stewartson

I (DSI) equation

[5] of

the form

i9,+455 +qso - 2A9

+ uz(C> r))I = 0.

It may be noted that this equation has been already shown to have the Painleve [h, 71 and is known to be a completely integrable system admitting exponentially solutions including dromions for nonvanishing boundaries [3, 4, 8, 91.

(4)

property localized

When a, = 1, /3 = 0, we obtain the integrable equation called DSIII, 5 + U~(TI, t) i9, - 4g + 4011- 2A9 -miq@E’ )d;dq’ + uz(E, r))] = 0 (5) 1 (i [ii obtained by Boiti et al. [lo] by using the ideas of the direct linearising method [ll, 121. However, as noted by Fokas [I], it was derived earlier by Schulman [13] and also by Santini and Fokas [14, 151 using the symmetry approach. Equation (5) again supports certain localized solutions. However, it should be mentioned that eqn (1) is not the only simplest integrable generalization of NLS equation in (2 + 1) dimensions. There exists another generalization of NLS equation in (2 + 1) dimensions introduced by Strachan [16] of the form Case (iii).

v, = 23,lqj”. k, = qxy + V9, (6) This equation is again completely integrable [17] and is similar to eqn (3), except for the second derivative term. Unlike the previous equations, it does not seem to possess localized solutions. The goal of this paper is to explore the integrability aspects of the generalized NLS equation (1) and to bring out the existence of explicit localized coherent structures such as dromions and breathers using Hirota bilinearized form derivable from the Painleve analysis. The plan of the paper is as follows. In Section 2, we carry out the Painleve singularity analysis of eqn (1) and show that the system is free from movable critical manifolds. Then in Section 3, we generate its Hirota bilinear form. Constructing the line soliton solutions, we point out the existence in general of two nonintersecting ghost solitons along the boundaries. (However, the presence of only one ghost soliton is itself sufficient to generate an exponentially localized structure in eqn (3)). These are then used to generate exponentially localized multidromions and breathers in Section 4. Finally, in Section 5, we give a brief discussion of our results. 2. SINGULARITY

To explore transformation

the singularity

structure

v, = 191;, and

STRUCTURE

ANALYSIS

[18] of eqn (l), V =

we introduce

’ )q1;dE’ + u,(v, t) I -c.z

the following

(74

Localized

coherent

structures

and integrability

in eqn (l), where ui( q, t) and u2(E, t) are boundary coupled equations - 2Aq[(a

iq, - (a - B)~E + (a + Pk,,

1Y

terms, and convert it into a system of + p>v - (cv - /3)U] = 0,

@a>

v, = IsI’,

(8b)

u, = k&

(8~)

and the complex conjugate form of (8a). In the above equation, V and U represent some real potentials. It is also instructive to compare the form of (8) with the Nizhnik-NovikovVeselov (NNV) equation [19] which is a symmetric generalization of the KortewegdeVries equation in (2 + 1) dimensions. Now, we rewrite the above set of equations, by letting q = a and q* = h, as ia, - (a - P)+ -ib,

+ (a + PIa,,

- (a - IWEE + (a + PP,,

- 2Aa[(cx + @)V - (a - @)U] = 0,

Pa)

- 2Ab[(a

Pb)

+ @)V - (a - o)U]

= 0,

VE = (ab),,

(9c)

U, = (ab)E.

(94

We now effect a local Laurent expansion in the neighbourhood of a noncharacteristic singular manifold Q(&!, q, t) = 0, (GE, &,, 4, # 0). Assuming the leading orders of the solutions of eqn (9) have the form a = a&P,

b = b&f’,

v = V&Y,

u = u@#f,

(10)

where ao, bo, V,, and U, are analytic functions of (5, rl, t) and m, n, y and 6 are integers (if they exist) to be determined, we now substitute (10) into (9) and balance the most dominant terms to get m=n=-1 6 = y = -2, (11) with the condition AU, = cp;;2. nvo = hj2, %bo = #&,7 Now, considering the full Laurent expansion of the solutions in the neighbourhood singular manifold [20] a = ao@’ + . . . + aj$lml + . . ., b = b,+-’

