Jizong Lu- A New Dromion Solution of the Davey-Stewartson Equation

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T T7 5"'

IC/95/322

INTERNAL REPORT

(Limited Distribution)

International Atomic Energy Agency

and

United Nations Educational Scientific andCultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

A NEW DROMION SOLUTIONOF THE DAVEY-STEWARTSON EQUATION1

Jizong Lu2

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

A new dromion solution is obtained for a (2+1)-dimensional integrable model: the

Davey- Stewartson equation. Some interesting questions which emerge in the procedureof getting the solution arealso discussed.

MIRAMARE - TRIESTE

October 1995

'Submitted to II Nuovo Cimento.2Permanent address: Shanghai Teachers University, Shanghai 200234, People's Repub-

lic of China.

1 Introduction

By using the Backlund transformation, Boiti etal [1] found that for the Davey-Stewartson

(DS) equation there existed a kind solution which localized at the intersection of two plane

waves. This kind solution was later named as the dromion solution. After then dromions

were also found by using other methods, for instance, the inverse scattering transformation

(1ST) method [2] and the bilinear (BL) method [3]. Why call them dromions? This is

because they can be thought to form tracks (dromos in Greek).

The system for which dromion solutions have been mostly studied is the DS equa-

tion. Using the BL method, dromion solutions for other types of (2+l)-dimensional

integrabte models like the generic (2-M)-dimensional nonlinear equations of Schrodinger

(NLS), Korteweg-de Vries (KdV) [4] and the Nizhnib-Novikov-Veselov (NNV) equations[5]

were also found.

These dromions are constructed from plane waves or ghost solitons. Besides ghost soli-

tons, however, there also exist curved line soliton solutions which are finite on a curved line

and exponentially (or algebraically) decaying apart from the curve andperiodic solitons

which are periodic on one direction anddecaying in others for some (2+l)-dimensional

integrable systems. Can one construct dromions from these solitons? Most recently, it

has been shown that dromion solutions can be also made out of curved line solitons for

the potential breaking soliton (BS) equation and a (2+l)-dimensionaI KdV type equation

[6]. In this letter, the similar kind dromions for the DS equation will be discussed.

2 Special solutions of th e DS equat ion

ou t an

The DS equation describes the evolutions of surface waves ofslowly varying amplitude

and has been widely applied to many fields, such as water waves, plasma physics and

nonlinear optics etc. In solution which localized at the intersection of two plan was

named as th e dromion solution. It can be written in several variant butequivalent forms.

Here we use the same form as in [3] [4]:

. -I- uYV - 4 u |u |2- 2uu = 0,

vxx+v Yy+A(\u\2}xx=0 .

Introducing new dependent variables F (real) andG (complex) by

u~j, v = -2d\\ogF

and rotating the coordinate axes by45°, eqs. (1) and (2) can be written as

D xDyF-F = 2|G|2,

where D is the usual bilinear operator defined by

= (dt - dr)m{dx - dx,r(dv -

(2)

(3)

(4)

(5)

•,*=V,«=f) . (6)



In order to solve eqs. (4) and (5), let us expand G an d F in the forms of power series

where e is a small para mete r. Subs titu ting eqs. (7) and (8) into eqs. (4) and (5) and

comparing the coefficients of various powers of t, we have

f° : g0 0,

41' = 0,

This equation system can be solved in principle due to its linearity. It is very difficulty

to do so, however, because there are infinite equations. In order to look for its special

solutions, we cut off the equation series by setting if>W and ^ *' = 0 for all k > N(N =

2,3, - • •). Thus the equation system becomes

< = 0, (10)

0, (U)

(12)

(13)

(14)

where p'1= fx{x,t)gv(y,t) and £ is an arbitrary function of x,y and t. Substituting

eqs. (13) and (14) into eqs. (9) and (11) and vanishing both real and imaginary parts

respectively yield only three equations to restrict functions f,g and £:

The solution of eq. (10) is

<l >M

=f{x,t)-g(y,t).

Substituting eq. (13) into eq. (12), we have

fx9 vt

= 0, (16)

Separating variables in eq. (15) and integrating with respect to x and y respectively,

we get:

(18)

(19)

where C is a constant introduced in the variable separation procedure and Ci(t), C2(i) areintegral functions.

On the other hand, eq. (17) implies that the only possible form of f(x, t/,t) is

, y, t) = y, 0 + c'(t) f fl-»dz f (20)

where £i(z, t) = ^ i, ^(j/,*) = £2 and c'(t) = d axe functions of indicated variables. Thuseq. (17) is separated to two equations:

(21)

9t = 2c' f gyldy + c(t), (22)

where c(t) is another function of (. S ubstituting eqs. (20) ~ (22) into eq. (15), we find

that it is satisfied only for d = d{t) = 0 and there is no further restriction on / , g, £.1

and £j. Substituting eq. (20) into eqs. (18) and (19) and comparing them with eqs. (21)

and (22), we get

/« = c{t) - 2 / ^ , , , (23)

gt = c(t) - 2^ 6 , . (24 )

Particularly, a special case for c(t) = 0 is interesting. In that case eqs. (23) and (24)

become

F {2S)

2g s

(26)

If we substitute eqs. (25) and (26) into eq. (16) and separate variables, then eq. (16)

will be reduced to two trilinear equations:

+ fL = 0. (27)

\ l 9m ~ °- (2 8

)The integrability of eq. (27) (or eq. (28)) is guaranteed by the one of the DS equation.

