Chern–Simons solitons in quantum potential

Chaos, Solitons and Fractals 11 (2000) 2193±2202

www.elsevier.nl/locate/chaos

Chern±Simons solitons in quantum potential O.K. Pashaev a,b,*, Jyh-Hao Lee b,1 a b

Joint Institute for Nuclear Research, 141980 Dubna (Moscow), Russian Federation Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan, ROC Accepted 28 July 1999

Abstract The self-dual Chern±Simons solitons under the in¯uence of the quantum potential are considered. The single-valuedness condition for an arbitrary integer number N P 0 of solitons leads to quantization of Chern±Simons coupling constant j ˆ m…e2 =g†, and the integer strength of quantum potential s ˆ 1 ÿ m2 . As we show, the Jackiw±Pi model corresponds to the ®rst member …m ˆ 1† of our hierarchy of the Chern±Simons gauged nonlinear Schr odinger models, admitting self-dual solitons. New types of exponentially localized Chern±Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found. Ó 2000 Published by Elsevier Science Ltd.

1. Introduction The Chern±Simons theory attracted much attention recently as a gauge theory in 2 ‡ 1 dimensions [1], describing non-local interaction between matter particles, a€ecting the phase of the wave function, and leading to the fractional statistics phenomena [2]. In this connection, the Chern±Simons coupling constant plays the role of a statistical parameter and relates to the spin of particles. At the classical non-relativistic level, with the Chern±Simons coupling strength determined by overall strength of a quartic scalar potential, it leads to the existence of the self-dual Chern±Simons solitons [3,4]. They are static solutions with ®nite charge and ¯ux of exactly solvable self-dual equations [5]. Unfortunately, the full dynamics of Chern± Simons solitons according to the nonlinear Schr odinger equation is still in¯exible due to the lack of exact integrability, even without external forces, although it may be realized in the framework of the Davey± Stewartson-II equation [6]. In the present paper we show that the ``external'' force produced by the so-called ``quantum potential'' U …x† ˆ …ÿh2 =2m†Djwj=jwj, leads to an additional statistical transmutation of Chern±Simons solitons. The quantum potential, introduced by de Broglie [7] and explored by Bohm [8] does not depend on the strength of the wave but only on its form, and therefore its e€ect could be large even at long distances. Then, it satis®es the homogeneity property [9], this is the reason why it appears in attempts of a nonlinear extension of the quantum mechanics [10±16]. An pproach has been developed for the stochastic formulation of quantum mechanics, where the quantum ¯uctuations are represented by superimposing a classical trajectory and an additional random motion generated by quantum potential [12]. The nonlinear extension of the Schr odinger equation with the quantum potential non-linearity has been considered in connection with several problems [13]: (a) in allowing formally the di€usion coecient of the stochastic process in a

*

Corresponding author. E-mail addresses: [email protected], [email protected] (O.K. Pashaev), [email protected] (J.-H. Lee). 1 Fax: 886-2-27827432 0960-0779/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 3 9 - 3

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stochastic quantization to di€er  h=2m, related to the di€erence in the Plank constant [14] or the inertial mass [15], (b) in corrections from quantum gravity [16]. As was shown by Auberson and Sabatier [17,18], depending on intensity of the quantum potential it can be linearized in the form of the Schr odinger equation with rescaled potential or as the pair of time-reversed di€usion equations, and in both cases does not admit 2 soliton solutions. Recently we found that for self-consistent potential U ˆ gjwj in 1 ‡ 1 dimensions the theory has soliton solutions with rich resonance dynamics [19]. In the present paper we ®nd exactly soluble case for 2 ‡ 1 dimensional nonlinear Schr odinger (NLS) model interacting with Chern±Simons ®eld under the in¯uence of the quantum potential. In the above mentioned interpretation of quantum mechanics, the quantum particle moves under the action of a force which along with classical potential includes a contribution from the quantum potential. If instead of classical particle we consider Chern±Simons solitons, then subject to the in¯uence of intensity s quantum potential, it could represent the stochastically quantized anyons. In this case we can expect quantization condition on the Chern±Simons coupling constant, an e€ect on the anyonic parameter and the appearance of the zero point ¯uctuation for the statistical ¯ux. 2. Chern±Simons solitons hierarchy We consider the Chern±Simons gauged Nonlinear Schr odinger model (the Jackiw±Pi model) with nonlinear quantum potential term of strength s: Lˆ

