Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation

In tegrable h ydro dynamics of Calogero-Sutherland mo del: Bidirectional Benjamin-Ono equation. Alexander G. Abano v Departmen t of Ph ysics and Astronomy , Ston y Bro ok Univ ersity , Ston y Bro ok, NY 11794-3800. Eldad Bettelheim Racah Institute of Ph ysics, The Hebrew Universit y of Jerusalem, Safra Campus, Giv at Ram, Jerusalem, Israel 91904. P aul Wiegmann James F ranck Institute of the Universit y of Chicago, 5640 S.Ellis Aven ue, Chicago, IL 60637. Abstract. W e dev elop a h ydro dynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland mo del with an infinite num b er of particles and a non-v anishing densit y of particles. The h ydro dynamic equations, b eing written for the density and v elo cit y fields of the liquid, are shown to be a bidirectional analogue of Benjamin- Ono equation. The latter is known to describ e internal wa ves of deep stratified fluids. W e show that the bidirectional Benjamin-Ono equation app ears as a real reduction of the mo dified KP hierarch y . W e derive the Chiral Non-linear Equation whic h app ears as a c hiral reduction of the bidirectional equation. The conv entional Benjamin-Ono equation is a degeneration of the Chiral Non-Linear Equation at large densit y . W e construct m ulti-phase solutions of the bidirectional Benjamin-Ono equations and of the Chiral Non-Linear equations. CONTENTS 2 Con tents 1 In tro duction 3 2 P articles as p oles of meromorphic functions 4 3 Hydro dynamics of Calogero-Sutherland liquid 6 3.1 Densit y and velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Hydro dynamic form of 2BO. . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Bidirectional Benjamin-Ono equation (2BO). . . . . . . . . . . . . . . . . 8 4 Hamiltonian form of 2BO 9 5 Bilinearization and relation to MKP1 equation 10 6 Chiral Fields and Chiral Reduction 12 6.1 Chiral fields and currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Chiral Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3 Holomorphic Chiral field . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.4 Benjamin-Ono Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 Multi-phase solution 15 7.1 Multi-phase and m ulti-soliton solutions of MKP1 . . . . . . . . . . . . . 15 7.2 Multi-phase solution of 2BO . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.2.1 Sc hw arz reflection condition . . . . . . . . . . . . . . . . . . . . . 16 7.2.2 Multi-soliton solution of 2BO . . . . . . . . . . . . . . . . . . . . 17 7.2.3 Analyticit y condition . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.3 Multi-phase solution of the Chiral Non-linear Equation . . . . . . . . . . 19 7.4 Multi-phase solution of the Benjamin-Ono equation . . . . . . . . . . . . 20 7.5 Mo ving P oles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8 Conclusion and discussion 21 9 Ac knowledgmen ts 22 App endix A Hilb ert transforms 23 App endix B Conserv ed integrals of 2BO 24 App endix C Geometrical in terpretation of the c hiral equation 24 App endix D Determinant identities 25 App endix E Non-degeneracy condition of matrix (107) 25 CONTENTS 3 1. In tro duction The Calogero-Sutherland mo del (CSM) [1, 2] describes particles moving on a circle and in teracting through an in v erse sin-square potential. The Hamiltonian of the model reads H C S M = 1 2 N X j =1 p 2 j + 1 2  π L  2 N X j,k =1; j 6 = k g 2 sin 2 π L ( x j − x k ) , (1) where x j are co ordinates of N particles, p j are their momenta, and g is the coupling constan t. W e took the mass of the particles to b e unit y . The momenta p j and co ordinates x j are canonically conjugate v ariables. The mo del (classical and quantum) o ccupies an exceptional place in physics and mathematics and has b een studied extensiv ely . It is completely in tegrable. Its solutions can b e written down explicitly as finite dimensional determinan ts (for review see [3]). In the limit of a large p erio d L → ∞ the CSM degenerates to its rational v ersion – Calogero (ak a Calogero-Moser) mo del (CM) where the pair-particle interaction is 1 /x 2 . ‡ The CSM itself is a degeneration of the elliptic Calogero model, where the pair particle interaction is giv en b y the W eierstrass ℘ -function of the distance. In this pap er w e discuss the classical trigonometric mo del (1) commen ting on the rational limit when appropriate. W e are in terested in describing a Calogero-Sutherland liquid , i.e., the system (1) in thermo dynamic limit when N → ∞ and L → ∞ while the a verage densit y N /L is k ept constan t. W e assume that the limit exists and that in this limit a microscopic densit y and current fields ρ ( x, t ) = N X j =1 δ ( x − x j ( t )) , (2) j ( x, t ) = N X j =1 p j ( t ) δ ( x − x j ( t )) (3) are smo oth single-v alued real p erio dic functions with a p erio d L equal to the p erio d of the p oten tial § . In this case the system will b e describ ed b y h ydro dynamic equations written on the densit y field ρ ( x, t ) and the v elo city field v ( x, t ). The velocity is defined as j = ρv . The hydrodynamic approac h is a p o w erful to ol to study the ev olution of smo oth features with t ypical size m uch larger than the inter-particle distance. Apart from application to the CSM, the hydrodynamic equations obtained in this pap er are in teresting integrable equations. W e show that they are new real reductions of the mo dified Kadomtzev-P etviashv ili equation (MKP1). ‡ In the rational case one usually adds a harmonic p otential, 1 2 ω 2 P i x 2 i , to the Hamiltonian to prev ent particles from escaping. This addition do es not destroy the integrabilit y of the system [2]. § It is likely that there are classes of solutions of the CSM, whose thermodynamic limit consists of a n um b er of in teracting liquids. In this case the microscopic density give rises to a num b er of functions in the contin uum - the densities of the distinct interacting liquids. In this pap er w e consider a class of solutions which leads to a single liquid. CONTENTS 4 In this pap er w e consider a classical system, how ev er the approach dev elop ed b elow can b e extended to the quan tum case { p j , x k } = δ j k → [ p j , x k ] = i ~ δ j k almost without c hanges. F or a brief description of the hydrodynamics of the quantum system see Ref. [4]. The h ydro dynamics of the quan tum Calogero mo del has b een studied previously [5, 6] in the framework of the c ol le ctive field the ory and some of the results b elo w can b e obtained in a classical limit (see [7]) of the quantum coun terparts of Refs. [5, 6]. The outline of this pap er is the follo wing. In Sec. 2 we parameterize the particles of CSM as p oles of auxiliary complex fields so that the motion of particles is enco ded b y evolution equations for fields. In Sec. 3 w e derive a h ydro dynamic limit of these equations - con tinuit y and Euler equations with a particular form of specific en thalpy . W e will refer to these equations as to the bidirectional Benjamin-Ono equation or 2BO. W e presen t the Hamiltonian form of 2BO in Sec. 4. I n Sec. 5 we discuss the bilinear form of 2BO and its relation to MKP1. In Sec. 6 we obtain the Chiral Non-Linear equation (CNL) - chiral reduction of 2BO and discuss some of its prop erties. In Sec. 7 w e construct multi-phase and m ulti-soliton solutions of 2BO and CNL as a real reduction of MKP1. These solutions corresp ond to collectiv e excitations of the original man y-b o dy system. Some technical p oin ts are relegated to the app endices. 