Acta Applicandae Mathematicae 39: 445--455, 1995. (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

445

New Features of Soliton Dynamics in 2 + 1 Dimensions* E PEMPINELLI Dipartimento di Fisica dell'UniversitY, Lecce, Italy LN.EN., Sezione di Lecce, Italy (Received: 12 September 1994)

Abstract. Exponentially localized soliton solutions have been found recently for all the equations of the hierarchy related to the Zakharov-Shabat hyperbolic spectral problem in the plane. In particular the N2-soliton solution of the Davey-Stewartson I equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The interacting solitons can have, asymptotically, zero mass and can simulate quantum effects as inelastic scattering, fusion and fission, creation and annihilation. Mathematics Subject Classifications (1991): 35Q51, 35Q55, 58F07. Key words: bidimensional solitons, Davey-Stewartson equations.

It has recently been shown that localized solitons exist in two spatial and one temporal dimensions (2 + 1). More precisely, in 1988 Boiti, L6on, Martina and Pempinelli [1] discovered that all the equations in the hierarchy related to the Zakharov-Shabat (ZS) hyperbolic spectral problem in the plane have exponentially localized soliton solutions. The most representative equation in the hierarchy is the Davey-Stewartson I (DSI) equation, which provides a two dimensional generalization of the non-linear Schr6dinger equation. Moreover, as a relevant application of the spectral method proposed by Sabatier [2] and further developed by Boiti, Pempinelli and Sabatier [3], it has been stated that an additional nonlinear evolution equation, called that Davey-Stewartson III equation, also admits localized soliton solutions with properties similar to those of the DSI equation [4]. Localized soliton solutions for the DSI equation were first found by using gauge B/icklund transformations (BT) [1, 5]. Later these solutions were also obtained by means of the inverse spectral transform [6-8] and direct methods [9, 10]. These coherent structures display a richer phenomenology than the one dimensional solitons. Different effects have been reported successively, due to the fact that the soliton solution is structurally unstable with respect to special choices of * Work supported in part by M.U.R.S.T.

446

F. PEMPINELLI

the parameters [7]. Some quantum-like effects, such as the nonconservation of the number of solitons, have also been discovered. In particular there exist soiltons with zero mass at large time (virtual solitons), which interact with the other visible solitons with nonzero mass (real solitons). This special effect is possible because, although the total mass of solitons is conserved, the mass of the single soliton, in general, is not preserved by the interaction. Solitons can simulate inelastic scattering processes of quantum particles as creation and annihilation, fusion and fission, and interaction with virtual particles. An extended review of multidimensional localized solitons and a large reference list are given in [11]. The solitons we are considered are solutions of the so-called Davey-Stewartson I (DSI) equation, which can be written in characteristic coordinates u = x + y and v --- x - y as follows

iQt + ~r3(Q~ + Qvv)+

[A, Q] = 0

(1)

where Q is a 2 + 2 off diagonal matrix field

( 0 q(u,v,t)) Q= r(u,v,t) 0

(2)

and the diagonal 2 • 2 matrix field A, called the auxiliary field, is chosen to have arbitrary boundary values al and a2 according to the formula

A (-89176176

0 g1 ~Voo _ dv,(Q2)u +

0

=

) a2(u,t) "

(3)

Of interest is the reduced case r = r (the overbar means complex conjugation and ~2 = 1), describing physical situations as in hydrodynamics or in plasma physics. Originally it was proposed as a model of the evolution of shallow water waves that are weakly nonlinear and nearly monochromatic, when the effects of the surface tension are important. The field q is the complex envelope of the free surface wave and the auxiliary field a3A is the real velocity for the mean motion generated by the surface wave. However, more generally, the DSI equation is obtained as a universal multiscale limit of nonlinear dispersive wave equations. The DSI equation can be obtained [3, 4] as the compatibility condition between two Lax operators T1 and T2 which commute in the 'weak' sense [12] Tl~b = 0,

[T1, T2]r = 0.

(4)

T1 is the ZS hyperbolic operator in the plane

0 &

+Q

r

(5)

447

NEW FEATURES OF SOLITON DYNAMICS

and T2 has the form 2T2r =

lot + a2v - O2 + A +

r~

0

~

= --]g2~O" 3.