j lmk i  4     Al om Ak ‡ …wD 0 w ÿ wD0 w† ÿ DwDw ‡ s$jwj$jwj ‡ gjwj ; 2 2

…1†

where Dl ˆ ol ‡ ieAl …l ˆ 0; 1; 2†. Classical equations of motion are 2

iD0 w ‡ D2 w ‡ 2gjwj w ˆ s

Djwj w; jwj

…2a†

e  o1 A2 ÿ o2 A1 ˆ ww; j

…2b†

e   k w ÿ wD  k w† o0 Aj ÿ oj A0 ˆ ÿ ijk …wD j

…j; k ˆ 1; 2†:

…2c†

As we mentioned in Section 1 the extension of the linear Schr odinger equation with s 6ˆ 0 term has been considered before [13±16], and was linearized for any s [18]. But, when one adds the nonlinear self-interaction and the gauge ®eld, integrability of the model is allowed only under some restrictions on the coupling parameters. For s < 1, and this is only the case where we ®nd the self-duality condition, we decompose w ˆ eRÿiS and introduce new rescaled variables t ˆ …1 ÿ s†ÿ1=2~t;

~ S ˆ …1 ÿ s†1=2 S;

A0 ˆ …1 ÿ s†A~0 ;

A ˆ …1 ÿ s†

1=2 ~

A:

…3†

~ we get the gauged NLS system (the Jackiw±Pi model) Then for new function w~ ˆ exp…R ÿ iS†, ~ 2 w~ ˆ 0; ~ 2 w~ ‡ 2g jwj ~ ~0 w~ ‡ D iD 1ÿs o1 A~2 ÿ o2 A~1 ˆ

e j…1 ÿ s†

o~0 A~j ÿ oj A~0 ˆ ÿ

1=2

~ w~w;

e 1=2

j…1 ÿ s†

 ~ ~ ~ ~ ~ ijk …w~D …j; k ˆ 1; 2† k w ÿ wDk w†

…4a† …4b†

…4c†

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but with new coupling constants j~ ˆ j…1 ÿ s†1=2 ; g~ ˆ g=…1 ÿ s†. Since invariance of the density q ˆ ~w, ~ the Gauss law (4b) modi®es the value of magnetic ¯ux, which depends now of s. Only when the  ˆw ww coupling constants are restricted by the condition gj 1=2 ˆ …1 ÿ s† ; e2

…5†

static solutions of the system (4a)±(4c) with the gauge potential A~0 ˆ

e j…1 ÿ s†1=2

 ww

require the self-dual Chern±Simons equations, ~ ÿ w~ ˆ 0; D

…6a†

o1 A~2 ÿ o2 A~1 ˆ

e j…1 ÿ s†

1=2

~w: ~ w

…6b†

Condition (5) extends for s 6ˆ 0, the well-known Jackiw±Pi constraint. Namely, self-duality equations in the presence of quantum potential survive only if its strength is restricted by s ˆ 1 ÿ g2 j2 =e4 . To solve the system (6a) and (6b) we insert A from the ®rst equation to the second one and will get for  the Liouville equation the density q ˆ ww D ln q ˆ ÿ2

e2 j…1 ÿ s†1=2

q:

…7†

The radially symmetric solutions, which have been constructed by Walker [20], and discussed in [3,4], 1=2

j…1 ÿ s† N 2 q…r† ˆ 4 e2 r 2

"

r r0

N ‡

 r N 0

r

#ÿ2 ;