2. P articles as p oles of meromorphic functions The Equations of motion of the CSM are readily obtained from the Hamiltonian (1) ˙ x j = p j , (4) ˙ p j = − g 2 ∂ ∂ x j N X k =1 ( k 6 = j )  π L cot π L ( x j − x k )  2 . (5) W e rewrite this system in an equiv alen t wa y as i ˙ w j w j = g 2  2 π L  2   N X k =1 w j + u k w j − u k − N X k =1 ( k 6 = j ) w j + w k w j − w k   , j = 1 , . . . , N (6) − i ˙ u j u j = g 2  2 π L  2   N X k =1 u j + w k u j − w k − N X k =1 ( k 6 = j ) u j + u k u j − u k   , j = 1 , . . . , N , (7) where w j ( t ) = e i 2 π L x j ( t ) are complex co ordinates lying on a unit circle, while u j ( t ) = e i 2 π L y j ( t ) are auxiliary co ordinates. Indeed, differen tiating (6) with respect to time and using (6,7) to remov e first deriv atives in time one obtains equations equiv alent to (4,5). W e note that while the co ordinates x j are real, i.e., | w j | = 1, the auxiliary co ordinates, y j ( t ), are necessarily complex. Giv en initial data as real p ositions and v elo cities x j (0) and ˙ x j (0) one can find complex y j from (6) and then initial complex v elo cities ˙ y j (0) from (7). Once x j and ˙ x j are chosen to b e real they will stay real at later times, ev en though co ordinates y i are moving in a complex plane. CONTENTS 5 The co ordinates w j ( t ) and u j ( t ) determine an evolution of tw o functions u 1 ( w ) = g π L N X j =1 w + w j w − w j = − ig N X j =1 π L cot π L ( x − x j ) , w = e i 2 π L x , (8) u 0 ( w ) = − g π L N X j =1 w + u j w − u j = ig N X j =1 π L cot π L ( x − y j ) , w = e i 2 π L x . (9) The latter functions play a ma jor role in our approach. These are rational functions of w regular at infinit y and ha ving particle co ordinates as simple poles with equal residues 2 π g /L . The condition that the co ordinates of particles x j are real yields Sc h warz reflection condition for the function u 1 with resp ect to the unit circle u 1 ( w ) = − u 1 (1 / ¯ w ) or u 1 ( x ) = − u 1 ( ¯ x ) , (10) where bar denotes complex conjugation. The v alues of u 1 ( w ) in the in terior and exterior of a unit circle are related b y Sc hw arz reflection. Comparing (6), (4) and (9) we notice that while the function u 1 ( w ) enco des the p ositions of particles w j , the function u 0 ( w ) encodes the momen ta of particles as its v alues at particle p ositions w j p j = u 0 ( w j ) + g π L N X k =1 ( k 6 = j ) w j + w k w j − w k . (11) W e notice here that the p ositions of the particles fully determine the imaginary part of the field u 0 on a unit circle. Indeed, w e hav e from (11) Im u 0 ( x j ) = g X k 6 = j π L cot π L ( x j − x k ) . (12) W e no w introduce complex functions u = u 0 + u 1 , ˜ u = u 0 − u 1 . (13) One can sho w that they ob ey the equation u t + ∂ x  1 2 u 2 + i g 2 ∂ x ˜ u  = 0 . (14) Indeed substituting the p ole ansatz (8,9) into (14) and comparing the residues at p oles w j and u j one arrives at (6,7). The equation (14) connects tw o complex functions u 0 and u 1 . The equation is equiv alent to the mo difie d Kadomtzev-Petvisashvili equation (or simply MKP1). W e will discuss its relation to MKP1 in Sec. 5. Ho wev er, being complemen ted b y the Sc hw arz reflection condition (10), analyticit y requiremen ts, and an additional reality requiremen t it b ecomes an equation uniquely determining u 0 and u 1 through their initial data. The analyticit y requirements read: u 0 ( w ) is analytic in a neighborho o d of a unit circle | w | = 1, while u 1 is analytic inside | w | < 1 and outside | w | > 1 of the unit circle, CONTENTS 6 approac hing a constan t at w → ∞ . An additional realit y requirement is the relation b et w een the imaginary part of u 0 on a unit circle and u 1 stemming from the condition (12). W e formulate and discuss these conditions in Sec. 3.3 and Sec. 5. W e will refer to the equation (14) as the bidirectional Benjamin-Ono equation (2BO). It is a bidirectional (ha ving b oth right and left mo ving w av es) generalization of the con ven tional Benjamin-Ono equation (BO) arising in the h ydro dynamics of stratified fluids [8]. W e discuss its hydrodynamic form in the next section. The solution of (14) giv en b y (8,9) is the CSM many b o dy system with a finite n umber of particles (1). Other solutions describe CSM fluids. They are the cen tral issue of this pap er. T o conclude this section we mak e the follo wing comment. The function u 1 can b e expressed solely in terms of the microscopic density of particles (2) as u 1 ( w ) = − π g I dζ 2 π iζ ζ + w ζ − w ρ ( ζ ) . (15) The in tegral in this formula goes ov er the unit circle ζ = exp  i 2 π L x  . In the following w e will denote for brevit y ρ ( ζ ) as ρ ( x ), when ζ lies on a unit circle ζ = e i 2 π L x . The density itself can b e obtained as a difference of limiting v alues of the field u 1 at the real x (on the unit circle). The discon tinuit y of u 1 on the unit circle gives a microscopic densit y (2) of particles u 1 ( x + i 0) − u 1 ( x − i 0) = − 2 π g ρ ( x ) , Im x = 0 , 0 < Re x < L. (16) 3. Hydro dynamics of Calogero-Sutherland liquid 3.1. Density and velo city W e assume that in the thermo dynamic limit N , L → ∞ , N /L = const the p oles of the function u 1 are distributed along the real axis with a smo oth density ρ ( x ) and consider a complex field u 1 ( w ) given b y formula (15). Notice that u 1 ( w ) defined b y (15) is analytic ev erywhere outside of the real axis of x (ev erywhere off the unit circle in z -plane) approac hing a constan t as z → ∞ . It also satisfies the realit y condition (10) (the density ρ ( x ) is real). In the thermo dynamic limit the function u 1 is not a rational function an ymore. It is