+ . . . + b,$‘-’

v = v,~-2

+ vl$-l

(12)

of the

+ . . .,

+ . . . + vj+j-2

(13)

+ . . .,

u = ,yoqm2 + u,q -1 + . . . + ujf2

+ . . .,

the resonances (powers at which arbitrary functions enter in (13)) can be determined by substituting (13) into (9) and comparing the coefficients of (#je3, c#J-~, #jv3, +jm3) to give i(i

-. 3X(@

+ iy’

- (a - PME21 i(i

(i - Wvh

(i - 2)bo@E

- 3)[(~

+ B)iq2 - Cm- P)@E~I (i -

(i -

2bo@q WO@E

-2Ano(cY -2kbo((u

+ p) + /3)

-(i ; W#q

2Ano(a - P)

2,1bo((u

- /3)

-(i _” WJ, = 0. (14)

20

R. RADHA

and M. LAKSHMANAN

Solving eqn (14), one gets the resonance values as j = -1, 0, 2, 2, 3,4.

(1% The resonance at j = -1 represents as usual the arbitrariness of the singularity manifold 9(5, rl, t> = 0. T o b ring out the existence of arbitrary functions at the other resonance values, we now substitute the full Laurent series (13) into (9). Now, collecting the coefficients of (c$-~, c#-~, @-3, 4-3) and solving them, we obtain essentially the three relations of eqn (12) for the four coefficients ao, b,, V. and Uo. From the relation Aaobo = &&,, we deduce that one of the coefficients a0 or b. is arbitrary, corresponding to the resonance j = 0. Now, collecting the coefficients of (@-*, c$-~, c$-~, @*), we have 2[(a - @@s2- (a + P)&*h

- 2A(a + P>aoV, + 2X& - B)aoUl = A,,

2[(a - PME2 - (a + m+J2P1 - wa

+ BPOV, + 34a - PPOU, = &,

~bO&Ql + ~~094)b, - My1

abo@p, + Aao4@, - W$J, = D,,

= Cl,

(16)

where 4

=

iwh

-

(a

4 = -ibd+

-

+ uo#nl + (a + PM2aoll& + ao@,J,

PP~o&

- Cc - B)Pb&

Cl = (P&/7) - h+&,>

+ bo#EEl + (a + P)[2bo,#q + boQfas],

4

= #&,

(17)

- +&q.

Solving equation (17) with (18), we obtain a1

=

ia

-

(a

-

P)@o&

-

ao@&

+

(a

+

/3)(2~~~4+

-

uo~,,)

7

2[(& - mQ2 - (a + PM,21 b = --iboA - (a - P@h&

- b&)

1

+ (a + PWbo,@, - bo$,,)

>

W)

ata - PMs2 - (a: + PM,*1 Vl = -M?p

Wa)

u1 = -a@&.

W’d)

Again, collecting the coefficients of (#-’ , #-’ , $-I, @I), we get 2[(@ + 8)4,*

- (a - /4&*1u2

+ 24a

+

Pbov2

-

2&a

-

3.t~ + PMq2 - (a - 8M~21b2 + 2%~ + P)boV, - %a

PboU2

=

4,

- P)boU, = &,

(204 Gob)

VIE = (aoh + bo4,,

WC)

UI, = (aoh + boa&>

(204

where A2 = ia B2 = -ibot

- (a -

P)UOEE

+

(a

+

16)aoqll

-

2A(a

+

P)alVl

+

- (a - @bog, + (a + /3)bo,l,l - 2A(a + p)b,V,

2A(~u - &Ul,

+ 2A(a - p)b,U,.