…8†

would be regular if N P 1. Then, like in [3,4], from regularity of the gauge potential A we can ®x the phase ~ ~ as S~ ˆ …N ÿ 1†h; h ˆ arctan x2 =x1 , and restrict N to be an integer for single-valued w. of w~ ˆ exp…R ÿ iS† ~ However, the auxiliary function w is not the physical one, this is why ®xing an integer N is not sucient for single valuedness of the original function w ˆ exp…R ÿ i…1 ÿ s†1=2 …1 ÿ N †h†. It would be now that the phase is restricted as 0 < S 6 2p…1 ÿ s†1=2 , and in general we have the problem with angular defect in the plane 1=2 2p…1 ÿ …1 ÿ s† †, describing a cone . It is easy to avoid this complication and integer valued must be the product …1 ÿ s†

1=2

…N ÿ 1† ˆ n;

…9†

valuedness of which for any integer N, requires an integer valuedness of …1 ÿ s†

1=2

ˆ m;

…10†

and as a consequence of (5), we obtain the quantization condition gj2 ˆm e2

…m ˆ 1; 2; 3; . . .†:

…11†

The last one means that in the presence of quartic self-interaction, the Chern±Simons coupling constant and the quantum potential strength must be quantized jˆm

e2 ; g

s ˆ 1 ÿ m2 :

…12†

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When m ˆ 1, the quantum potential vanishes, s ˆ 0, while the ®rst constraint in (12) reduces to the Jackiw± Pi self-dual condition. It is worth noting that in our relation (12) g and j coupling constants play a role similar to ones in the electro-magnetic duality condition derived by Dirac. The self-dual system (6a) and (6b) for original …w; A† has the form, depending on the quantum potential with a coecient proportional to the angular defect Dÿ w ‡ ……1 ÿ s†

1=2

ÿ 1†

oÿ jwj w ˆ 0; jwj

…13a†

e  o1 A2 ÿ o2 A1 ˆ ww j

…13b†

and turn the energy Z   4   wDw ÿ s$jwj$jwj ÿ gjwj ; H ˆ d2 r D

…14†

to vanish H ˆ 0. It shows that under the in¯uence of the quantum potential Chern±Simons solitons continue to be a zero energy con®guration. Due to (9) and (10), the ¯ux for solution (8) is quantized: Z 2p 2p 1=2 2N …1 ÿ s† ˆ …2m†N ; …15† U ˆ d2 rB ˆ e e where N ˆ 1; 2; 3; . . . and value of m is ®xed. Thus, the magnetic ¯ux of our vortex/soliton is an even multiple m of the elementary ¯ux quantum, which generalizes the Jackiw±Pi result related to the particular value m ˆ 1. Here it is worth noting that the single-valuednes condition (9) may be satis®ed also for particular real values for s and N. Thus, if …1 ÿ s†1=2 ˆ p=q is the rational number, then single-valuedness of w is allowed only for the speci®c sequence of integers N ˆ 1; 1 ‡ q; 1 ‡ 2q; . . . or N ˆ 1 ‡ ql, where l ˆ 0; 1; 2 . . . In this case the ¯ux is quantized as   2p 1 …2p† l ‡ ; …16† Uˆ e q with 1=q playing the role of the zero-point ¯ux. For an irrational value …1 ÿ s† Uˆ

2p 2…n ‡ a† e

1=2

ˆ a the ¯ux is quantized

…n ˆ 0; 1; 2; . . .†;

…17†

only for the irrational sequence N ˆ 1 ‡ n=a. The decomposition w ˆ exp…R ÿ iS† is known in quantum mechanics as the Madelung ¯uid representation and has been explored for description of superconductivity [21]. In problem (2a)±(2c) the velocity of associated Madelung ¯uid is de®nite as v ˆ ÿ2…$S ÿ eA† and satis®es the conservation law ot q ‡ $qv ˆ 0:

…18†

For the self-dual ¯ows, from Eq. (6a) we ®nd the canonical Hamiltonian equations x_  v1 ˆ

ov ; oy

y_  v2 ˆ ÿ

ov ; ox

…19†

where the stream function v, playing the role of a Hamiltonian, has the form v ˆ …1 ÿ s†

1=2

ln q:

…20†

For the soliton solution (8), function q is regular everywhere and has 2…N ÿ 1†th order zero at the beginning of coordinates. This zero determines singularity of v and v, such that near the singularity v ˆ 2…N ÿ 1†…1 ÿ s†

1=2

ln r  ÿ

b ln r; 2p

…21†

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and the ¯ow corresponds to the line vortex, the strength of which according to (9) must be quantized b ˆ ÿ4pn ˆ 4pm…1 ÿ N †. The velocity ®eld near the center ÿ2n  y x  ÿ ; ; …22† vˆ r r r is the gradient of multivalued function / ˆ ÿ2n arctan…y=x†, which ®xes the value of the phase of the wave function w. As is well-known, functions v and / are conjugate harmonic functions, providing holomorphicity condition for the gauge potential near the center. The above consideration shows that Chern± Simons soliton can be interpreted as a planar vortex in the Madelung ¯uid, having the form of a linear vortex near the rotation point. This interpretation allows us to give the physical meaning for the vortex ¯ow in the Madelung quantum liquid and answer the question posed in [22]. 3. Exponentially localized Chern±Simons solitons In the previous section we considered self-duality reduction for s < 1. However, no solution for s > 1 has been found. Now we show that in a special case of hyperbolic energy surface in the dynamics of Bloch electrons under the in¯uence of Chern±Simons and quantum potentials, the problem admits an exact treatment also for s > 1. In the dynamics of Bloch electron in a solid a central role is played by the inverse e€ective mass tensor 1=m ! o2 E…k†=ok2 , which is not necessary positive-de®nite. When the Fermi surface is near the band boundary, the sign of the mass in orthogonal direction is negative. In this hyperbolic case, due to strong Bragg re¯ection from the boundary of the band, the electron propagates along trajectories which are parallel to the planes of the lattice, and E…k† has ``saddle points'', where the curvature of the surface may be positive in one direction and negative in another. For two-dimensional motion, in simplest case of the energy surface E…k† ˆ k12 =2m ÿ k22 =2m, the e€ective mass matrix is M ÿ1 ˆ diag…1=m; ÿ1=m† and the Bloch electron subject to the Lorentz force, imitated by Chern±Simons interaction, has a ``dissipative'' character [23], with the constant magnetic ®eld playing the role of the damping. Then, in Lagrangian (1) and equations of motion (2a)±(2c) we replace the positive space metric gij ˆ diag…1; 1† with the inde®nite one gij ˆ diag…1; ÿ1† as follows: j i  4 a   a   L ˆ lmk Al om Ak ‡ …wD 0 w ÿ wD0 w† ÿ Da wD w ‡ soa jwjo jwj ‡ gjwj ; 2 2 iD0 w ‡ Da Da w ‡ 2gjwj2 w ˆ s

…23†

oa oa jwj w; jwj

…24†

where oa oa ˆ gab oa ob ˆ o21 ÿ o22 . Now we rescale t ˆ …s ÿ 1†ÿ1=2~t;

~ S ˆ …s ÿ 1†1=2 S;

A0 ˆ …s ÿ 1†A~0 ;

A ˆ …s ÿ 1†

1=2 ~

A;

…25†

~ satisfying the system and instead of w ˆ exp…R ÿ iS†, introduce two real functions Q ˆ exp…R  S†, 

