Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems

Nonc hiral In termediate long-w a v e equation and interedge effects in narrow quan tum Hall systems Bjorn K. Ber nt son, 1 , ∗ Edwin Langma nn, 2 , † and Jona tan Lenells 1 , ‡ 1 Dep artment of Mathematics, KTH R oyal Inst itute of T e chnolo gy, SE-100 44 Sto ckholm, Swe den 2 Dep artment of Physics, KTH R oyal Institute of T e chnolo gy, SE-106 91 Sto ckholm, Swe den (Dated: Octob er 27, 2020) W e p resen t a nonchiral versio n of the In termediate long-wa ve (IL W) equation that can model nonlinear wa ves propaga ting on tw o opp osite edges of a quantum Hall system, taking in to accoun t interedge interactions. W e obtain exact soliton solutions gov erned by the hyperb olic Calogero - Moser-Sutherland (CMS) mo del, and w e give a Lax pair, a Hirota form, and conserv ation la ws for this new equation. W e also present a p eriodic nonc h iral IL W equation, together with its soliton solutions go verned by the elliptic CMS model. I. INTRO DUCTION One impo rtant feature of the F ractional Quantum Hall E ffect (F QHE) is the strikingly high accura cy by which the Hall conductance, σ H , is mea s ured in units of the in verse v on Klitzing constant, e 2 /h . 1 Therefore, satisfactory explana tions of these F QHE mea surements, σ H h/e 2 = 1 3 , 2 5 , 3 7 , . . . , must b e based on exact ana lytic arguments, and theor ie s of the FQHE hav e clo se con- nections to integrable systems. Two imp orta n t classe s of integrable systems which are seemingly very differ- ent but which a re b oth co nnected with the FQHE are (i) Caloger o-Moser -Sutherland (CMS) 2 mo dels describ- ing FQHE edge states, 3–9 and (ii) soliton equations of Benjamin-Ono (BO) type describing the dynamics of nonlinea r w aves pr opagating alo ng FQHE edges 10–12 (background on the soliton eq ua tions app earing in this pap er can be found in Section VI B). These systems are related by a fundamental cor resp ondence b etw een CMS sys tems and BO-type s oliton equatio ns , which pr o- vides the bas is for a mathematically precise deriv a tion of hydrodynamic descriptions of CMS systems. 13–16 It is worth noting that this sub ject has recently received considerable attention in the con tex t of no nequilibrium ph ysics. 17–20 While the CMS-BO c o rresp ondence has been success- fully used to under stand FQHE physics, it is incomplete. Indeed, CMS systems come in four t yp es: (I) ra tional, (II) trigo nometric, (II I) hyperb olic, and (IV) elliptic, 21,22 and while the so liton equations r elated to the ratio nal and trigonometric cases are w ell-understo o d since a long time, 14–16 soliton equations related to the hyperb olic and elliptic cases w ere o nly recen tly iden tified as the In ter- mediate lo ng-wa ve (IL W) equation and the perio dic IL W equation, resp ectively . 23–25 How ever, as w e will show in this pap er, the latter tw o soliton equa tion are not unique: there are other equations which a re more in teres ting in that they are of a different k ind and describ e new ph ys ic s. The corresp ondenc e b etw een CMS a nd BO systems ex- ists b oth a t the clas sical 15,16 and at the quantum 10,14 level, and w e consider b oth. As will b e explained, w e dis - cov er ed the quantum elliptic v ersion of the soliton equa- tion presented in this pap er fr om a second quan tization of the quan tum elliptic CMS mo del. 26,27 How ever, the ex- act results on the solution of this equation presen ted in this pap er are restr ic ted to the classical case for simplic- it y . W e fir st give a nd prov e our results in the hyperb olic case; the generaliza tion to the elliptic case is surprisingly easy , a s will b e shown later on. Plan. In Section II A, w e give a heuris tic a rgument motiv ating a g eneralization of the BO equation that de- scrib es c o upled no nlinear wa ves propa gating in oppo- site directions, a nd we present this so-ca lled nonchiral IL W equation in Section I I B . Our quantum results can be found in Section I I I: First, the relation betw ee n the quantum elliptic CMS model and a quantum version of the nonchiral IL W equation is presented (Section I II A); second, a detailed motiv ation of our pro po sal that the quantum nonchiral IL W equation can describ e nonlinear wa ves propa g ating on tw o opp osite edges of a FQHE sys- tem b oundary and taking in to account in tera ctions b e- t ween different edges is given, including a review of the relev ant bac kground (Section II I B). Results that prov e that the classica l nonchiral IL W equa tions are exa ctly solv able can b e found in Sections IV (hyper b o lic case) and V (elliptic case). In Section VI, we shortly r ecall the application of the BO- and IL W equations to nonlinear water wav es, and we presen t a simple physical argument suggesting that the nonchiral IL W equation is also rel- ev ant in that context. W e conclude with final remarks in Sec tio n VI I. Appendix A pro vides mathematical de- tails, and Appendix B shortly explains numerical compu- tations we p e rformed to test o ur exact ana lytic s olutions. II. CLASSICAL PHYSICS DESCRIPTION As explained in the Section I I I, we discovered the quan- tum version o f the nonchiral I L W in the context of the F QHE. Ho wever, for simplicity , we first present in this section a simpler heur istic a rgument on the cla ssical level which lea ds to the cla ssical version of this equation. As elab orated in Section VI in o ne e x ample, this heuristic argument can b e straightforw ardly adapted to other sit- uations, sugges ting that the nonchiral IL W will also find other applica tions in physics. 