Using (19) and (12), we easily check that eqns (2Oc, d) are identically satisfied. Then, we are left with only two equations for four unknown functions u2, b2, V,, U, and hence two of them must be arbitrary. Now, gathering the coefficients of (Go”, Go”, Go, c$‘), we get 2Auo[(a

+

P)V3

- (a - P)U,l = p,

(214

2aboKa

+

P)V,

-

@lb)

(a

-

PI U31=

R,

21

Localized coherent structures and integrability 4%@ob3

+ boa,)

- V3@5 = s,

WC)

~&ob3

+ boa,)

-

(214

u3&,

= T,

where P = ial,

+ ia2h

- (a - B)[~I~~

+ 2a2pPp

+ a2+551 + (a + P)E4,,

- 2h((u + P)[UlV, + UzVJ + 2A((u - @[u,U, R = -ih,

- GA

- (a - P)[bg~

- 2k(a + P)[b,V,

+ 2b&

+ b2&1

+ bzV,] + 2A(a - P)[b,U,

+ 2a2+?& + a:!d+J

+ u,!U*],

(224

+ (a + PMb,,,

+ 2b2+&

+ bzUJ,

+ b24v111

(22b)

S = V2t: - [uo,&

+ a&

+ wh,b~

+ azDbo + aobzq + 01,

+ &~v

+ a2bo,J

(22c)

T = U,,, - [ao$z

+ a&

+ w&b,

+ @o

+ &GE

+ a,bo,l.

(224

+ ao&

+ &E

Analysing the above set of four equations (21a-d), one can easily check that they can be reduced to a set of only three equations in four unknowns, thereby implying that one of the functions a3, b3, V3 and U3 is arbitrary. Proceeding further, we have checked, by collecting the coefficients of ($‘, 4’) @‘, #J’), that in a similar manner one of the functions a4, b4, V4 and U4 is arbitrary. Thus the general solution (a, b, V, U) (E, q, t) of eqn (9) admits the required number of arbitrary functions, without the introduction of any movable critical manifold, thereby satisfying the Painleve property and hence the system (1) is expected to be integrable.

3. BACKLUND

TRANSFORMATION

AND BILINEAR

FORM

Now, to generate the Backlund transformation of eqn (l), we truncate the Laurent series at the constant level term, that is Uj = bj = 0 for j 2 2 and Vj = Uj = 0 for j 2 3. Thus from (13) we have a = uop

+ al,

v = voc#-2 + vp#-l

+ v,,

b = bo@-’ + bl,

u = uor#-2 + &p-l

+ u,,

(23)

where (a, aI), (b, b,), (V, V2) and (U, U2) satisfy eqns (9) with (ao, bo, V,, U,) and (V,, U,) satisfying eqns (12) and (19), respectively. Without loss of generality, we consider the vacuum solution al = bl = V2 = U2 = 0 in the above Backlund transformation (23) to yield a = a&,

(244

b = bo@-‘,

Wb)

v = voq2

+ V&l

= - A8,,log $,

(24~)

u = u,$-2

+ ulqF’

= - Aa,,log 4.

(244 (24) in eqn (9), we get

Assuming now that C#is real and inserting the above transformation the Hirota bilinear form of eqn (9) as [iDt - (a: - @DE2 + ((Y + @D,~]u,.#= [-iD,

- ((u - /3)DE2 + (a + /3)D,2]bo.#= DED,+$.@ = -2Auobo.

0,

(254

0,

G-1 WC)

Since (25a) and (25b) can be treated as conjugate to each other in view of the form of (8)

22

R. RADHA

and M. LAKSHMANAN

and (9) (b, = ao*), they can be put together in a simpler form by choosing a,, = g, b0 = g* as [iDt - ((u - p)DE2 + (LY + p)D,2]g.# DED,,+$ which is the desired Hirota

bilinear

= 0,

(264

= -2Agg”,

Wb)

form for eqns (8).

4. LINE SOLITONS,

DROMIONS

AND BREATHERS

To generate line solitons, we now expand g and C#Iin the form of power series as g = Egl + E3g3 + * . .) cp = 1 + E%#$ +

&4qb4

(274

+

. . *,

W’b)

where E is a small parameter. Substituting (27) in (26) and comparing E, we obtain the following set of equations, E: igl, - Cm - I%155 + (a + Pk,,,

the various powers of

= 0,

@a)

E2: qQIJ = -~g1g,“, and so on. Now, solving equation

(28b)

(28a), we get

N

xj = PjE f sjrl - i[(& - p)pj” - (Ly +

+ X,‘,

(29)

where pi, sj and Xi0 are complex constants. To construct the one soliton solution, N = 1 in (29) and so we have

we take

b3 = Cexpx,,

/3)Sj2]t

j=l

gl = Substituting

expxl,

Bb~~lt + XI’.