(6)

Let us remember that the boundary conditions of the auxiliary field A can be arbitrary chosen. The DSI equation in his standard version, i.e. with the boundary written as in (3), admits localized soliton solutions [1]. The one-soliton solution has the form 2 q = - -~ )ki~]e iO,

2

r = -- -~ # i p e -i~

(7)

where: D = 27(cosh~1 + cosh~2) + e ~z

(8)

~1 = - # x u - ,~iv + 2()~R~I + # R # I ) t

(9)

~2 -'~ I-tI u - - /~I v "~

0 = .Ru

2(/~R/~1 -- # R # I ) t

+ ),Rv +

-

1

+ .2 _

(10) (11) (12)

The complex parameters • ) k R q - i ) ~ i , # = #R+it~ I are the discrete eigenvalues of the associated ZS spectral problem and p, ~7 are arbitrary complex constants satisfying the conditions 7 C It~ and ~/(1 + 3') > 0. The elastic scattering of two solitons is shown in Figure 1, where Iql of the first soliton, in its reference frame, is represented during the interaction with the second (outside) soliton. The only memory of the interaction is a bidimensional shift of the position. These results were first obtained by using Backlund gauge transformations, but can be reobtained by defining a new spectral transform (ST). Our choice, in analogy with the 1 + 1 dimensional case, is that it must satisfy the following requirements: =

1. the discrete part of the spectrum corresponds to solitons and the continuous part to the radiation; 2. the time evolution can be explicitly integrated. Alternative STs with different requirements are allowed in multidimensional case. The boundaries al and a2 can be considered as potentials in two different time-dependent Schr6dinger equations (iOt q- 0 2 -q- al)qS1 ----0

(13)

448

F. PEMPINELLI

d. t = -12

t = -10

t=-6

t=O

t=4

I=6

t=lO

t=12

Fig. 1. The time sequence of ]qll of one soliton in its reference frame describes the interaction with another outside soliton. The soliton, during the interaction, for t < 0 generates a new bump, which grows and the old bump decreases. At t = 0 there is only the new bump. For t > 0 again a new bump is generated and it grows and the previous bump decreases. At the end of the interaction there is only one bump of the same form of the original bump, but shifted in its position.

449

NEW FEATURES OF SOLITON DYNAMICS

(io~ - o~ + a2)r

(14)

= O.

The complex spectral parameter k is introduced by requiring that the eigenfunctions ~b and r satisfy, as k ---+oe, the asymptotic properties Cexp[-ik(cr3x-y)]

(1) (1) (1)

=1+O

~

(15)

r exp [ - ikv -k- ik2t] = 1 + 0

-~

(16)

r exp [iku - ik2t] = 1 + (.9 ~ .

(17)

The ST is defined as the measure of the departure from analyticity of the eigenfunctions 0r N

=

ff dea dgr [[deAdgr JJ

0r Ok

g)

(18) i=

1,2.

(19)

The integral equation for r is of Volterra type and therefore the singularities of r in the complex plane are those of the homogeneous term and of the sectionally holomorphic Green function defining the integral equation. Consequently, r has simple poles only if the homogeneous term, which is related to the eigenfunctions of the time dependent Schr6dinger equations, has simple poles. We deduce that solutions Q of the DSI equation have a discrete ST only for boundaries ai which are wave-solitons of the Kadomtsev-Petviashvili I equation. We therefore need to consider the most general discrete spectral transforms of al and a2 rl (k, g) = V" r n(1) exp [ - i(k 2 - g2)t] 5(g - Am)5(k - An) m

(20)

n~?Tt

~ ( k , e) = ~

~(~) n m exp [~(k ~ - e~)t] e(t - p m ) 6 ( k --

..).

(21)

n~TTI,

They are furnishing real a 1 and a2 if the constant matrices r (i) are Hermitian. It is convenient to parametrize these matrices as follows r(~) = 27ri exp [ - i A m V o m

+ i)~nVon]Dnm~mm

?~(2)nm 27ri exp [i#mUom - i#nuon] Cnm~m~. =

(22) (23)

The complex parameters An and #n are the discrete eigenvalues, the real parameters v ~ and u ~ fix the position of the nth wave soliton in the corresponding

450

F. PEMPINELLI

plane, Amn = Am -- An, ~mn = ~m - #n and the two matrices D ~ , Gn~ satisfy D2nn = C2n = 1,

D n m ~ m m = Dmn)~nn,

F o r definiteness w e c h o o s e D n n

C n m ~ m m -~ -Cmn~nn.(24)

-= Cnn = 1, /~nI < O, # n I

> 0 and the

same number N of As and #s. By solving the O-equation one gets the r and, consequently, the following boundaries al ---- 202 In

det B,

a2 =

-202 In det.A

(25)

where B = 1 + Dr,

flnm = -Ann Anm

exp [/3n + tim]

(26)

A = I + Ca,

exp[6n + am] a n m - /2nn _

(27)

#nm

with fin -~ - - i ~ n ( V -- Yon) ~-

lASt,

Cgn -~ i~n(U -- Uon ) -- i~2n t.