2 A. Heuristic motiv ation The CMS mo dels can b e de fined by Newton’s equa tio ns ¨ z j = − N X k 6 = j 4 V ′ ( z j − z k ) ( j = 1 , . . . , N ) (1) where the tw o-b o dy interaction p otential is V ( r ) = r − 2 in the rational case and V ( r ) = ( π /L ) 2 sin − 2 ( π r/ L ), L > 0 , in the trigonometr ic ca se 21 (the arg ument s in this paragr aph apply to b oth case s). 28 Eq. (1) describ es an arbitr a ry num b er, N , of in ter acting par ticles with p o- sitions z j ≡ z j ( t ) at time t . While one often res tr icts to real p ositions when interpreting the CMS mo del as a dynamical system, one has to allow for complex z j when studying the relation to the BO equa tion; 15,16 this gener- alization preser ves the in teg r ability . 29 The CMS mo del is inv ariant under the par it y transfor ma tion P : z j → − z j for all j . Ho wev er , the co rresp onding BO equation is not parity-in v a riant: it is given b y u t + 2 uu x + H u xx = 0, where u ≡ u ( x, t ) a nd H is the Hilb ert transfor m (in the rational case, ( H f )( x ) = (1 / π ) − R R ( x ′ − x ) − 1 f ( x ′ ) dx ′ , where − R denotes the usual C a uch y principle v alue inte- gral), a nd under the parity transformation P : u ( x, t ) → u ( − x, t ) ≡ v ( x, t ), it changes to v t − 2 v v x − H v xx = 0. This mismatch of sy mmetry is pa radoxical at first sight, but the paradox is resolved by interpreting u as a w av e propaga ting on one edge of a FQHE sys tem and noting that, in general, there is another edge far aw ay carrying another w ave v . Thus, a ctually , the rational CMS mo del corres p o nds to tw o uncoupled BO equations for u and v . This system of equa tions is in v ar iant under a parity transformatio n in terchanging u and v , P : [ u ( x, t ) , v ( x, t )] → [ v ( − x, t ) , u ( − x, t )] . (2) It is p eculiar that these tw o BO equations are uncoupled, and it is for this reason that one can r educe the system to a single equation, ignoring the other. While this un- coupling is r easonable if the t wo edges are infinitely far apart, it is natural to ask what would happ en if the tw o edges are parallel and clo se together; see Fig. 1 . In this case, one would exp ect that the nonlinear wav es pro pa- gating on the tw o edges interact. W e now give a simple heuristic argument to suggest that the hyperb olic CMS mo del ca n describ e this situation. v u x L y FIG. 1. Schematic picture of a narrow FQHE system with tw o edges carrying the t wo nonlinear wa ves u ( x, t ) and v ( x , t ). The hyperb olic CMS mo del can b e defined by New- ton’s equatio ns (1) with the in teraction potential V ( r ) = X n ∈ Z 1 ( r + 2i δ n ) 2 =  π 2 δ  2 sinh − 2  π 2 δ r  , (3) where δ > 0 is an a rbitrary leng th par ameter. D ividing the particle positio ns z j int o t wo groups and shifting the ones in the second group by the imaginary half-p erio d, w k ≡ z k − N 1 + i δ for k = 1 , . . . , N 2 ≡ N − N 1 , with 1 < N 1 < N , w e can wr ite these Newton’s equations as ¨ z j = − N 1 X j ′ 6 = j 4 V ′ ( z j − z j ′ ) − N 2 X k =1 4 ˜ V ′ ( z j − w k ) , ¨ w k = − N 2 X k ′ 6 = k 4 V ′ ( w k − w k ′ ) − N 1 X j =1 4 ˜ V ′ ( w k − z j ) (4) for j = 1 , . . . , N 1 and k = 1 , . . . , N 2 , with ˜ V ( r ) ≡ V ( r − i δ ) = −  π 2 δ  2 cosh − 2  π 2 δ r  . (5) This can be in terpr eted as a model of t wo kinds of par- ticles, z j and w k , in which particles of the same kind int eract via the singular repulsiv e t wo-b o dy p otential V , wherea s pa rticles of different kinds interact via the weakly attractive nonsingular p otential ˜ V . W e interpret δ as a parameter of the same order of magnitude as the distance L y betw een the t wo edges of the F Q HE s y stem; see Fig. 1. In the rational limit δ → ∞ , we hav e ˜ V → 0, so par ticles of differen t t yp es do not interact and the tw o corres p o nding so liton equations for u and v decouple; for finite δ , the system is coupled. B. Nonchiral IL W e quation. In the hyperb olic case, the tw o -comp onent gener aliza- tion of the BO equation we present in this pa p er is giv en by u t + 2 uu x + T u xx + ˜ T v xx = 0 , v t − 2 v v x − T v xx − ˜ T u xx = 0 (6) for u = u ( x, t ) a nd v = v ( x, t ), with ( T f )( x ) ≡ 1 2 δ − Z R coth  π 2 δ ( x ′ − x )  f ( x ′ ) dx ′ , ( ˜ T f )( x ) ≡ 1 2 δ Z R tanh  π 2 δ ( x ′ − x )  f ( x ′ ) dx ′ . (7) The s tandard IL W equatio n is u t + 2 uu x + T u xx = 0; 30–32 it reduces to the BO equation in the limit δ → ∞ . Thus, if one dro ps the ˜ T -terms, (6) corres po nds to a system of uncoupled IL W equations generalizing the system o f uncoupled B O e quations