XI = PIE + sir - i[(a - P)PI~ - (a +

(30)

(30) in (28b), we obtain

exp(xl

@2=

(314

+ XI* + WI,

where exp (299 = -W~PIRSIR

Plb) (31) and (30) in

and PrR and SrR are the real parts of p1 rind sl, respectively. Substituting (24), the physical field q and the potentials V and U take the form q(t-, 7, t) = -;

exp

(--

111) se&

V(E,

17, f)

=

+lR2=h2

(xlR

U(~Y

V,

=

-@,R2=h2

(xlR

t)

(xlR + +

+

VI

exp

(ixd,

(324

q),

Pb)

‘$$,

PC)

where XlR = p1Rt + slRq + [2(a - fi)plRp11 - 2(cx + P)S~RS~I]~. Here plI and slI are the imaginary parts of p1 and sl, respectively. This treatment can be extended to generate N-line soliton solutions as well. A simple analysis of the above solutions (32) reveals the fact that, as the parameter pra + 0, both q and U tend to zero, but V takes the form V = +,R2sech2

(&,

+ $) = U,(n, t),

where G is a new phase constant. Similarly, while both q and V vanish:

%R

=

SIR[r]

-

the other potential

u = -Ap,n2sech2 (IrR + IjJr) = U2(& t),

%R

=

PIR[g

+

(33) U survives when srR -+ 0,

2(a

2(a

+

Pb,,t]

-

~>plI~l

+

cl,

+

c25

(34)

Localized coherent structures and integrability

where r/+ is another phase constant. Thus the solution is composed of two ghost solitons u,(q- t) and ~~(5, t) driving the potentials V and U respectively in the absence of the physical field q. To generate a (1, 1) dromion solution [21], we take the ansatz @1lD = 1 + jexp(x1 + xl*) + kexp (x2 + x2*) + Zexp (xl + x1* + x2 + x2*>* (35) where j, k, I are real, positive constants such that (I - jk)e-2w > 0, and allow x1 and x2 to assume the forms

Substituting

x2 = s,q + i(& + p)sr2t + c2.

(36b)

- jk)exp(xl

+ xl* + x2 + x2*) = -b*.

(37)

(37) yields gllD

Thus, a (1, 1) dromion

=

q,

l)

P exp

(xl

+

x2),

lPl2

=

(38)

%wdjk-W.

solution has the form

pexp(plR5^ qllD(E,

(364

(35) with (36) in (26b), we get ~PIRG~~

Equation

xl = plE - i(a - P)p12t + cl,

+

slRq

+

ih5^

+

@

+

(@

+

,@li2

-

ta

-

/-%i2)tl>

= 1 +

jexd2plR8

+

kexp(2slRfl)

+

zexp(2hR~

+

SlRql)

’

(39) where 9 = rj - 2(a + B)SlIf.

ii = 6 + 2(a - P)PlIt>

(40) It is not difficult to see that the solution (39), which is bounded, is localized exponentially in all directions in the 5, rl plane and moves with the velocity (-2[a - /3]p,,, ~[CY-t /3]srr) in the (E, q) coordinates.

Limiting

cases

(i) In the case of eqn (3) for LY= /3 = -i, (with the identification q + x, c + y) to Pexp(plRx^ hD@>

Y,

t,

=

+

1 + jexp(2plRx^)

the above (1, 1) dromion $lRY^

+

iblIx^

+

slIY^

+

solution

reduces

ld2d)

$ k exp(2slRy^) + lexp(2[plRx^

+ s&I)

’

(41)

where 2 = x - 2s,rt,

y^ = y.

(42)

We note that in this case, due to the form of eqn (3), only one ghost soliton ur(x, present here corresponding to eqn (33). (ii) In the case of DSI equation for (Y= 0, p = 1, we have Pexp(plR~ qllD(k

q,

l)

+

SlRq

8

+

+

i[PlI$

+

%Ifl

fi)

+

+

did2

+

bli2)f])

(43)

= 1 +

j exp

(2PlR

k exp

&lR

1 exp

(2blR8

+

t) is

SlRql)

’

where fj = rj - 2s11t, z = 5 - 2p11t, which agrees exactly with the solution given by Fokas and Santini [4].