That means that special solutions of the DS equation can be obtained by those from

two new (l+l)-dimensional integrable models (eqs. (27) and (28)). However, to give outdetail solutions of eqs. (27) and (28) is still very difficult. Here we only write down some

special solutions. The simplest solutions possess exponential forms

/ =

g =

(29)

(30)

with arbitrary constants OQ, bo. 1, b.



3 Dromion solutions

If we were able to obtained the dromion solution by substituting eqs. (29), (30) with

eqs. (7), (8), (13) and (14) into eq. (3), no doubt it would be a new kind of dromions.

Unfortunately, thesolution made out in this way decays inmost directions butone, so it

is not a dromion solution. We should consider higher order terms of e. After doing so, F

and G aremodified as follows:

F = f'(x, t) - g'{y, t) + Cf(x, t)g'{y, t), (31)

G =p(x, y,t}expi[£i(x, t) + £.2(y,t)\} (32)

where p2— fx(x,t)g'y(y,t) andC" is an arbitrary constant. After substituting eqs. (31)

and (32) into eq. (4) and doing straight calculation like in Section 2, we obtain the

following equations,

= 0, (33)

= 0, (34)

• /2 -2 / ; / ;M -c ' 2( j ) / ;2

= o, (35)

+ g'y\ - 2g'ygy = 0, (36)

where c[(t) andc2(t) introduced in the variable separation procedure are two arbitrary

functions of t. Comparing eqs. (33) and (34) with eqs. (25 and(26), we find the relation

between / , j and f\g'

(38)

where B = exp J' C'd(t')dt' andc'(t) is another arbitrary function of time t. From eqs.

(35) and (36), we get the equations satisfied by / and g. They are just eqs. (27) and

(28). Therefore eqs. (27) and(28) do probably possess some universality, which will make

them quite meaningful.

Now we can construct solutions of the DS equation byusing solutions (29) and (30).

The solutions made in this way are completely a new kind ones. Inorder to see this point,

let us further assume

Sl{x,t) = kx + at> &(!/,'} = <& + »*• (39)

Substituting eqs. (31) and (32) with eqs. (29), (30) and (39) into eq. (3), we have

{akbl)i expi(-<^r/2fc - Uy/2l + c3{t) + ct(t)}

aexp [-(6 + 6)/2] + /?exp[(£ - fc)/2] +7«P[( 6 - fi)/2] + ««p[(£, + &)/2]

(40)

with a = aoB-l-boB + (2/c} +aobo, 0= B~

1a(l + bo), -y= bB(a0B~

l- I), 6 = cab. The

solution (40) decays exponentially in all directions after selecting a, j3, 7 andS to possess

same sign andabkl > 0, so it obviously is a dromion solution.

4 Conclusion

In summary, we would like to point out:

1) Usually one uses thebilinear form (4) and (5) to construct thedromion solutionby

assuming £, = fox + Uy +W + 6^ In our discussion, d(£2) is only an arbitrary function

of {x,t}({y,t}). This will make our solutions aremore general than usual ones. Even

in equation (40), u is different from that given in refs. [1], [2], [3] and [4] due to the

arbitrary functions of time t. In deed, if we restrict c3(t) and c4(t) being linear in t and

dx(t) = 0 {B = 1) in u, then « identify to them. No doubt, this is a new kind dromion

solution of the DS equation, although it does notpossess the obvious meaning of curved

line dromions as in ref. [6j.

2) Both eqs. (27) and (28) are of trilinear form. This is a new kind of (1+1)-

dimensional integrable model. Although some other types of trilinear equations have

been discussed [7], the knowledge about them arestill much less than those of bilinear

ones. The further studies about trilinear equations must bevery important andvaluable.

3) Ineqs. (27) and (28) thevariables are completely separated. / (or g) is a function

of {i,t } (or {y,i}). This implies that the variable separation approach may be applied

in non-linear partial differential equations. Recently, some mathematical physicists have

paid their attention to this subject. Forinstance, byusing symmetry constrain, Chang

and Li [8] got the solution of the (2+l)-dimensional K-P equation from two solutions:

one of the (l + l)-dimensional NLS equation with variables {x,y} and the other of the

modified (l + l)-dimensional KdV equation with variables {x, t}. However, in that case

the variables are not separated completely. We have discussed this subject in ref. [9].

Acknowledgments

The author would like to thank Prof. S. Randjbar-Daemi for hospitality at the Inter-

national Centre for Theoretical Physics, Trieste, where the work was done. The author

would also like to thank Prof. S.-y. Lou for hisvery helpful discussion.



References

[lj M. Boiti, J, J. -P. Leon, L. Martina and F. Pampinnelli, Phys. Lett. A 13 2 (1988)

432

[2] A. S. Fokas and P. M. Santini. Phys. Rev. Lett. 63 (1989) 1329

[3] J- Hietarinta and R. Hirota, Phys. Lett. A 14 5 (1990) 237

[4] J. Hietarinta, Phys. Lett. A 14 9 (1990) 113

[5] R. Radha and M. Lakshmanan, J. Math. Phys. 35 (1994) 4746

[6] S-y. Lou, Preprint (1995)

[7] J. Matsukolaira, J. Satsuma and W. Strampp, Phys. Lett. A 14 7 (1990) 467; B.

Grammat.ices, A. Ramam and .1 . Hietarinta, Phys. Lett. A 19 0 (1994) 6

[8] Y. Chang and Y-s Li, Phys. Lett. A 15 7 (1991) 22

[9] S-y, Lou and J. Lu, Preprint (1995)

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Jizong Lu- A New Dromion Solution of the Davey-Stewartson Equation