discussed above. How ever, due to the presenc e of the ˜ T -terms, the nonlinear wa ves u and v in teract. F o r this reason, and since equation (6) is inv ariant under the pa rity tra ns formation (2), we call it the nonchir al IL W e quation ; another motiv atio n for this name is its relation to a nonc hir al conformal field theory explained in Section II I A. 3 F or la ter refer ence, we also give the nonchiral version of the p er io dic IL W equation: 33 it is defined b y (6) but with the int egra l o per ators ( T f )( x ) = 1 π − Z L/ 2 − L/ 2 ζ 1 ( x ′ − x ) f ( x ′ ) dx ′ , ( ˜ T f )( x ) = 1 π Z L/ 2 − L/ 2 ζ 1 ( x ′ − x + i δ ) f ( x ′ ) dx ′ , (8) where ζ 1 ( z ) = π L lim M →∞ M X n = − M cot  π L ( z − 2i nδ )  (9) is equal to the W eierstras s elliptic ζ -function with p erio ds ( L, 2i δ ), up to a term pro po rtional to z . 34 T o see that the op erato rs in (8) are natural p erio dic generalizations of the ones in (7), we recall that π 2 δ coth  π 2 δ z  = lim M →∞ M X n = − M 1 z − 2i δ n . (10) II I. QUANTUM PHYSICS DESCRIPTION It is kno wn that the edge excita tio ns in a F Q HE sys- tem can b e des crib ed by a conformal field theo r y (CFT) of chiral b osons, 35 and that this CFT a ccommo dates a quantum v ersion of the BO equation 10,14 which, at the same time, pr ovides a second quantization of the trigono- metric CMS sys tem. 3–9 This CFT is a nonlinear , ex- actly so lv able system that can desc rib e universal fea- tures of FQHE physics; in pa r ticular, as prop osed by Wiegmann, 12 this descr iption implies that the dynam- ics of FQHE e dge states is essential ly nonline ar, and it fe atur es fr actional ly char ge d solitons with cha r ges deter- mine d by the fil ling level, ν . In this section, we explain how these results generalize to the elliptic case, and how this led us to the nonchi- ral IL W equatio n (Section I I I A). W e also substantiate our pro po sal that the (qua nt um version of the) no nchiral IL W equation ca n describ e the in tera ction of nonlinear wa ves o n t wo edg es in a F QHE system, ta k ing into ac- count in teredge effects (Section II I B). This sectio n c a n be skipp ed without los s of co n tinuit y . A. CFT and nonchiral IL W equation The (quantum) elliptic CMS system is defined b y the Hamiltonian H N ( x ) = − 1 2 N X j =1 ∂ 2 ∂ x 2 j + X 1 ≤ j 0 is the coupling constant, and ( g − 1) is to b e interpreted as ( g − ~ ), i.e., g ( g − 1) → g 2 in the classical limit. Thu s, for g = 2, the Hamiltonian in (11) defines the quan tum analogue of the classica l model defined by Newton’s equations in (1 ) for V ( x ) = ℘ 1 ( x ). It is imp ortant to note that g is an essential parameter in the quantum cas e, differen t from the cla s sical ca se where w e c an set g = 2 without lo ss of generality . 28 The CFT corr esp onding to the elliptic CMS system can b e defined by tw o chiral b oso n o per ators ρ 0 ( x ) (r ight- mov ers ) and σ 0 ( x ) (left-movers) lab eled b y a co o rdinate x ∈ [ − L/ 2 , L/ 2] o n the circle with circumference L > 0 and satisfying the the commutator relations [ ρ 0 ( x ) , ρ 0 ( x ′ )] = − 2 π i ν ∂ x δ ( x − x ′ ) , [ σ 0 ( x ) , σ 0 ( x ′ )] = 2 π i ν ∂ x δ ( x − x ′ ) , (13) and [ ρ 0 ( x ) , σ 0 ( x ′ )] = 0, with ν the filling factor o f the F QHE system; 35 the latter ca n b e ident ified with the inv erse of the coupling parameter in the cor resp onding CMS Hamiltonian: ν = 1 /g . 26,27 F or s implicit y , we re- strict our discussion to FQHE states where g = 3 , 5 , . . . , even thoug h the mathematica l results discuss ed here hold true for arbitrar y (rationa l) g > 0 ; 36 we us e the subscript 0 to distinguish these ba re fields from dressed b oson fields ρ ( x ) and σ ( x ) obtained from them by a Bogoliub ov trans - formation, a s describ ed below. The linear dynamics of these fields is given by the Hamiltonian (in this section a nd only here, we write R short for R L/ 2 − L/ 2 , to simplify nota tio n) H 2 = g 4 π Z dx :  ρ 0 ( x ) 2 + σ 0 ( x ) 2 + Z dx ′ h U 2 ( x − x ′ )[ ρ 0 ( x ) ρ 0 ( x ′ ) + σ 0 ( x ) σ 0 ( x ′ )] − U 1 ( x − x ′ ) ρ 0 ( x ) σ 0 ( x ′ ) i : (14) with colo ns indicating normal order ing and U j ( x ) = ∞ X n =1 4 q j n 1 − q 2 n cos(2 π nx/L ) ( j = 1 , 2) (15) int eraction p otentials deter mined by the para meter q = e − 2 π δ/L ( δ > 0) . (16) The op era tor H 2 is a sp ecia l case of a Luttinger Hamil- tonian which, a s is w ell- k nown, can b e diagona liz ed b y a Bogoliub ov transfor ma tion. 37 This case is specia l in that 4 the Bogo liub ov tra ns formed Hamiltonian has the same form as for q = 0, exc e pt that the bare field oper ators are r eplaced by Bogoliub ov transformed ones: 27 H 2 = g 4 π Z dx :  ρ 2 + σ 2  : . (17) This is a co nsequence of the sp ecial form o f the inter- actions in (15), and it corresp onds to the fact that the Bogoliub ov transformed fields ρ and σ pr ovide tw o com- m uting r epresentations of the Vir a soro alg ebra by the Sugaw a r a constructio n, as in the s pecia l case q = 0 where this is ob vious; this is a manifestation of the fact that we are dealing with a no nch iral CFT (see e.g. Ref. [38 and 39] for background on CFT). Ho wev er, for nonzero q , the bare v acuum | 0 i is not a highest w eight state for the dressed fields ρ and σ , and this has importa n t co nse- quences. The CFT describ ed a bove accommo dates the following t wo kinds of vertex oper ators, φ ( x ) = : e − i g R x ρ ( x ′ )d x ′ : , ˜ φ ( x ) = : e i g R x σ ( x ′ )d x ′ : . (18) Moreov er, using the b o son opera to rs ab ov e, one can con- struct a self-adjo int op erato r, H 3 , pro viding a second quantization of the elliptic CMS mo del in the following sense: this o pe rator sa tis fie s the r elations [ H 3 , φ ( x 1 ) · · · φ ( x N )] | 0 i = H N ( x ) φ ( x 1 ) · · · φ ( x N ) | 0 i , (19) for ar bitrary par ticle num b er N . 