(28)

By solving the 0-equation (18) one gets the corresponding eigenfunction and, then, by expanding the obtained r in powers of 1/k in the two spectral equations (5), (6), one derives the following explicit algebraic formulae for Q and A 0 f ~ ) exp[an] ) 0 f~(l) exp[f,~]

Q -- 2io'3

Q2 = -40~Ov In A

A__ 2 ( ~ 0 -02 0)

(29)

(30) In A

(31)

where (32)

+

1

1

(33) (34)

with 7n = ~ m

Cnm#mm eXp am,

(35)

NEW FEATURES OF SOLITON DYNAMICS

451

and p and r/ are arbitrary complex constant matrices. One can prove that the reduced case r = eq,

e 2 = 1,

(36)

the one we consider in the following, is obtained if and only if

PnmCmsfZss = e ~ DnmAmm~/s,~. m

(37)

m

In the so-called focusing case e = - 1 one can prove that if the Hermitian matrices iCnmftram and -iDnraAmm have positive eigenvalues then the N 2soliton solution and the auxiliary field A are regular. One can show that the solution q describes N 2 localized coherent structures interacting in a complicated way at finite times, but moving in the far past and in the far future with constant velocities Vin = (2AIR, 2#nR) (n, i = 1,2,..., N). It is therefore natural to call this solution the N2-soliton solution. One can also show that this N2-soliton solution is structurally unstable at t = - o c and at t = + o o when any couple of discrete eigenvalues An, Am or #n, #m have the same real part or are equal [8]. If both coefficients Pin and r/hi are equal to zero the corresponding soliton moving with velocity Vin = (2)kiR, 2#nR) at large times disappears. In particular, if the matrices p and rj are diagonal and C = D = 1 one recovers the N-soliton solution described in [7], where the explicit expressions of q and A are given. Let us examine more in detail the N-soliton solution in the cases when is structurally unstable at t = - o o and at t = +co. In the first case (AiR = AiR or PiR = #jR) the solitons after the interaction: 1. have a bidimensional shift; 2. change their form but do not change their mass. Figure 2 shows the evolution of ]q] in the two soliton case. Note the form of these new localized solitons which have two bumps. In the second case (Ai = Aj or #i = #j) the solitons after the interaction: 1. have a bidimensional shift; 2. change their form; 3. change their mass but the total mass is conserved. Figure 3 shows the evolution of ]ql in the two soliton case. Moreover, in both cases, the form and the mass of solitons depend on the initial position u0n, v0n of every soliton. A relevant information in the global dynamical behaviour of the solitons is furnished by the mass (energy, charge or number of particles according to the physical context) of the solution q M -- H JJ

Iql a dudv

(38)

452

F. PEMPINELLI

t--6e

l--Z

PJ

+P

l

@

I

-U

Ii

,.IJ

..+,

•8

"~-l' "","" :IJ'"P'"

.14

+~.'~- "2.-'" :I$'" "-'--" :%" """:" "~,~'""

;~" "" ":" +','+"+ ,,

I--48

m

rm

++

I i u

l o ou

I

.i

.... ~i..~I....i...Ii_.~ --IJ .~ .U I

-w .

--:W-+.;---:U+~--{----6----I----,% --, ,.

~ I A A ~ . J . . ~ A I .... YJ I Wl II

1o11

A-o24 ,i

i

,0

la

u

d

4

-'+.~'" :~i'" ".',"" -LL" "" :"" :'i" "" ~"" ",i'"

-u

1~

l

I II

,)

I -al

4

~m .N

Fig. 2. T h e case #IR = ]~2R ()`1 = - 0 . 1 - i, J~l -~- i, /91 ~-- 2.004, >,2 = - 1 . 0 1 i , #2 = 1.3i, p2 = 2.108). The time sequence o f the level contours o f Iql is described for two soliton solution in the frame o f reference of the soliton 1.

NEW FEATURES

OF SOLITON

I--II

@

i

453

DYNAMICS

m

I~

+++

i

+J

t...,+js + . . . , h ~ . L t ~ i u . % . . . +;..- ,,,,...+

-u

+.

t-B|

I--II

|

.i

*N

P'-~ r--~'-'~. I t ~ I ' - ~ ? ~ " I - -+",'+....

I--f4

t-$6 Bm

|

I

II

KI

I

U

~8

•8

=•_

"~-t|"" "-~"" :11"~ t " " ~ " "

I " ~ ' ? t "~

m

I--4

P~

i

I

+LI

~FJ

,.t+

~m

~N

F i g . 3. The time sequence )~2 = - 1 . 0 1 i , p 2 =-- 2 . 1 0 8 ) .