(44)

:!4

R. RADHA

(iii) The dromion form

solution

and M. LAKSHMANAN

for DSIII

equation corresponding

to a = 1, /3 = 0 assumes the

where

8= 5+

q = r] - 2s,1t,

%ht,

(46)

which is identical to the one given by Boiti et al. [lo].

Discussion

Looking at the nature of the above solutions, one is really surprised to see that even though the simplest complex scalar equation (3) consists of only one ghost soliton moving with the velocity 2su, it possesses dromions which is reminiscent of the behaviour of the equation of Boiti et al. [19]. On the other hand, the DSI and DSIII equations contain two nonparallel ghost solitons moving with the velocities (2p,,, 2s,r) and (-2p,,, 2s,i), respectively. It is also of interest to note that one can obtain localized breathers, which oscillate in time, from the above analysis. For example, a (1, 1) breather is obtained by putting slI = pll = 0 in (39) to give Pexp(plRE qllB@~

Y?

r,

+

SlRv

+

d(@

+

blR2

-

(m

-

P)PlR’lt)

=

1 + jexp(2&?Rt)

+ kexpC2slRr7) + lexd2bIRg

+ +Rvl)'

(47)

One can proceed in a similar fashion and construct multidromion solutions also, though in the present case the analysis is much more intricate compared with the case of NNV equation [19]. For example, to obtain a (2, 1) dromion, we make the ansatz &ID = 1 + Aexp(xI

+ x1*) + Bedx2

+ x2*) + Cev(x3

+ DEexp(xl

+ x2*) + exp(x2 + xl*)1

+ Ebp(x1

+ x2* + x3 + x3*) + e&x2

+ x3*)

+ XI* + x3 + x3*)1

+ Fexp (xl + XI* + x2 + x2*) + Gexp (x2 + x2* + x3 + x3*) + Hexp(xl

+ xl* + x3 + x3*)

+ Zexp(xl + x1* +

x2

+

x2*

+

x3

+ x3*),

(48)

where

xl = PIE - i(a - P)PI’~ + kl, x2 = ~25 - i(a - Ph2t

+ h

(49)

x3 = slq + i(a + P)s12t + k3,

where k,, k2, k3 are positive constants and A, B, C, D, E, F, G, H, and Z are some positive real constants. Substituting (48) with (49) in (26b) and choosing pzl = pII, we obtain g2lD

=

Pll

exp

(xl

+

x3)

+

P12

exp

(x2

+

x3)

+ ~2~exp(x~ + x1* + x2 + x3) + h2exp(xl

where the real parameters

+ x2 + x;T + x3),

pn, p12, p21 and pz2 can be chosen appropriately

(50)

in terms of the

25

Localized coherent structures and integrability

parameters A, B, C, D, E, F, G, H, 1, Pm, pZR, SIR and il. Thus, the (2, 1) dromion solution can be expressed as q21D

=

g21DhD

(51)

from eqns (48) to (50). The above procedure can be extended in principle to generate (N,M) dromions though the calculations are very cumbersome. One can also generate other exotic coherent structures [22] like dromion-kink, kink-kink bound states, and so on. Interaction properties of these localized structures which are expected to be quite similar to that of the DSI equation [4, 8, 91 can be analysed in the same way. 5. CONCLUSION

In this paper, we have carried out the singularity structure analysis of a generalized (2 + 1)-dimensional NLS equation introduced by Fokas and shown that the system satisfies the Painleve property. We have generated its bilinear form from the Painleve analysis and constructed its line solitons. By allowing the two nonparallel ghost solitons to interact, we have successfully generated multidromions and breather solutions. Acknowledgements-The first author wishes to thank the Council of Scientific and Industrial Research for providing a Senior Research Fellowship. The work of the second author forms a part of a research project of the Department of Atomic Energy Commission.

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