26,27 W e recently obs e rved that it is poss ible to generaliz e H 3 so that one has r elations s imilar to the ones in (19) also for the v ertex op erators ˜ φ . 40 This generalize d op er- ator can b e wr itten as H 3 = Z :  g 2 12 π  ρ 3 + σ 3  + g ( g − 1) 8 π ×  ρT ρ x + σ T σ x + ρ ˜ T σ x + σ ˜ T ρ x   : dx (20) with the integral op erato r s T , ˜ T in (8)–(9). Th us, the op erator H 3 defines the following q uantum version o f the per io dic nonchiral IL W-equa tion, ˆ u t + 2 : ˆ u ˆ u x : + 1 2 ( g − 1)[ T ˆ u xx + ˜ T ˆ v xx ] = 0 , ˆ v t − 2 : ˆ v ˆ v x : − 1 2 ( g − 1)[ T ˆ v xx + ˜ T ˆ u xx ] = 0 . (21) T o s ee this, we compute the Heisenberg equations o f mo- tion A t = i[ H 3 , A ] for A = ρ, σ and res c ale, ρ → ˆ u ≡ g ρ/ 2 and σ → ˆ v ≡ g σ / 2, to obtain (21). Mor eov er , b y taking the classical limit where the b oso n opera tors ( ˆ u, ˆ v ) b e- come functions ( u, v ) and ( g − 1) is replaced by g , and sp ecializing to g = 2, (21) reduces to (6). It is interesting to note that the ope r ator in (20) satis- fies the following generaliz ation of (19), allowing for b oth kinds of vertex opera tors, φ a nd ˜ φ , at the same time: [ H 3 , φ ( x 1 ) · · · φ ( x N 1 ) ˜ φ ( ˜ x 1 ) · · · ˜ φ ( ˜ x N 2 )] | 0 i = H N 1 ,N 2 ( x , ˜ x ) φ ( x 1 ) · · · ˜ φ ( ˜ x N 2 ) | 0 i (22) where H N 1 ,N 2 ( x , ˜ x ) = H N 1 ( x ) + H N 2 ( ˜ x ) + N 1 X j =1 N 2 X k =1 g ( g − 1) ℘ 1 ( x j − ˜ x k + i δ ) , (2 3) for arbitra r y particle n umbers N 1 , N 2 . This is a ge neral- ization of the elliptic CMS Hamilto nia n (11) describing t wo types of particles, where par ticles of the same t yp e int eract with the s ing ular t wo-bo dy p otential ℘ 1 ( x ), and particles of different types interact with the nonsingu- lar attractive p otential ℘ 1 ( x + i δ ). It can b e obtained from a standar d elliptic CMS Hamiltonian (11) by di- viding the particles int o tw o gr oups and shifting the p o- sitions in o ne group by i δ , similarly as in the classical case discussed in Section I I A; s ee (4) ff . This argument prov e s that the Hamiltonian H N 1 ,N 2 defines a qua nt um int egrable mo del. Ho wev er , the physically relev ant eigen- functions of H N 1 ,N 2 can not b e obtained from the ones of the corre s po nding standar d elliptic CMS Hamiltonian by this shift tr ick. This suggests that the generalized model can describe new ph ysics which would b e interesting to explore, but this is b eyond the sco pe o f the pres en t pap er. W e finally men tion that, to g enerate the full Hilber t space of t he CFT, one needs to co nsider tw o further kinds of v ertex op erators representing ho le ex c itations, and there is a generalization of the result in (2 3) a llow- ing for ar bitrary num b ers, N 1 , M 1 , N 2 , M 2 , o f all four t yp es of vertex op era tors and with an interesting cor - resp onding Hamilto nian H N 1 ,M 1 ,N 2 ,M 2 , 40 in ge ne r aliza- tion of a known r esult in the trigonometr ic ca se. 9 Thu s, H 3 is a ctually the second quantization o f these opera tors H N 1 ,M 1 ,N 2 ,M 2 generalizing the elliptic CMS Hamilto nia n. B. Nonchiral CFT and FQHE W e motiv a te and explain o ur pro p o s al that the nonchi- ral IL W equatio n can describ e the interactions of nonlin- ear wav es propag ating on the tw o bo undaries of a narrow F QHE s ystem, in generaliza tion of pr evious pr op osals for F QHE systems wher e the b oundaries a re w ell-separ ated and interboundar y interactions can b e ignored. 12 T o pre- pare for this, we review known facts about the FQHE, bo sonization and quantum h y dr o dynamics. 1. Pr oje ction to l owest L andau level W e recall the quantum mechanical descr iption of a charged particle confined to the xy - plane in the presence 5 of a constant magnetic o rthogona l to the plane (Lan- dau problem): Assuming p erio dic b oundary conditions in the x -dire ction: − L/ 2 ≤ x ≤ L/ 2 with L = L x > 0, and y ∈ R , the exact eigenfunctions in the lowest Landau level (LLL) hav e the form ψ k ( x, y ) = e i kx e − ( y − k ) 2 / 2 , (24) using the Landau gauge and units where the magnetic length is se t to 1, with k (short for k x ) an ar bitrary inte- ger multiple of 2 π / L . In such a s tate, the particle has the behavior of a plane- wa ve in the x -directio n but is well- lo calized in the y -direction, and the quantum num b er k therefore has a