discontin uous across the real axis with the discon tin uit y related to the densit y of particles b y (16). The v alue of the field u 1 ( x ) on a real axis (on a unit circle in z plane) dep ends on whether one approac hes the real axis from abov e or b elo w (unit circle from the in terior z → e i 2 π L ( x + i 0) or from the exterior z → e i 2 π L ( x − i 0) ). More explicitly , w e hav e from (15) u 1 ( x ± i 0) = π g ( ∓ ρ + iρ H ) . (17) The sup erscript H in the second term of (37) denotes the Hilbert transform and is defined as (see App endix A for definitions and some prop erties of the Hilb ert transform) f H ( x ) = − Z L 0 dy L f ( y ) cot π L ( y − x ) . (18) CONTENTS 7 W e also assume that in N → ∞ limit the complex field u 0 ( w ) remains analytic in the vicinity of the real axis in x -plane (i.e., in the vicinity of a unit circle in z -plane). The 2BO (14) do es not explicitly dep end on the n umber of particles N . It holds also in thermo dynamic limit N , L → ∞ , N /L = const , how ev er solutions describing a liquid are not rational functions any longer. W e can use 2BO to define velocity through the c ontinuity e quation ρ t + ∂ x ( ρv ) = 0 . (19) The discon tinuit y of the complex field u ( x ) (13) across the real axis, as w ell as a discon tinuit y of the field u 1 (see (16)) is the densit y u ( x + i 0) − u ( x − i 0) = u 1 ( x + i 0) − u 1 ( x − i 0) = − 2 π g ρ ( x ) . (20) Differen tiating (20) with resp ect to time and using 2BO (14) w e obtain the con tin uity equation and iden tify the velo city field v ( x ) as v ( x ) = u 0 ( x ) + 1 2 ( u 1 ( x + i 0) + u 1 ( x − i 0)) − ig ∂ x log √ ρ ( x ) = u 0 ( x ) + ig  π ρ H ( x ) − ∂ x log √ ρ ( x )  (21) or u 0 ( x ) = v − ig  π ρ H − ∂ x log √ ρ  . (22) Since v ( x ) is a real field (22) provides a reality condition analogous to (12). Indeed, one can see from (22) that Im u 0 ( x ) = − g  π ρ H − ∂ x log √ ρ  , (23) i.e., the imaginary part of u 0 ( x ) is completely determined by the densit y of particles or equiv alently b y the field u 1 . It is also conv enien t to hav e an expression for u ( x ) on a real axis u ( x ± i 0) = v + g ( ∓ π ρ + i∂ x log √ ρ ) . (24) It has the same discon tinuit y across the real axis as u 1 ( x ). 3.2. Hydr o dynamic form of 2BO. No w w e are ready to cast the equation (14) in to h ydro dynamic form. T aking the real part of 2BO (14) on the real axis and using identifications (17,22) and the con tinuit y equation, (19), after some algebra w e arrive at the Euler e quation v t + ∂ x  v 2 2 + w ( ρ )  = 0 , (25) with sp ecific (p er particle) en thalp y or chemical p otential k giv en b y w ( ρ ) = 1 2 ( π g ρ ) 2 − g 2 2 1 √ ρ ∂ 2 x √ ρ + π g 2 ρ H x . (26) k The sp ecific enthalp y and chemical p otential are identical at zero temp erature. CONTENTS 8 Equations (19,25) are the contin uit y and Euler ¶ equations of classical Calogero- Sutherland mo del. They are the classical analogues of quantum hydrodynamic equations that hav e b een obtained for the quan tum CSM in Refs. [5, 6, 9] first using collective field theory approac h [10, 11, 12] and later b y the p ole ansatz similar to the one used ab o v e [4]. It was noticed in [12] and then in [7] that the system (19,25,26) has a lot of similarities with classical Benjamin-Ono equation [13]. The similarities and differences with Benjamin-Ono equation are discussed b elo w. W e will refer to (19,25,26) as to a h ydro dynamic form of the bidir e ctional Benjamin-Ono e quation (2BO). 3.3. Bidir e ctional Benjamin-Ono e quation (2BO). Let us no w summarize the 2BO equation: u t + ∂ x  1 2 u 2 + i g 2 ∂ x ˜ u  = 0 , (27) u = u 0 + u 1 , ˜ u = u 0 − u 1 . (28) The functions u 0 and u 1 are sub ject to analyticit y conditions u 1 ( x ) - analytic for Im ( x ) 6 = 0 , (29) u 0 ( x ) - analytic for | Im ( x ) | <  for some  > 0 , (30) and to realit y conditions u 1 ( x ) = − u 1 ( ¯ x ) . (31) In addition, the fact that the equation (27) holds in the upp er half plane and in the lo wer half plane (inside and outside of the unit circle) yields the condition Im [ u ( x ± i 0)] = g 2 ∂ x log Re [ u 1 ( x ± i 0)] . (32) It also follows from (17,23,24). The condition (32) lo oks more “natural” in the bilinear form ulation (see eq. (56) b elow). These reality and analyticit y conditions reduce t w o complex fields u 0 and u 1 to tw o real fields - densit y ρ ( x ) and v elo cit y v ( x ) as (17,22). Then, a complex equation (14) defined in b oth half planes immediately yields the hydrodynamic equations (19,25,26). In versely , knowing real p erio dic fields ρ ( x ) and v ( x ) one can find fields u 0 , u 1 ev erywhere in a complex x -plane. Mo de exp ansion The analyticit y and realit y conditions can b e recast in the language of mo de expansions. It follows from (15) that u 1 ( w ) = ( − π g ( ρ 0 + 2 P ∞ n =1 ρ n w n ) , | w | < 1 π g  ρ 0 + 2 P ∞ n =1 ρ † n w − n  , | w | > 1 (33) ¶ The eq. (25) has a form of an Euler equation for an isentropic flow. Because of the long range c haracter of interactions the en thalpy cannot be replaced by the con ven tional pressure term ∂ x w ( ρ ) → ρ − 1 ∂ x ( p ( ρ )) - the standard form of the Euler equation. CONTENTS 9 where ρ n = ρ † − n = R L 0 dx L ρ ( x ) e − i 2 πn L x are F ourier comp onents of the densit y . The v alues of the field u 1 ( w ) in the upp er and lo wer half-planes are then automatically related b y Sc hw arz reflection (10). Con versely , the field u 0 ( x ) b eing analytic in a strip around the unit circle is represen ted b y Laurent series u 0 ( w ) = V 0 + ∞ X n =1  a n w n + b n w − n  , | Im log w | < 2 π /L. (34) The 2BO equation remains in tact in the case of rational degeneration. Rational degeneration of formulas of the Sec. 2 are obtained b y a direct expansion in 1 /L . In this limit fields are defined microscopically as u 1 ( x ) = − ig P j 1 x − x j and u 0 ( x ) = ig P j 1 x − y j . 4. Hamiltonian form of 2BO The 2BO is a Hamiltonian equation. Let us start with its Hamiltonian formulation in the h ydro dynamic form ρ t = { H , ρ } , v t = { H , v } with the canonical Poisson brack et of densit y and velocity fields { ρ ( x ) , v ( y ) } = δ 0 ( x − y ) . (35) Equations (19,25,26) follo w from H = Z dx  ρv 2 2 + ρ ( ρ )  , (36)  ( ρ ) = g 2 2 ( π ρ H − ∂ x log √ ρ ) 2 . (37) Here the “in ternal energy” (37) and the en thalp y (26) are related b y a general form ula w ( ρ ) = δ δ ρ ( x ) R dx ρ ( ρ ). F or references w e will giv e alternative expressions for the Hamiltonian. Let Ψ = √ ρe iϑ where v = g ∂ x ϑ then H = g 2 2 Z   ∂ x Ψ − π ρ H Ψ   2 dx, (38) where ρ = | Ψ | 2 . The Poisson’s brack ets for Ψ( x ) are canonical: { Ψ( x ) , Ψ( y ) } = 0, and { Ψ( x ) , Ψ ? ( y ) } = i g δ ( x − y ). The equations of motion for Ψ and Ψ ? are i g ∂ t Ψ =  − 1 2 ∂ 2 x + π 2 2 | Ψ | 4 + π  | Ψ | 2  H x  Ψ (39) and its complex conjugate. A simple c hange of a dep enden t v ariable Φ = Ψ e iπ R x dx 0 | Ψ( x 0 ) | 2 leads to i g ∂ t Φ =  − 1 2 ∂ 2 x + i 2 π  | Φ | 2  + x  Φ , (40) where f + denotes the function analytical in the upper half-plane of x defined as f + = f − if H 2 . One can recognize in (40) the in termediate nonlinear Sc hr¨ odinger equation (INLS) which app eared in Ref.