'~:r i" " W ' " ~t l ' ~ ' t ~ " 5 ~ ' " I ~ ' ~

i n t h e c a s e ]~1 = i/~2 ( ) ~ t = - 0 . 1

-i,

#1 = / ~ 2

"~

= i, p l = 2 . 0 9 4 ,

454

F. PEMPINELLI i0

10

7.5'

[ = -2

5' 2.5' 0' -2.5 -5

t=O

7. ~.

<>

2.

51 O,

-2.5 -5 -7,5

-7.5

-10

-I0 -8

-6

-4

-2

0

2

4

6

8

7.

-8

-6

-4

-2

0

2

4

6

-6

-4

-2

0

2

4

6

8

7.

2:t 21 -8

2:t 2:I -6

-4

-2

0

2

4

6

8

-8

Fig. 4. The fission of a soliton flashed at four different times. The contour levels are drawn for Iql ranging between 0 and 0.1. The soliton parameters are Al = --1, A2 = 0.5 -- 1.5< (0.1 0.1)

]-b2 = i, C = D = 1, UOl = u 0 2 = VOl = v02 = 0 a n d p = 0.2 0.3 ,q = the time is inverted, the sequence s h o w s the fusion of two solitons. I-Q =

/0.1 4/30] t o.1 0.2 i -

If

and by the masses of the solitons at t = •

M}ni) = ]]" qin (+) 12du dv

(i = l, 2, ..., N)

(39)

-(+) is the asymptotic behaviour of q at t = + e c computed in the rest where ~i~ reference frame o f the (i, n) soliton. In general, it results that

M = ~ in

M(n -) = z_..,S-"M (+)in9

(40)

in

However, the mass o f the single soliton is not conserved and in particular it can be zero at t -- + c ~ or at t = - c ~ . For a special choice of the parameters the mass of the (i, n) soliton can be zero at t = 4-oo also when the coefficients Pin and r/~i are not both equal to zero. We call these solitons with zero mass virtual solitons and they generate peculiar effects.

NEW FEATURES OF SOLITON DYNAMICS

455

For different choices of the parameters the four soliton solution can describe the creation or the annihilation of a soliton, the creation or the annihilation of a pair of solitons, two interacting solitons and inelastic scattering of solitons. All these dynamical behaviours can be obtained by choosing the same boundaries. Boundaries do not give any information on the dynamics of solitons, but fix only their asymptotic kinematics, i.e. their possible locations in the plane and their velocities at large times. Of special interest is the case in which two spectral data are equal, say #1 = #2. In general, this solution describes the interaction of two solitons and the masses M } f ) (i = 1,2) in this degenerate case depend also on the initial position of the solitons. With a special choice of the parameters one can get a solution describing the fission of a soliton and the fusion of two solitons (see Figure 4). If in addition one chooses det(pr/) = 0 the solution describes a single soliton that by the interaction with a virtual soliton is forced to change velocity. These two dynamical processes can be obtained by choosing the same boundaries and different matrices p and 77. References 1. Boiti, M., Lron, J., Martina, L., and Pempinelli, E: Phys. Lett. A 132 (1988), 432. 2. Sabatier, E C.: Inverse Problems 8 (1992), 263; Sabatier, E C.: Phys. Lett. A 161 (1992), 345. 3. Boiti, M., Pempinelli, E, and Sabatier, E C.: Inverse Problems 9 (1993), I. 4. Pempinelli, E: Localized soliton solutions for Davey-Stewartson I and Davey-Stewartson III equations, in: Clarkson, E A. (Ed.) Application of Analytic and Geometric Methods to Nonlinear Differential Equations, NATO ASI Series C 413, Kluwer Acad. Publ., Dordrecht, 1993, pp. 207-215. 5. Boiti, M., Lron, J., and Pempinelli, E: J. Math. Phys. 31 (1990), 2612. 6. Fokas, A. S. and Santini, E M.: Phys. Rev. Lett. 63 (1989), 1329; Fokas, A. S. and Santini, E M.: Physica D 44 (1990), 99. 7. Boiti, M., Lron, J., and Pempinelli, E: Inverse Problems 6 (1990), 715. 8. Boiti, M., Martina, L., Pashaev, O. K., and Pempinelli, F.: Phys. Lett. A 160 (1991), 55. 9. Hietarinta, J. and Hirota, R.: Phys. Lett. A 145 (1990), 237. 10. Hemandez Heredero R., Martinez Alonso L., and Medina Reus E.: Phys. Lett. A 152 (1991), 37. 11. Boiti, M., Martina, M., and Pempinelli, E: Multidimensional localized solitons, Chaos, Solitons and Fractals 4 (1994). 12. Boiti, M., Lron, J., Manna, M., and Pempinelli, E: Inverse Problems 2 (1986), 271; Boiti, M., Lron, J., Manna, M., and Pempinelli, E: Inverse Problems 3 (1987), 25.