tw o-fold physical int erpretatio n: it can b e int erpreted as momentum in x -direction and, at the same time, it c o rresp onds to the loc ation of the wa ve packet in y -directio n. As is well-known, the wa ve functions in the LLL are all degener ate: the ener gy is k -indep endent . W e now consider the situation where, in addition to the magnetic field, we also have a p o tent ial, V conf ( y ), confin- ing the charged particle to a region − L y / 2 ≤ y ≤ L y / 2 for some L y > 0; this p otential is zero at pos itio ns y far- ther a wa y than some distance ℓ b > 0 from the b oundary: V conf ( y ) = 0 for | y ∓ L y / 2 | > ℓ b , and it g rows smo othly to very la rge v a lues in the b ounda ry regions | y ∓ L y / 2 | < ℓ b . In this situation, the degeneracy of the eigenfunctions in the LLL is lifted, a nd the energ y E 0 ( k ) of the par ticle as a function of k is qualitatively similar to the function V conf ( y ) with y identified with k ; se e Fig. 2. Thus, to describ e noninteracting such particles pr o jected to the LLL, one can use the quantum many-bo dy Hamiltonian H LLL = X k ( E 0 ( k ) − µ ) ˆ ψ † ( k ) ˆ ψ ( k ) (25) with fermion field op era to rs ˆ ψ ( † ) ( k ) ob eying canonical anticomm utator relations, { ˆ ψ ( k ) , ˆ ψ † ( k ′ ) } = δ k,k ′ etc. ( µ is the chemical po ten tial). By symmetry , we ca n a s sume E 0 ( − k ) = E 0 ( k ). y , k Energy E n ( k ) lo w er edge upp er edge E 2 ( k ) E 1 ( k ) E 0 ( k ) µ − k F k F FIG. 2. Sc h ematic p icture of the low est Land au lev el E 0 ( k ) in the presence of a p otentia l confining t h e c harged particles to a region − L y / 2 < y < L y / 2, as illustrated in Fig. 1 . The grey lines indicate h igher Landau lev els that we ignore. Thu s, even though w e cons ider a t wo dimensional sys- tem, it is mo delled by a o ne-dimensional Hamilto nia n that can be trea ted by the b os onization metho d pio - neered by Haldane. 41 This b osonized description is useful since it allows to find int eractions tha t ca n be a dded to the Hamiltonian witho ut sp oiling integrabilit y; as dis- cussed in the introduction, such interactions ar e pa rticu- larly interesting in the context of FQHE physics. 2. Bosonization W e recall so me p er tinen t facts a bo ut b osonizatio n. 41,42 Consider the free fermion mode l defined b y the Hamilto- nian (25). Its gro undstate is the Dirac sea wher e all states − k F < k < k F are filled and all others a re empty , with the F er mi momentum k F > 0 determined by E 0 ( k F ) = 0. It is convenien t to decomp os e the (inv er se) F ourier trans- form of the fermion field, ψ ( x ) = P k (2 π /L ) ˆ ψ ( k )e i kx , as follows, ψ ( x ) = ψ + ( x )e i k F x + ψ − ( x )e − i k F x (26) with fermion field op era tors ψ ± ( x ) repre s ent ing the lo w- energy excitations in the vicinity of the F er mi surface po int s ± k F . As explaine d in Section I I I B 1, these F er mi surface points ca n b e identified with the t wo b oundar ies, y = ± L y / 2, o f a FQHE system, as illustrated in Fig. 1. The fermion fields o n the RHS in (26) can b e r epre- sented b y vertex oper ators, ψ ± ( x ) = : e ∓ i R x ρ ± ( x ′ )d x ′ : (27) where ρ ± ( x ) are op erator s satisfying the c ommut ator re- lations o f c hiral b os ons, [ ρ ± ( x ) , ρ ± ( x ′ )] = ∓ 2 π i ∂ x δ ( x − x ′ ) (28) and [ ρ + ( x ) , ρ − ( x ′ )] = 0. These b oson op era tors ca n be ident ified w ith the corresp onding fermion densities, ρ ± ( x ) = 2 π : ψ † ± ( x ) ψ ± ( x ) : . (29) W e note in passing that the b oso n fields ρ + ( x ) a nd ρ − ( x ) are e q ual, up to a factor √ g and zero mo de deta ils, 9 to the bare b os on fields ρ 0 ( x ) and σ 0 ( x ), resp ectively; s ee Section I I I A. By T aylor-expanding the dispers io n relation in the vicinity of the F er mi surface points: E 0 ( ∓ k F + k ) = ± v F k + k 2 2 m ∗ + . . . (30) with the F ermi v elo city v F = E ′ 0 ( k F ) a nd the effective mass m ∗ = 1 /E ′′ 0 ( k F ), one can expand H LLL = v F ( H (0) 2 , + + H (0) 2 , − ) + 1 m ∗ ( H (0) 3 , + + H (0) 3 , − ) + . . . (31) with H (0) 2 , ± = 1 4 π Z : ρ ± ( x ) 2 : dx, H (0) 3 , ± = 1 12 π Z : ρ ± ( x ) 3 : dx (32) etc. This provides a bas is for the quantum hydro dy- namic description of such systems prop os e d b y Abanov and Wiegmann. 10 6 3. Chir al Luttinger liquids and FQHE W e reca ll W en’s chiral Luttinger liquid desc ription of F QHE s ystems. 