[14] as an evolution of the mo dulated in ternal w a ve in a CONTENTS 10 deep stratified fluid. Therefore, one can alternativ ely think of 2BO as the h ydro dynamic form of (40) identifying h ydro dynamic fields ρ and v to b e Φ = √ ρ exp  i g Z x dx 0 ( v + π g ρ )  (41) or with the field u ( x ) from (24) as ig ∂ x log Φ ? = u ( x − i 0) . (42) The Hamiltonian (36) or (38) can b e rewritten in terms of Φ as H = g 2 2 Z   ∂ x Φ − i 2 π ρ + Φ   2 dx, (43) where ρ = | Φ | 2 . Ho w ever, the Poisson’s brac kets for Φ are not canonical anymore. + 2BO is an integrable system. It has infinitely many integrals of motion. The first three of them follo w from global symmetries. They are conv en tional the num b er of particles N = R dx ρ , the total momen tum P = R dx ρv , and the total energy H = R dx  ρv 2 2 + ρ ( ρ )  . They are conv eniently written in terms of the fields u and ˜ u as I 1 = N = 1 2 π g I C dx u, (44) I 2 = P = 1 2 π g I C dx 1 2 u 2 , (45) I 3 = 2 H = 1 2 π g I C dx  1 3 u 3 + i g 2 u∂ x ˜ u  , (46) where the integral is taken ov er the both sides of the unit circle. (“double” contour C sho wn in Fig. 1). F or more details on conserved in tegrals see App endix B. The P oisson brac k et for the fields u 0 ( w ) and u 1 ( w ) can be easily obtained from from (17,22) and (35) b y analytic contin uation. W e find that { u 0 ( w ) , u 0 ( w 0 ) } = { u 1 ( w ) , u 1 ( w 0 ) } = 0 and { u 0 ( w ) , u 1 ( w 0 ) } = ig  2 π L  2 w w 0 ( w − w 0 ) 2 = ig ∂ x π L cot π L ( x − y ) . (47) 5. Bilinearization and relation to MKP1 equation The equations describ ed in the previous section, their in tegrable structures and their connection to in tegrable hierarc hies are the most transparent in bilinear form. Let us in tro duce tau-functions τ 0 and τ 1 as u 0 = ig ∂ x log τ 0 , (48) u 1 = − ig ∂ x log τ 1 . + Simple calculation using (35) giv es { Φ( x ) , Φ( y ) } = π g Φ( x )Φ( y ) sgn ( x − y ), { Φ( x ) , Φ ? ( y ) } = i g δ ( x − y ) − π g Φ( x )Φ ? ( y ) sgn ( x − y ) and similar expressions for complex conjugated fields. One should think of Ψ( x ) as a canonical b osonic field while of Φ( x ) as a classical analogue of a field with fractional statistics. CONTENTS 11 τ −1 τ +1 R L τ 0 Figure 1. Con tour C surrounding the unit circle is sho wn together with our con v en tions in defining right and left fields. It can be easily c heck ed that the 2BO (14) can b e rewritten as an elegant bilinear Hirota equation on τ -functions:  iD t + g 2 D 2 x  τ 1 · τ 0 = 0 . (49) Here we used the Hirota deriv ativ e sym b ols defined as D n x f ( x ) · g ( x ) ≡ lim y → x ( ∂ x − ∂ y ) n f ( x ) g ( y ) . (50) F or example, D t f · g = ( ∂ t f ) g − f ( ∂ t g ) , D 2 x f · g = ( ∂ 2 x f ) g − 2( ∂ x f )( ∂ x g ) + f ( ∂ 2 x g ) , (51) etc. W e emphasize that the bilinear equation holds on b oth sides of the unit circle. In tro ducing notations τ ± 1 = τ 1 ( x ± i 0) (52) w e can rewrite the equation as  iD t + g 2 D 2 x  τ +1 · τ 0 = 0 , (53)  iD t + g 2 D 2 x  τ − 1 · τ 0=  − iD t + g 2 D 2 x  τ 0 · τ − 1 = 0 . (54) Equation (49) is the mo dified Kadom tsev-P etviashvili equation (MKP1). MKP1 con tains tw o independent functions τ 1 and τ 0 and is formally not closed. The analyticity and realit y conditions (29-32) stemming from the fact that all solutions are determined b y tw o real functions ρ ( x, t ) and v ( x, t ), close the equation. Under these conditions the equations can b e seen as a real reduction of MKP1. Let us form ulate these conditions in terms of tau-functions. CONTENTS 12 The first requiremen t is that τ ± 1 is analytic and do es not hav e zeros for Im x > 0 ( < 0) after analytic con tinuation. Also τ 0 should b e analytic and should not ha v e zeros in the vicinit y of the real axis, i.e., for | Im x | <  for some  > 0. The second requiremen t is that τ ± 1 should b e related by Sc hw arz reflection (10). In terms of tau-functions it b ecomes on the unit circle (for real x ) τ − 1 = τ +1 e i Θ( t ) , (55) where a phase Θ( t ) can b e an y time-dep endent function. The third requiremen t is related to the fact that Im u 0 is a function of density only and, therefore, can b e expressed in terms of u 1 as can b e easily seen from (17,22). This condition (32) can b e written in a bilinear form as follo ws iD x τ +1 · τ − 1 = 2 π τ 0 τ 0 . (56) The multiplicativ e constant in the r.h.s of (56) fixes the relative normalization of τ 0 and τ 1 and is arbitrary . W e ha ve c hosen it to b e 2 π . Finally , we note that the p ole ansatz solution (8,9) corresp onds to the p olynomial form of tau-functions with zeros at w j and u j τ 1 ( w , t ) = w − N/ 2 N Y j =1 ( w − w j ( t )) , (57) τ 0 ( w , t ) = w − N/ 2 N Y j =1 ( w − u j ( t )) . (58) 6. Chiral Fields and Chiral Reduction 6.1. Chir al fields and curr ents The 2BO equation can b e conv enien tly expressed through y et another right and left handed chiral fields J R,L = v ± g  π ρ + ∂ x (log √ ρ ) H  . (59) These fields are real. ∗ In terms of them, the 2BO equation (14) reads ∂ t J R,L + ∂ x  J 2 R,L 2 ± g 2 ∂ x J H R,L  ∓ g ∂ x  J R,L ∂ x (log √ ρ ) H − ( J R,L ∂ x log √ ρ ) H  = 0 . (60) Here ρ is a function of J R and J L implicitly giv en by (59). The Hamiltonian acquires a Suga wara-lik e form H = 1 8 Z dx ρ h ( J R + J L ) 2 + ( J H R − J H L ) 2 i (61) ∗ J R,L can b e expressed solely in terms of u 0 field. It is easy to chec k that (59) is equiv alent to J R,L = Re  u 0 ∓ iu H 0  . CONTENTS 13 with Poisson brac kets { J R,L ( x ) , J R,L ( y ) } = ± 2 π g ∂ x δ ( x − y ) (62) ± g 2 L ∂ x ∂ y  1 ρ ( x ) + 1 ρ ( y )  cot π L ( x − y )  , { J R ( x ) , J L ( y ) } = − g 2 L ∂ x ∂ y  1 ρ ( x ) − 1 ρ ( y )  cot π L ( x − y )  . (63) W e note that Poisson brac kets b ecome canonical and left and right fields decouple in the limit of a constan t densit y . 