2g ‡ ÿ ‡ ‡ ~ ÿ ~ aÿ ‡ ~ÿ Q Q Q ˆ 0; ÿD ~0 Q ‡ Da D Q ÿ sÿ1

…26a†

2g ‡ ÿ ÿ ÿ ~ ‡ ~ a‡ ÿ ~‡ Q Q Q ˆ 0; D ~0 Q ‡ Da D Q ÿ sÿ1

…26b†

o1 A~2 ÿ o2 A~1 ˆ o~0 A~j ÿ oj A~0 ˆ

e j…s ÿ 1†1=2 e

Q‡ Qÿ ;

 g …Q 1=2 jk kk

j…s ÿ 1†

…26c† ‡

~ ‡ Qÿ ÿ Qÿ D ~ ÿ Q‡ † D k k

…j; k ˆ 1; 2†;

…26d†

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where D l ˆ ol  eAl . This system represents the gauged version of the pair time-reversal invariant di€usion equations with the reaction term proportional to Q‡ Qÿ . It is worth noting that switching on the Chern± Simons term leads to switching o€ the time-reversal invariance without changing the geometry of the classical trajectories. Moreover, instead of the local phase transformations for the original model (23), in the system (26a)±(26d) the Chern±Simons gauge ®eld corresponds to the local rescaling of Q . For static 1=2 con®gurations, when A~0 ˆ ÿ…e=j…s ÿ 1† †Q‡ Qÿ , and …s ÿ 1†1=2 ˆ ÿ

gj ; e2

…27†

we have the self-duality equations ~ ÿ Q‡ ˆ 0; D ÿ ~‡Š ˆ ÿ ~‡; D ‰D ‡ ÿ

~ ‡ Qÿ ˆ 0; D ‡ 2e2 j…s ÿ 1†

1=2

…28a†

~ÿ; D ~ ÿ Š; Q‡ Qÿ ˆ ÿ‰D ‡ ÿ

…28b†

with Da ˆ Da1  Da2 . Expressing A~ ˆ …1=e†o ln Q and substituting to (28b) for q ˆ Q‡ Qÿ we obtain the Liouville equation …o21 ÿ o22 † ln q ˆ 2

e2 1=2

j…s ÿ 1†

q:

…29†

As is well-known this equation admits the general solution qˆ

8 A0 …x‡ †B0 …xÿ † ; a0 ‰A…x‡ † ‡ B…xÿ †Š2

…30†

written in terms of two arbitrary functions A…x‡ † and B…xÿ † of x‡ ˆ x1 ‡ x2 and xÿ ˆ x1 ÿ x2 respectively, where a0 ˆ 2e2 =j…s ÿ 1†1=2 . It has been considered before as 1 ‡ 1 dimensional evolution equation with x coordinate considered as a time variable. However, the known regular soliton solutions [24,25] decay in all directions except the soliton world line, which leads to the divergent integral for q. This is why they have no physical meaning in our problem. Instead, we choose A and B functions in the form A ˆ …a ‡ 1† coth2k‡1

a x‡ ; 2

B ˆ …a ÿ 1† tanh2l‡1

b xÿ ; 2

…31†

and we get qˆ

21ÿ2…k‡l† …1 ÿ a2 †…2k ‡ 1†…2l ‡ 1†ab sinh2k ax‡ sinh2l bxÿ a0 ‰…a ‡ 1† cosh2k‡1 12ax‡ cosh2l‡1 12bxÿ ‡ …a ÿ 1† sinh2k‡1 12ax‡ sinh2l‡1 12bxÿ Š

2

:

For parameter a > 0 this solution is nonsingular in the whole …x1 ; x2 † plane. The sign de®niteness of q requires k; l to be integer, while regularity at the ``light-cone'', x‡ ˆ 0 or xÿ ˆ 0 is valid for k P 0, l P 0. In contrast to the algebraic Chern±Simons solitons (8), solution (31) is exponentially decreasing in the plane for the ``future'' and ``past'' null in®nities as eÿajxj or eÿbjxj , while for the ``time-like'' and ``space-like'' in®nities as eÿ…a‡b†jxj (here without loss of generality we put a > 0, b > 0). It is worth to note that our solutions are related with exponentially localized planar solitons …EL2 † for Davey±Stewartson-I (DS-I) equation [26] (also known as dromions [27]). These solutions have a rich integrable dynamics [28], and are reducible to the Liouville equation for jwj, and to the d'Alembert equation o‡ oÿ arg w ˆ 0 for arg w, like in the DS-II case [29]. This reduction provides an integrable dynamics for our exponentially localized Chern±Simons solitons [30]. Moreover, comparing to (8), having for N > 1 the zero at the beginning of coordinates, the 2l zeroes for solution (31) q  x2k ‡ xÿ are located along the ``light cone'' x‡ ˆ 0, xÿ ˆ 0 for k > 0 and l > 0 correspondingly. For k ˆ 0 and l ˆ 0, we have exponentially localized soliton located at the beginning of coordinates, without zeroes and the rotation invariance at this point (Fig. 1). If only one of the k; l values is

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Fig. 1. Contour and 3D plot of one soliton solution …k ˆ 0; l ˆ 0† for a ˆ 0.7 in …x; y† plane.

vanishing, then we have two soliton solution represented in Fig. 2 and symmetrical under one of the light cone directions. When both numbers do not vanish we get four-soliton solution, represented in Fig. 3, with zeroes along the whole light cone. For k > 1, l > 1, number of solitons remains four, however the order of zeroes on the light cone is changing. As is well known, zeroes of q, which produce singularities forln q, lead to singularities of the gauge potential [3,4]. In the case (8) these singularities, corresponding to the Aharonov±Bohm type potential, can be removed by ®xing the phase of the wave function at the singular point. So the phase becomes the angle variable on the plane with ®nite range of values. For solution (31), the singularities of potential A are located along the light cone and can be compensated by derivation of function S~ as

Fig. 2. Contour and 3D plot of two soliton solutions …k ˆ 1; l ˆ 0† for a ˆ 0.7 in …x; y† plane.

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Fig. 3. Contour and 3D plot of four soliton solutions …k ˆ 1; l ˆ 1† for a ˆ 0.7 in …x; y† plane.

o‡ S~ ˆ 2k

1 ; x‡

oÿ S~ ˆ ÿ2l

1 xÿ

…32†

or 1 x2k S~ ˆ ln ‡2l : 2 xÿ This allows us to de®ne exact solutions of Eqs. (26a)±(26d) and original system (24)   ip sÿ1 p xlÿ : wˆ q k x‡

…33†

…34†

For particular values k ˆ l, the phase S~ ˆ k ln j xx‡ÿ j de®nes the hyperbolic analog of the singular Aharonov± Bohm potential ai ˆ

2k xj k ij ˆ ij gjj oj ln r2 ; e x21 ÿ x22 e

…35†

where r2 ˆ x21 ÿ x22 in the ``time-like'' quadrants II …x1 > 0† and IV …x1 < 0† of the plane, while r2 ˆ x22 ÿ x21 in the ``space-like'' quadrants I …x2 > 0† and III …x2 < 0†, parameterized as x2 ˆ r cosh h, x1 ˆ r sinh h in I and III, and x1 ˆ r cosh h, x2 ˆ r sinh h in II and IV, correspondingly. Here 0 < r < 1, ÿ1 < h < ‡ 1. The potential (35) has singularities not only at the beginning of coordinates, as the Aharonov± Bohm potential, but also along the ``light-cone'', x‡ ˆ 0, xÿ ˆ 0 and may be presented as a singular pure gauge 1 2k ai ˆ ÿ oi S~ ˆ ÿ oi h; e e ~ where S ˆ 2kh, and ( tanhÿ1 xx12 in I and III; hˆ tanhÿ1 xx21 in II and IV:

…36†

…37†

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The magnetic ¯ux associated with soliton solutions (23) has the form Z Z e 2 q d2 x ˆ ln a; B~ d2 x ˆ 1=2 e j…s ÿ 1†

2201

…38†

independent of k and l, and is not quantized. Moreover, the phase of w includes the hyperbolic rotation angle h which is valued on the whole real line. So no restriction of single-valuednees arises, and continual parameters s and j must be restricted only by relation (27). At the end of this section we represent another problem related to our self-dual system (28a) and (28b). Namely, the hyperbolic self-duality equations (28a) and (28b) are equivalent to SO…2; 1†=O…1; 1† self-dual r model: o 1 s ÿ s ^ o2 s ˆ 0

…39†

written in the tangent space representation for moving frame r a0   Q s; Dl n ˆ 2 2 l  ol s ˆ

r  a0 ÿ ÿ n n Q‡ ÿ Q ; ÿ ‡ l l 2

…40a†

…40b†

ÿ to the one sheet hyperboloid s2 ˆ ÿs21 ‡ s22 ÿ s23 ˆ ÿ1 as the constraints Q‡ ÿ ˆ 0, Q‡ ˆ 0, with the following ‡ ÿ ÿ  Q , Q  Q . In terms of the stereographic projection of the hyperboloid identi®cation Q‡ ‡ ÿ

S ˆ

2n ; 1 ‡ n‡ nÿ

S3 ˆ

1 ÿ n‡ nÿ ; 1 ‡ n‡ nÿ

…41†

Eq. (39) are just the chirality conditions o‡ n‡ ˆ 0;

oÿ nÿ ˆ 0;

…42†

having the general solution n‡ ˆ n‡ …xÿ †, nÿ ˆ nÿ …x‡ †. These functions correspond to the general solution of the Liouville equation (30) with identi®cation nÿ ˆ Aÿ1 …x‡ †, n‡ ˆ B…xÿ †. Then, our vortex con®gurations (31) generate solution of (39), (42) with nÿ ˆ …a ‡ 1†

ÿ1

tanh2k‡1

a x‡ ; 2

n‡ ˆ …a ÿ 1† tanh2l‡1

b xÿ : 2

…43†

which for a > 0 are regular everywhere on the plane with S3 > 0. The last condition means that the solutions for a > 0 are non-topological. To have topologically non-trivial con®gurations we need to consider a < 0 case, which would produces singularity for q. For the regular solution s with k > 0 or (and) l > 0 the plateau along the light cone appears, leading to the zeroes for q in (31). 4. Conclusions In conclusions we stress that two long-range interactions considered in the present paper, the Chern± Simons gauge interaction introduced to physics by Deser et al. [1] and the quantum potential introduced by de Broglie [7] and Bohm [8], are compatible in supporting static soliton solutions in 2 ‡ 1 dimensions, with arbitrary N, only when the coupling constants for both interactions are quantized. A further remark concerns the special case s ˆ 1, when the Madelung hydrodynamical formulation of quantum mechanics becomes the Euler equation and the continuity one [18]. For a special form of the nonlinearity an additional higher symmetry of the equations, related to the membrane theory, has been described recently [34,35]. It would be interesting to extend this work to include a Chern±Simons gauge ®eld, in attempt producing completely integrable model with in®nite number of conservation laws.

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Finally, we note existence of another integrable reduction of models (1) and (23), when the ®elds are independent of one of the space directions. In this case for s < 1 the model reduces to the BF gauged NLS, equivalent to NLS [31], and for s > 1, to BF gauged reaction±di€usion analog of NLS, equivalent to the reaction±di€usion system. The last one appears in the Jackiw±Teitelboim gravity and admits dissipative analog of solitons, called dissipatons and related to the black holes of the model [32,33]. The interaction of dissipatons shows the resonance character [19]. Acknowledgements The authors are grateful to Roman Jackiw for his interest to this paper and useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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