35 The lea ding term in (31), H (0) 2 = v F ( H (0) 2 , + + H (0) 2 , − ) , (33) provides a g o o d s ta rting po int to des crib e FQHE sys- tems, but the low-energy excita tions are not fermions but rather collective excita tions tha t can b e describ ed by v er tex op era tors φ ± ( x ) = : e ∓ i √ g R x ρ ± ( x ′ )d x ′ : (34) with g = 3 , 5 , . . . at filling level ν = 1 /g ; the fermion case g = 1 corr e spo nds to the integer Hall effect and, for g > 1, the vertex op era tors (3 4) describ e co mpo site fer mions. If the tw o b oundar ie s are far apar t, it is natural to a s sume that the lo w-energy excita tions at distinct b oundaries do not interact, and one can r e strict the discussion to one bo undary or, e q uiv alently , to one chiral sector, + or − . This is W e n’s chiral Luttinger liquid mo del. 35 4. Boundary waves in FQHE systems The dynamics o f the bo son fields provided by the Hamiltonian H (0) 2 via the Heisenber g equations o f mo- tion is ∂ t ρ ± ± v F ∂ x ρ ± = 0 . (35) These linear equations descr ibe wav es pro pagating at the t wo b oundar ies of a FQHE system: 12 at each b oundar y , the wa ve pack ets mov e in o ne direction, rig ht (+) or left ( − ), with cons tant sp eed v F and without changing shap e. The Hamiltonian H (0) 2 is highly deg enerate, and it is natural to ask if one can lift this deg e neracy b y adding int eractions that fulfill the following r e quirements: (i) they do not sp oil integrability , (ii) they provide nonlin- ear co r rections to the linear wa ve eq uations (35), (iii) they are compatible with the vertex o pe r ators (34) de- scribing comp osite fermions. 12 An interesting Hamilto- nian obtained by adding such terms to H (0) 3 , ± (32) is 10 H 3 , ± = Z :  √ g 12 π ρ ± ( x ) 3 + g − 1 8 π ρ ± H ( ρ ± ) x  : dx (36) with the Hilb ert trans fo rm H (obtained from T in (8 ) by taking the limit δ → ∞ ): the dynamics for the bo son fields provided by this Hamiltonian is a quantum ver- sion of the BO equation which is in teg r able, 10 and this Hamiltonian is compatible with the comp osite fermion op erators in that it also provides a second quan tization of trigonometric CMS model; 3–9 using the latter and the known eigenfunctions o f the trigonometric CMS sy s tem, one can construct the exact eigenstates of H 3 , ± . 9 5. Pr op osal W e now are ready to mo tiv a te and expla in our pro- po sal that the nonchiral IL W equation can descr ib e wa ves propaga ting on para llel bo undaries o f FQHE systems. W e recall that, in g eneric applicatio ns of b oso niz a - tion, the mo st impo rtant interactions to b e added to the Hamiltonian H (0) 2 are quadr atic in the boson o p- erator, and thus, generically , one obtains a Luttinger Hamiltonian a s in (14), for so me p otentials, U 2 ( x ) and U 1 ( x ); 41 these p otentials describ e interactions b etw een the same ( U 2 ) and opp osite ( U 1 ) chiral degrees of free- dom. Moreover, one often assumes that these interac- tions a re lo cal since this guara nt ees that the resulting mo del is conforma lly inv aria n t. In the context o f the F QHE, such Luttinger interactions are usua lly ignored by the following a rguments: (i) the t wo c hiral degre es of freedom describ e e x citations at tw o separa ted b oundaries of the system, and U 2 ( x ) ther efore describ es int eredge in- teractions which a re negligible if the b o undaries are s uffi- ciently fa r apart; (ii) a loca l interaction U 1 ( x ) within the same boundary only renor malizes the F ermi velocity a nd th us can be taken into account b y redefining v F . How- ever, since the one-particle eigenfunctions of the La ndau Hamiltonian are spatia lly ex tended, and Co ulo mb in ter- actions in a FQHE sy stem are long- r ange, there is no reason to ex clude nonlo cal interactions which pres e rve conformal symmetry . Mor eov er , it is known that trans- po rt co efficients in Luttinger liquid a r e universal even if the interactions mix the chiral degrees of freedo m and are nonlo cal, 43 i.e., the ac c urate qua n tization o f the Hall con- ductance observed in real FQHE s y stems is compatible with generic Luttinger mo del in teractions; see Ref. [44] for a recent construction of the p ertinent genera l Lut- tinger mo del for g eneral vertex ope r ators as in (34). As discussed in Section II I A, the Luttinger Hamilto- nian (14) with the fine-tuned in ter actions in (15) is con- formally inv ar iant, for arbitrary fixed δ > 0, and ther e are natur al co rresp onding gene r alizations of the comp os - ite fermio n op era tors and the op era to r in (36 ) sa tis fy- ing the requirements stated in Section II I B 4 : they are given in (18) a nd (20), r esp ectively . Our prop osa l to mo del a FQHE system at filling 1 / g , g = 3 , 5 , . . . , and with the geometry illustrated in Fig. 1 is therefore a s fol- lows: The b oson field op er ators ρ 0 and σ 0 , satisfying the c ommutator r elations in (13) , describ e low-ener gy ex cita- tions lo c ate d at the upp er and lower e dge, r esp e ctively, of the F Q HE system b oundary; the low-ener gy description of the system is by t he Hamiltonian H 2 in (14) – (15) , with the p ar ameters v F and δ determine d by system de- tails lik e the e dge distanc e, L y , and the c onfin ing p oten- tial, V conf ( y ) ; the line ar- and nonline ar dynamics of the b oundary waves is describ e d by the op er ators in (14) and (20) , r esp e ctively; the vert ex op er ators in ( 18) describ e quasip article excitations in the s ystem. 