6.2. Chir al R e duction W e first note that the righ t and left curren ts J R,L are not separated in eq. (60). The equations for J R and J L are coupled through the density ρ which should b e found in terms of J R,L from (59). How ev er, it is p ossible to find the chir al r e ductions of 2BO assuming that one of the currents is constant. W e explain this reduction in some detail in this section. The 2BO (14) or (60) admits an additional reduction to a c hiral sector [15] where one of the chiral currents (59), say left current, is a constan t J L ( x, t ) = v 0 − π g ρ 0 . W e can alw ays c ho ose a co ordinate system mo ving with velocity v 0 . This is equiv alent to setting the zero mo de of velocity to zero v 0 = 0. The condition J L = − π g ρ 0 b ecomes v = g  π ( ρ − ρ 0 ) + ∂ x (log √ ρ ) H  . (64) Then the curren ts can b e expressed in terms of the density field only J L ( x ) = J 0 , J R ( x ) = J 0 + J ( x ) , (65) J 0 = π g ρ 0 , J ( x ) = 2 g  π ( ρ − ρ 0 ) + ∂ x (log √ ρ ) H  . (66) It follows from Eq. (60) that once the current J L is chosen to b e constant J L ( x ) = J 0 at t = 0 it remains constant at any later time. The condition (64), therefore, is compatible with 2BO. Then the density ρ ( x, t ) evolv es according to the con tin uity equation (19) with v elo city determined by the densit y according to (64). W e obtain an imp ortant equation (written in the co ordinate system mo ving with velocity v 0 ) ρ t + g  ρ  π ( ρ − ρ 0 ) + ∂ x (log √ ρ ) H  x = 0 . (67) W e refer to this equation as the Non-Linear Chiral Equation (NLC). A substitution of the chiral constrain t (65) to (61) gives the Hamiltonian for NLC H = 1 8 Z dx ρ  J 2 + ( J H ) 2  (68) with Poisson brack ets for J ( x ) follo wing from (62). This equation constitutes one of ma jor results of this pap er. CONTENTS 14 NLC can b e written in several useful forms. One of them is: ϕ t + g  π ρ 0 (2 e ϕ − ϕ ) + 1 2 ϕ H x  x + g 2 ϕ x ϕ H x = 0 , (69) where ρ ( x ) = ρ 0 e ϕ ( x ) . 6.3. Holomorphic Chir al field Under the c hiral condition (64) the field u 0 b ecomes analytic inside the disk. Indeed, com bining (64) and (22) we obtain u 0 ( w ) = 1 2 I dζ 2 π iζ ζ + w ζ − w J ( ζ ) , | w | < 1 . (70) In the chiral case it has only non-negative p o w ers of w in the expansion (34). Negative mo des v anish b n = 0. Con versely , the condition of u 0 to be analytic inside the unit disk is equiv alen t to J L = const . The curren t itself (65) is the b oundary v alue of the field Re u 0 harmonic inside the disk J ( x ) = 2 J 0 + 2Re u 0 = 2 J 0 + X n =1 ∞  a n w n + ¯ a n w − n  . (71) The fields u and ˜ u are in turn also analytic inside the disk. Let ϕ to b e a harmonic function inside the disk with the b oundary v alue log ( ρ/ρ 0 ). Then ϕ = φ ( w ) + φ ( w ), where φ ( w ) = (log ( ρ/ρ 0 )) + . Here f + ( w ) = 1 2 R ζ + w ζ − w f ( ζ ) dζ 2 π iζ is a function analytic in the in terior of a unit circle which v alue on the boundary of the disk is ( f ( x ) − if H ( x )) / 2. It follows from (24,64) that u = − J 0 + ig ∂ φ, | w | < 1 , (72) ˜ u = u + 4 π g ρ 0 ( e ϕ ) + , | w | < 1 , (73) Then 2BO (27) b ecomes an equation on an analytic function in the in terior of a unit circle ˙ φ + i g 2  ( ∂ φ ) 2 + ∂ 2 φ  + π g ρ 0 ∂ (2 e ϕ − ϕ ) + = 0 . (74) This is the “p ositive part” of (69) which is a direct consequence of (67). W e remark here that the c hiral equation (67) has a geometric in terpretation as an evolution equation describing the dynamics of a con tour on a plane. Within this in terpretation the term ∂ x (log √ ρ ) H of (67) is the curv ature of the contour (see App endix C). 6.4. Benjamin-Ono Equation Another form of the Chiral Equation (67) arises when one considers the fields u and ˜ u outside the disk. There neither u nor ˜ u are analytic, but their b oundary v alues are connected by the Hilb ert transform u ( x − i 0) = − J 0 + 2 g  π ρ + i∂ x (log √ ρ ) +  , (75) ˜ u ( x − i 0) = − J 0 − iu H ( x − i 0) . (76) CONTENTS 15 The bidirectional equation Eq. (27) complemented by this condition b ecomes unidirectional (chiral) u t + ∂ x  1 2 u 2 + g 2 ∂ x u H  = 0 . (77) This is just another form of the chiral equation (67). The c hiral equation (77) has the form of the Benjamin-Ono equation [13]. There are noticeable differences, how ever. Contrary to the Benjamin-Ono equation, Eq. (77) is written on a complex function, whose real and imaginary v alues at real x are related b y conditions (75) implemen ting the reality of the densit y: Re u = − J 0 + 2 g π ρ + g ( ∂ x log √ ρ ) H , Im u = g ∂ x log √ ρ. (78) One understands this relation as a condition on the initial data. Once it is imp osed b y choosing the initial data for the densit y ρ , the condition remains in tact during the ev olution. Ho wev er, in the case when the deviation of a density is small with respect to the a verage density | ρ − ρ 0 |  ρ 0 , the imaginary part of u v anishes in the leading order of 1 /ρ 0 expansion u ≈ J 0 + 2 π g ϕ ≈ 2 π g ( ρ − ρ 0 ) + J 0 and the condition (78) b ecomes non-restrictiv e. In this limit Eq. (77) b ecomes an equation on a single real function. It is the conv entional Benjamin-Ono equation. One can think of NLC (67) as of finite amplitude extension of BO. Similarly , 2BO is an integrable bidirectional finite amplitude extension of BO. It is in teresting that there exists another bidirectional finite amplitude extension of BO – the Choi-Camassa equation [16]. How ever, it seems that the latter is not in tegrable. 7. Multi-phase solution In this section w e describ e the most general finite dimensional solutions of 2BO. These are m ulti-phase solutions and their degenerations – m ulti-soliton solutions. In the former case the τ -functions are p olynomials of e ik i x , where k i is a finite set of parameters, the latter are just p olynomials of x . These solutions are given by determinan ts of finite dimensional matrices. They app eared in the arXiv v ersion of Ref. [17]. One can construct those solutions using the transformation (41) of 2BO to INLS (40). F or the latter m ulti-phase solutions w ere written in [14] (see also [18, 19]). W e use a different route in this section deriving m ulti-phase and m ulti-soliton solutions as a real reduction of corresp onding solutions for MKP1. 7.1. Multi-phase and multi-soliton solutions of MKP1 W e start from a general multi-phase solution of MKP1 equation and then restrict it to 2BO equation. CONTENTS 16 A general m ulti-phase solution of MKP1 equation  iD t + g 2 D 2 x  τ 1 · τ 0 = 0 . (79) is given b y the following determinan t form ulae [21, 22] τ a = e iθ a det  δ j k + c a,j e iθ j p j − q k  , a = 0 , 1 (80) c 1 ,j c 0 ,j = q j − K − v 0 p j − K − v 0 , (81) where the phases are g θ j ( x, t ) = ( q j − p j )( x − x 0 j ) − q 2 j − p 2 j 2 t, (82) g θ 0 ( x, t ) = K x − K 2 2 t −  ( v 0 + K ) x − ( v 0 + K ) 2 2 t  , (83) g θ 1 ( x, t ) = K x − K 2 2 t. (84) This solution is c haracterized by an integer n umber N (n umber of “phases”), and b y 4 N − 1 parameters p j , q j , c 0 ,j , x 0 j and mo duli K and v 0 . The solutions b ecome single- v alued on a unit circle if p j and q j are integers in units of g 2 π L . 