7 6. Inter-e dge effe cts in FQHE systems W e arg ue that the mo del propo sed in Section II I B 5 can descr ibe in teredge effects in narrow F QHE system. Our mo de l pr edicts that the quasiparticles of the sys- tem are the Bogoliubov transformed bo son fields, ρ and σ , diago nalizing H 2 in (14); see (17). Thu s, the linear dynamics is g iven by the same equations as for δ = ∞ , i.e., ρ t − v F ρ x = 0 and σ t + v F σ x = 0. Howev er, since ρ (say 45 ) is a s uper po sitions o f the fields ρ 0 and σ 0 lo calized at tw o dis tinct b oundar ies, a righ t-moving wa ve excited at the upper edge will alwa ys dev elo p in to a pair of w ell- defined corresp onding excita tio ns a t b oth edges mo ving in para llel. Thus, our pr op osal can b e tested already in exp eriments on rea l F QHE systems that can only r esolve line ar b oundary wa ves: our mo del predicts corr esp ond- ing ex citations, u 0 ( x − v F t ) and v 0 ( x − v F t ) pro po rtional to the exp ectatio n v alues of ρ 0 ( x, t ) and σ 0 ( x, t ), resp ec- tively , where u 0 ( x ) and v 0 ( x ) are deter mined by a single function, u ( x ), and the in verse of the Bogoliub ov trans- formation describ ed in Sectio n I I I A ( u ( x ) is prop or tional to the exp ectation v alue of ρ ( x, t = 0 )). F urthermore , o ur mo del pr edicts nonlinea r wav es desc r ib e d by the quan- tum nonchiral IL W equatio n (21), in g eneralizatio n of the Wiegma nn prop osa l quoted in the b eginning of this section. 12 It would b e interesting to elab orate these pr e- dictions in detail, and to prop ose sp ecific exp eriments on real FQHE s y stems to test them. Clearly , this is a resear ch pro ject in its o wn. Our r esults in Sections IV and (V) are a first step, g iving an indication of the new ph ysics that the nonchiral IL W equation can describ e. T o elabo rate predictions of our mo del, it would b e in- teresting to construct the exa c t eigenstates of the Hamil- tonian H 3 in (20), in genera lization of known results for δ = ∞ . 9 This is challenging. One reason is that, while the exact eigenstates of the trigo nometric CMS mo del hav e been known for a long time, the ones of the rele- v ant elliptic CMS-type sys tems are the sub ject of o ngoing resear ch. 46 IV. RESUL TS: HYPERBOLIC CASE A. Multisoli ton solutions The following fundamental r esult shows that (6) ad- mits multisoliton solutions whose dy na mics is described by the hyperb olic CMS mo del, thus g eneralizing a fa - mous result for the ra tio nal case: 47 F or arbitr ary inte gers N ≥ 1 and c omplex p ar ameters a j with imaginary p arts in the r ange δ / 2 < Im a j < 3 δ / 2 for j = 1 , . . . , N , t he fol- lowing is an exact sol ution of the nonchir al IL W e quation (6) :  u ( x, t ) v ( x, t )  = i N X j =1  α ( x − z j ( t ) − i δ / 2) − α ( x − z j ( t ) + i δ / 2)  + c . c . (37) wher e α ( x ) = ( π/ 2 δ ) coth( π x/ 2 δ ) and the p oles z j ( t ) ar e determine d by Newton ’s e quations (1) with V ( r ) given by (3) and with initial c onditions z j (0) = a j and ˙ z j (0) = 2i N X j ′ 6 = j α ( a j − a j ′ ) − 2i N X k =1 α ( a j − a k + i δ ) (38) (the bar denotes co mplex co njuga tion, c.c.). Thu s, to obtain an exact solution of (6), one choo ses complex pa- rameters a j satisfying δ / 2 < Im a j < 3 δ / 2; next, the time-evolution of z j ( t ) is obtaine d by so lving the hyper- bo lic CMS mo del with initial conditions determined by the a j ; fina lly , the solution of (6) is obtained from (37). Using the exact analy tic solution of the hyperb olic CMS mo del obtained by the pr o jection metho d, 21 the n umeri- cal effort to compute such an mult isoliton solutio n a t an arbitrar y time, t , is reduced to diago na lizing an explicitly known N × N matrix. As elabor ated in Appendix B, we tested this result by comparing with a numeric solution of (6). u , v x FIG. 3. Time ev olution of a tw o-soliton solution of the nonchiral IL W eq uation (6) with a u -channel dominated soliton (big blue an d small red humps) colliding with a v -channel dominated soliton (big red and small blue humps), as explained in the main tex t . The plots show u ( x, t ) (blue line) and v ( x, t ) (red line) at su ccessiv e times t = ( n − 1) t 0 , n = 1 , . . . , 5; the p arameters are δ = π , a 1 = − 4 + 1 . 2i δ , a 2 = 3 + 0 . 85i δ , and t 0 = 2 . 25. B. Example s. The o ne-soliton solution of (6) is given by  u ( x, t ) v ( x, t )  = i  α ( x − z ( t ) − i δ / 2 ) − α ( x − z ( t ) + i δ / 2)  + c . c ., (39) where the poles e volve linearly in time, with initial con- ditions determined b y a complex parameter a such that 8 y t/t 0 x x FIG. 4. (a ) Time evolution of th e p oles z j ( t ), j = 1 , 2, in the complex plane corresp onding to the t wo -soliton solution in Fig. 3. The times t = ( n − 1) t 0 , n = 1 , . . . , 5, defined in the caption of Fig. 3 are indicated by circles; th e arro ws mark circles correspond ing to n = 1. The dotted lines indicate the evol ution of p oles without interacti ons. ( b) Time evo lution of the cen t er-of-mass locations of the solitons giv en by Re z j ( t ). δ / 2 < Im a ≤ 3 δ / 2, z ( t ) = a + ˙ z (0) t, ˙ z (0 ) = 2i α ( a − ¯ a + i δ ) . (40) It is imp ortant to note that ˙ z (0) is real, a nd therefore, Im z ( t ) = Im a indep endent of t . Thus, the functions u ( x, t ) and v ( x, t ) b oth descr ibe h umps whose sha pes do not change with time. These hu mps are ce ntered at the same p oint and move