7.2. Multi-phase solution of 2BO Without further restrictions the parameters entering (80-84) are general complex n umbers. Realit y nature of 2BO equation restricts them to b e real. The real mo duli K and v 0 are ob viously zero mo des of the fields u 1 and u 0 resp ectiv ely , and therefore, they are zero mo des of the densit y ρ 0 = 1 L R ρ dx = − K / ( π g ) and velocity 1 L R v dx = v 0 . 7.2.1. Schwarz r efle ction c ondition W e hav e to restrict the co efficien ts c a,j , so that there exists another solution τ − 1 , τ 0 of Eq. (79) sharing the same τ 0 with the solution (86,87) and ob eying the Sc hw arz reflection prop erty (55). The Galilean symmetry of the equation (79) is here to help. If τ a ( x, t ) , a = 0 , 1 giv e a solution of (79) then the pair e iP a x − iE a t τ a ( x − g P a t, t ) is also a solution provided that E a +1 − E a = 1 2 ( P a +1 − P a ) 2 . Being applied to the solution (80-84) the Galilean inv ariance can b e utilized as follo ws. W e notice from (82) that θ j ( x − P t, t ; { p j , q j } ) = θ j ( x, t ; { p j + P , q j + P } ) . (85) P erforming the Galilean b o ost to (80), m ultiplying b oth tau-functions by e − iP v 0 t and shifting p j → p j − P , q j → q j − P w e obtain that τ − 1 = e iθ 1 − iP ( K + v 0 ) t + i g ( P x − P 2 2 t ) det  δ j k + b j e iθ j p j − q k  , (86) b j c 0 ,j = q j − K − v 0 − P p j − K − v 0 − P (87) CONTENTS 17 form a solution of (79) with the same τ 0 (80). No w w e are going to show that for a particular c hoice of co efficients c 1 ,j the Galilean b o osted solution (86,87) is a complex conjugate of τ +1 from (80). T o show this w e will emplo y the determinant iden tity (D.7). ] W e apply the determinan t identit y (D.7) to (86) and obtain τ − 1 = e iθ 1 − iP ( K + v 0 ) t + i g ( P x − P 2 2 t ) Y j e iθ j s b j ˜ b j ! det  δ j k + ˜ b j e − iθ j p j − q k  , (88) where ˜ b j b j = ( p j − q j ) 2 Y k 6 = j ( p j − q k )( q j − p k ) ( p j − p k )( q j − q k ) . (89) The Sch w arz reflection condition(55) τ − 1 = τ +1 e i Θ( t ) requires ˜ b j = c 1 ,j , (90) determines Θ( t ), and gives a relation P = X j ( p j − q j ) − 2 K . (91) Finally , com bining all relations (81,87,89,90) together we obtain the condition on co efficien ts c a,j  c a,j p j − q j  2 Y k 6 = j ( p j − p k )( q j − q k ) ( p j − q k )( q j − p k ) (92) =  p j − K − v 0 q j − K − v 0  1 − 2 a p j − K − v 0 − P q j − K − v 0 − P . Condition (92) is necessary to turn a general solution of MKP1 into a solution of 2BO. W e also hav e to find a condition that guarantees that τ 1 has no zeros inside the unit disk. Before turning to this analysis, w e first discuss degeneration of formulas (80) in to m ulti-soliton solution. 7.2.2. Multi-soliton solution of 2BO The m ulti-soliton solution of 2BO follows from the multi-phase solution in the limit p j → q j . W e introduce k j = p j − q j , (93) v j = 1 2 ( p j + q j ) (94) and consider the limit k j → 0 keeping v j fixed. After some straigh tforward calculations w e obtain τ a = e iθ a det  δ j k ( x − x 0 j − v j t + iA a,j ) + ig 1 − δ j k v j − v k  , (95) A a,j = g 2  1 v j − v 0 + K ± 1 v j − v 0 − K  , a = 0 , 1 . (96) ] A similar tric k w as used by Matsuno [18] to pro ve the realit y of a m ulti-phase solution for con ven tional Benjamin-Ono equation. CONTENTS 18 One notices that in the limit t → + ∞ the solution (95) asymptotically go es to the factorized form τ a → e iθ a Y j ( x − x 0 j − v j t + iA a,j ) , (97) describing separated single solitons. Eq. (97) giv es a large time v alue of zeros of τ 1 . Their imaginary part is − Re A 1 ,j = g K ( v j − v 0 ) 2 − K 2 . (98) It m ust b e negative in order for τ 1 to ha ve no zero es inside the unit disk. Since K < 0 w e m ust require ( v j − v 0 ) 2 > K 2 . (99) In the next paragraph w e argue that under this condition and additional restrictions on parameters p j , q j (see eq. (109) below) the mo ving zeros nev er cross the real axis, and therefore zeros sta y outside of the unit disk at all times. T o conclude this section w e note a unique prop erty of the 2BO equation (shared with the BO equation). Namely , there is a “quan tization” of the mass of solitons: each soliton of 2BO carries a unit of mass regardless of its velocity . W e ha ve for N -soliton solution Z dx ( ρ − ρ 0 ) = N . (100) Where K = − π g ρ 0 . The total momen tum, and the total energy of a multisoliton solution is giv en b y Z dx ( ρv − ρ 0 v 0 ) = X j v j , (101) Z dx  ρv 2 2 + ρ ( ρ ) − ρ 0 v 2 0 2 − ρ 0  ( ρ 0 )  = X j v 2 j 2 , (102) where  ( ρ ) is defined in (37). One-soliton solution has a form ρ = ρ 0 + 1 π A 1 ( x − x 01 − v 1 t ) 2 + A 2 1 , (103) v = v 0 + g A 0 ( x − x 01 − v 1 t ) 2 + A 2 0 , (104) where A 1 = Re A 1 , 1 = π g 2 ρ 0 ( v j − v 0 ) 2 − ( π g ρ 0 ) 2 , (105) A 0 = Re A 0 , 1 = g ( v j − v 0 ) ( v j − v 0 ) 2 − ( π g ρ 0 ) 2 . (106) This one-soliton solution has b een found first in Ref. [7] (see also [23]). CONTENTS 19 7.2.3. A nalyticity c ondition No w we can turn to the multiphase solution and derive conditions sufficient in order for u 1 to b e analytic in the upp er half-plane in complex x -v ariable (inside the unit disk). Analyticity in the low er half-plane follows from the Sc hw arz reflection condition (55). W e will follo w the approach of Dobrokhotov and Kric hever [24] developed for Benjamin-Ono equation. Analyticit y of u 1 means that τ 1 giv en by (80) has no zeros in the upper half plane, or that the matrix M j k = δ j k + c 1 ,j e iθ j p j − q k (107) is non-degenerate. F ollowing the approac h of Ref. [24] w e deriv ed in App endix E a sufficien t condition of non-degeneracy of the matrix M from (107). Let us now write that condition (E.10,E.9) with c j defined by (E.1,92). W e obtain (calculating f j ) P ˜ p j ( ˜ q j − P ) Y k ( k 6 = j ) ˜ p j − ˜ q k ˜ p j − ˜ p k same sign for all j, (108) where w e used shifted num b ers ˜ p j = p j − K − v 0 and similar for ˜ q j . Here P = − 2 K + P j ( ˜ p j − ˜ q j ). The set of conditions ˜ q 1 < ˜ p 1 < . . . < ˜ q m < ˜ p m < 0 < P < ˜ q m +1 < ˜ p m +1 < . . . < ˜ q N < ˜ p N (109) satisfies (108). Moreov er, (109) yields to (99), whic h in its turn means that at least at some v alues of parameters (large time and soliton limit) no zeros of τ 1 are inside the unit disk. Since they also can not b e on the circle they do not cross it while mo ving in time and in the space of parameters. Condition (109) suggests that a general solution is c haracterized b y a in teger n umber N − 2 m . This is c hirality – the difference b etw een the num b er of N − m righ t and m - left mo ving mo des 1 2 π g Z ( J R − v 0 − π g ρ 0 ) dx = N − m, (110) 1 2 π g Z ( J L − v 0 + π g ρ 0 ) dx = m. (111) Eqs. (80-84,92,109) summarize a general finite dimensional quasi-p erio dic solution. W e emphasize here that this solution is not chiral and con tains b oth right and left-mo ving mo des. 7.3. Multi-phase solution of the Chir al Non-line ar Equation The (righ t) c hiral case app ears when τ 0 has no zeros outside the unit disk. It naturally happ ens when the num b er of, say , left-moving mo des m in (109) v anishes m = 0. In this case all v j − v 0 < 0. In their turn, imaginary parts of zeros of τ 0 in the multi-soliton limit (as in (98)) − A 0 j = − g v j − v 0 ( v j − v 0 ) 2 − K 2 > 0 (112) CONTENTS 20 are p ositiv e. One can c hec k that in this case (107) with c 1 ,j → c 0 ,j is non-degenerate for arbitrary v alues of parameters satisfying (109) with m = 0 (and similarly for m = N ). Therefore, τ 0 ( x ) is non-zero in one of half-planes. This is a chiral m ulti-phase solution of 2BO. 7.4. Multi-phase solution of the Benjamin-Ono e quation The kno wn solutions of the Benjamin-Ono equation [20, 21] are obtained from the solutions of the Chiral Non-linear equation b y taking the limit ρ 0 → ∞ . In this case K → −∞ and conditions (109) allow for a go o d limit only if m = N (left sector) or m = 0 (righ t sector). Let us concentrate on the right sector. W e redefine p j → p j − K , q j → q j − K , v 0 → v 0 − 2 K , go to the frame moving with v elo city − K ( x → x + K t ), and obtain from (109,92) in the limit K → −∞ q 1 < p 1 < q 2 < . . . < q N < p N , (113)  c a,j p j − q j  2 Y k 6 = j ( p j − p k )( q j − q k ) ( p j − q k )( q j − p k ) =  p j − v 0 q j − v 0  1 − 2 a (114) with solution giv en b y (80,82) and with (83,84) (one should put K = 0 in latter tw o). This is nothing else but the m ultiphase solution of con ven tional Benjamin-Ono equation [21, 18]. 7.5. Moving Poles The 2BO equation (14) lo oks v ery similar to the classical BO equation. One of imp ortan t to ols in studying the classical BO equation is the so-called p ole ansatz - solutions in the form of p oles moving in a complex plane. [20] W e ha v e already seen that the pole ansatz (8,9) describ es the dynamics of the original Calogero-Sutherland mo del with finite num b er of particles N . In this section w e consider collective excitations of Calogero-Sutherland model in the limit of infinitely man y particles. These excitations are given by “complex” pole solutions of the 2BO. In the P ole Ansatz (8,9), the realit y conditions w ere satisfied by requiring x j to b e real (or w j ( t ) mo ving on a unit circle). One could generalize the P ole Ansatz (8,9) to case where w j ( t ) are aw a y from the unit circle and moving in a complex plane. The equations (6,7) describing the motion of p oles preserv e their form. How ev er, u − 1 ( w ) outside of the unit circle is not related to the u 1 ( w ) inside of the circle by analytic con tinuation but only b y Sch w arz reflection (10). The field u 1 ( w ) is analytic inside the unit circle and has mo ving p oles outside of the unit circle (and vice v ersa for u − 1 ( w )). Of course, ha ving obtained the solution of 2BO inside the unit circle do es not mean automatically that the Sc h warz reflected function (10) will solv e 2BO in the exterior of the circle with the same u 0 . The prop ert y (10) requires that (6,7) are satisfied not only b y u j and w j but also by u j and 1 / ¯ w j . This requiremen t will significan tly constrain the CONTENTS 21 p ositions of p oles w j and u j in a complex plane. It turns out that this constraint allows for non-trivial solutions. W e emphasize here once again that while real axis p oles x j of u 1 in the p ole ansatz represen t the original CS particles, the complex p oles x j represen t collectiv e excitations of the CS liquid mo ving in the bac kground of macroscopic n um b er of particles. Instead of looking for moving p ole solution in this section w e ha v e taken a different route. W e first construct the m uch more general solution of 2BO (14) with prop er realit y conditions and then obtain a mo ving p ole (i.e., multi-soliton) solution as a limit of the m ulti-phase solution. One can see from (95) that for soliton solutions the zeros of tau-functions mov e in a complex plane. It is esp ecially clear at large times when solitons are w ell separated (97). 8. Conclusion and discussion In this paper we hav e sho wn that the dynamics of the classical Calogero-Sutherland mo del in the limit of infinite num b er of particle is equiv alent to the bidirectional Benjamin-Ono equation (14). The bidirectional Benjamin-Ono equation (2BO) is an in tegrable classical in tegro-differential equation. Its in tegrability can b e deduced from the fact that it is a Hamiltonian reduction of MKP1 as it is shown in this pap er. As an alternativ e, one can use the equiv alence of 2BO to INLS (40). The integrabilit y of INLS w as pro ven and the sp ectral transform w as constructed for INLS in Ref. [25] (see also [19]). Therefore, one can use all techniques developed in the field of classical integrable equations for 2BO. It has m ulti-phase solutions (explicitly constructed in this pap er), bi-Hamiltonian structure, an asso ciated hierarc hy of higher order equations, etc. 2BO is intrinsically simpler than man y other classical in tegrable models. Its solitons hav e “quan tized” area indep endent of soliton’s v elo cit y . The collision of t wo solitons go es without an y time delay etc. This is a reflection of the fact that underlying Calogero- Sutherland mo del is essentially a mo del of non-in teracting particles in disguise. In particular, 2BO supp orts a phenomenon of dispersive sho ck w a ves. Some applications of this phenomenon to quasi-classical description of quantum systems w ere considered in [15]. Most of the results of this pap er can be ge neralized along t w o a ven ues: generalization to an elliptic case and generalization to a quantum mo del. The Calogero-Sutherland model (trigonometric case) can b e generalized to an elliptic case — elliptic Calogero mo del where the in teraction b etw een particles is either W eierstrass ℘ ( x | ω 1 , ω 2 )-function with purely real and purely imaginary p erio ds ω 1 , iω 2 , and to its h yp erb olic degeneration (hyperb olic case) with in ter-particle in teraction giv en b y sinh − 2 ( x/ω 2 ) (see [3] for review). In b oth cases most of form ulas remain unchanged if one substitutes the Hilb ert transform f H for a transform with resp ect to a strip 0 < Im x < ω , where ω is an CONTENTS 22 imaginary p erio d f H = Z ζ ( x − x 0 ) f ( x 0 ) dx 0 or Z 1 ω 2 coth 1 ω 2 ( x − x 0 ) f ( x 0 ) dx 0 . (115) In the first case the integration go es ov er a real p erio d of the W eierstrass ζ -function. The elliptic Calogero mo del allows one to study a crosso ver b et ween liquids with long range in ter-particle interaction to liquids with short range interaction. In the limit of a large imaginary p erio d ω 2 → ∞ the ℘ -function degenerates to 1 / sin( x/ω 1 ) 2 – the case of long range in ter-particle in teraction. The opp osite limit ω 2 → 0 giv es rise to a short range in teraction: ω 2 ℘ ( x ) → δ ( x ). In the latter case the the Hilbert transform (115) b ecomes a deriv ative f H → ω ∂ x f and the equations discussed in this pap er b ecome lo cal. In particular the Benjamin-Ono equation flows to the KdV equation, while the bidirectional BO-equation flo ws to NLS - the nonlinear Schr¨ odinger equation. 2BO in the limit of small amplitudes and in the c hiral sector b ecomes the con ven tional Benjamin-Ono equation. In elliptic case (and in the h yp erb olic one) the limit of small amplitudes in the chiral sector leads to a generalization of the Benjamin- Ono equation, known as the IL W (In termediate Long W av e) equation [8]. Con trary to the Benjamin-Ono equation and to 2BO, the latter and its bidirectional generalization 2IL W ha ve elliptic solutions. W e in tend to address the elliptic case in a separate publication. Probably , even more in teresting is a generalization of the results of this pap er to the quantum case. It is w ell known that the classical CSM mo del (1) can b e lifted to a quan tum in tegrable Calogero-Sutherland mo del [1, 2, 27]. The latter model is defined b y (1) with g 2 → ~ 2 λ ( λ − 1) and p i = − i ~ ∂ x i . The 2BO equation in the form (14) remains unc hanged, except for the c hange of the coefficient g → λ − 1 and for the c hange of P oisson brac kets (47) by a comm utator: { , } → i ~ [ , ]. The c hange g → λ − 1 v alid for eq. (14) is not correct for all formulas. F or example, the bilinear form of classical 2BO (49) is iden tical to its quan tum version with just a change of notations g → λ . F or some details see [4]. Multi-soliton solution of 2BO presen ted here corresp onds to exact quasiparticle excitations of quantum Calogero-Sutherland mo del [29, 7]. The more detailed study of the relations b etw een integrable structures of the classical 2BO and its quantum analogue is necessary . 9. Ac knowledgmen ts A GA is grateful to A. P olyc hronak os for the discussion of the c hiral case. PW thanks J. Shiraishi for discussions. The work of AGA w as supp orted b y the NSF under the gran t DMR-0348358. EB was supp orted b y ISF grant num b er 206/07. PW was supp orted b y NSF under the gran t NSF DMR-0540811/F AS 5-27837 and MRSEC DMR-0213745. W e also thank the Galileo Galilei Institute for Theoretical Ph ysics for the hospitality and the INFN for partial supp ort during the completion of this work. CONTENTS 23 App endix A. Hilb ert transforms Giv en a function f ( x ), f ( x ) → 0 as x → ±∞ , the Hilb ert transform is defined as f H ( x ) = 1 π − Z + ∞ −∞ dy f ( y ) y − x . (A.1) F or p erio dic functions with p erio d L w e define the transform as f H ( x ) = − Z L 0 dy L f ( y ) cot π L ( y − x ) . (A.2) The Hilb ert transform of the constan t function is zero. The Hilb ert transform is inv erse to itself H 2 = − 1 or ( f H ) H = − f and it commutes with deriv ative ( f H ) x = ( f x ) H . More generally , a function f ( x ) defined on the closed (directed) contour C surrounded the origin of a complex plane can b e decomposed in to a sum f = f + + f − , analytic functions f ± inside (outside) of the con tour, suc h that f + (0) = f − ( ∞ ). Then f H ≡ P I dζ 2 π ζ f ( ζ ) ζ + w ζ − w = i ( f + − f − ) . (A.3) Using (A.3) it is easy to deriv e the following prop erties: f H g + f g H = ( f g ) H − ( f H g H ) H (A.4) and some in tegration form ulas Z dx f H = 0 , (A.5) Z dx f H g = − Z dx f g H , (A.6) Z dx f H f = 0 . (A.7) F rom (A.3) we ha ve for functions analytic in one of half planes ( f ± ) H = ± if ± (A.8) and as an immediate consequence ( e ikx ) H = ie ikx sgn k , (A.9)  1 x − a  H = i x − a for Im a > 0 . (A.10) Generally , in F ourier space the Hilb ert transform is equiv alen t to a multiplication by i sgn k , i.e., for F ourier co efficien ts ( f H ) k = i ( sgn k ) f k . (A.11) It is easy to deriv e the following useful identities Z u 2 dx = Z ( u H ) 2 dx, (A.12) Z u 3 dx = 3 Z u ( u H ) 2 dx, (A.13) CONTENTS 24 Z u 4 dx = Z  ( u H ) 4 + 6 u 2 ( u H ) 2  dx, (A.14) Z u 5 dx = 5 Z  u ( u H ) 4 − 2 u 3 ( u H ) 2  dx. (A.15) App endix B. Conserv ed in tegrals of 2BO In this section w e present the conserv ed in tegrals of 2BO written in different forms. The con tour in tegrals b elow are taken along the con tour C defined in Figure 1. I 1 = I dx 2 π g u = Z dx ρ = Z dx 2 π i∂ x log τ 1 τ − 1 , (B.1) I 2 = I dx 2 π g 1 2 u 2 = I dx 2 π g u 1 u 0 = Z dx ρv = g 2 Z dx 2 π  D 2 x τ − 1 · τ 0 τ − 1 τ 0 − D 2 x τ 1 · τ 0 τ 1 τ 0  = i∂ t Z dx 2 π log τ 1 τ − 1 , (B.2) I 3 = I dx 2 π g  1 3 u 3 + i g 2 u∂ x ˜ u  = I dx 2 π g  u 1 2 u 0 + u 1 u 0 2 + ig u 1 ∂ u 0  = Z dx  ρv 2 2 + ρ ( ρ )  . (B.3) Higher in tegrals of motion for 2BO can b e constructed recurrently similarly to Benjamin- Ono equation [26] or can b e written using the integrals obtained for INLS [25, 19]. App endix C. Geometrical in terpretation of the c hiral equation: Contour dynamics Equation (67) can b e cast in the form of contour dynamics. Let us in terpret the unit disk as a uniformization of a simply-connected domain em b edded into the complex z -plane. In other words, z ( w ) is a conformal map of the in terior of the unit disk | w | = 1 to a b ounded domain suc h that the length elemen t of its b oundary is prop ortional to the densit y ds ≡ | z 0 ( x ) | dx = π ρ ( x ) dx. (C.1) (equiv alently 1 / √ ρ is a harmonic measure of the con tour). Then the equation (67) describ es the evolution of the planar domain. W e notice that the curv ature of the b oundary κ = − i∂ s log z s can b e expressed in terms of the density ρ ( x ) as κ = − ( π ρ ) − 1  ∂ x (log √ ρ ) H − 1  . 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