with constant velocity , Re z ( t ) = Re a + ˙ z (0) t , and their heights, max u > 0 and max v > 0, are determined by Im a . F or Im a close to 3 δ / 2, ma x u ≫ max v , a nd the solito ns mov e to the r ight, ˙ z (0) > 0. As Im a de c r eases, max u and ˙ z (0 ) decrease while max v incr eases un til, at Im a = δ , max u = max v and ˙ z (0) = 0. Thus, if Im a lies in the range δ < Im a < 3 δ / 2, then the one-soliton is mainly in the u -channel and mov es to the r ight; it is there- fore similar to the one-solito n s olution of the standard IL W equation u t + 2 uu x + T u xx = 0. Similarly , when δ / 2 < Im a < δ , the o ne-soliton is mainly in the v -channel and mo ves to the left, similar to a one-so liton so lution o f the P -tra nsformed IL W equation v t − 2 v v x − T v xx = 0. F or para meters a j such that Re ( a j − a k ) ≫ δ for all j 6 = k , the multisoliton s olution o f (6) is well- approximated by a sum of N one-s olitons of the form (39) where ˙ z j ( t ) ≈ 2i α ( a j − ¯ a j + i δ ) is time-indep endent for times such that Re ( z j ( t ) − z k ( t )) ≫ δ ; see Fig. 3 for a tw o -soliton solution, with the corr esp onding motion of po les in Fig . 4 (a). How ever, when tw o so litons meet, they int eract in a nontrivial wa y , a nd a fter the interaction they re-emerg e with the same shap e but with phase-shifts; see Fig. 4(b). Suc h nontrivial interactions betw ee n solitons can also b e mo deled by the sys tem of deco upled IL W equations obtained from (6) b y dro pping the ˜ T -terms. A qualitatively new effect stemming from the ˜ T -terms is that u -channel dominated solitons ( u -so litons) interact nontrivially with v -solitons, a s clearly seen in our exam- ple in Figs. 3 a nd 4 . It is interesting to note that the po les corresp onding to the u - and v -solito ns interc ha nge their ima ginary parts and directions during the collision and th us, in this sense, exchange their iden tities: while the first p ole corr esp onds to the u -soliton and the sec- ond to the v -soliton b efor e the c ollision, it is the other wa y round a fter the collision; see Figs. 4(a) and (b). W e note that such an identit y change of p o le s dur ing soliton collisions is k nown for the BO equation, 48 but only for solitons moving in one direction. C. Deriv ation of multisoliton solutions. W e explain the k ey difference b e t ween the deriv ation of so litons for (6) and the cor resp onding der iv ation in the rational case; 47 further details can b e found in Ap- pendix A 1. The Hilber t tra nsform, H , satisfies H 2 = − I , and this prop erty is crucial for the ex is tence of eigenfunctions of H needed in the deriv ation of the CMS-related soliton solutions of the BO e quation u t + 2 uu x + H u xx = 0. 47 How ever, while the trigono metric g e ne r alization of H also has this prop erty , the h yp erb olic gener alization of H is the op erato r T in (7), and T 2 6 = − I . This is the reason why the soliton solution o f the BO equation straig htfor- wardly generaliz e s to the trigonometric case, 47 but the naive genera lization to the hyperb olic case fails. How- ever, the nonchiral IL W equation (6) ca n be written in vector form as u t + ( u . u ) x + T u xx = 0 , u ≡  u v  , u . u ≡  u 2 − v 2  , T ≡  T ˜ T − ˜ T − T  (41) where the matrix o pe r ator, T , s atisfies T 2 = − I . Mo re- ov er , ( α ( x + z ± i δ / 2) , − α ( x + z ∓ i δ / 2 )) t are eigenfunctions of T with eigenv alue s ± i. The latter are the eigenfunc- tions needed to b e able to use the metho d developed for the ra tional cas e: 47 using well-kno wn identities for the function α ( x ), 49 as well a s a B¨ a cklund transfor mation for the hyperb olic CMS mo del, 50 it is stra ightforw a rd to adapt a known der iv ation o f m ultisoliton solutions of the BO equa tion 47 to the hyperb olic case. D. Int egrability . W e found a Lax pair , a Hirota bilinear form, a B¨ acklund tr ansformation, and an infinite num b er of con- serv ation la ws for (6). Thus, the nonch iral IL W equation is a so liton equation that is integrable in the same s trong sense as the standard IL W equatio n. 31 Below w e prese n t some of these res ults that can be c heck ed by straightfor- ward computations. The Lax pair we found is a s fo llows: L et ψ ( z ; t, k ) b e an analytic function on the u nion of the strips 0 < Im z < δ and δ < Im z < 2 δ and exten de d to C by 2i δ -p erio dicity, ψ ± 0 ( x ; t, k ) and ψ ± δ ( x ; t, k ) the b oundary va lues of t his function on R and R + i δ , r esp e ctively, and µ 1 , µ 2 , ν 1 , and ν 2 arbitr ary functions of the sp e ctr al p ar ameter k . Then the c omp atibility of the fol lowing line ar e quations 9 yields (6) : (i ∂ x − u − µ 1 ) ψ − 0 = ν 1 ψ + 0 , (i ∂ x + v − µ 1 ) ψ + δ,x = ν 2 ψ − δ ,  i ∂ t − 2 µ 1 i ∂ x − ∂ 2 x + T u x + ˜ T v x ± i u x + µ 2  ψ ± 0 = 0 ,  i ∂ t − 2 µ 1 i ∂ x − ∂ 2 x + T v x + ˜ T u x ± i v x + µ 2  ψ ± δ = 0 . Inspired by known re s ults for the BO equation, 16 we obtained the following Hirota bilinear form of (6), (i D t − D 2 x ) F − · G + = (i D t − D 2 x ) F + · G − = 0 (42) with u = i ∂ x log( F − /G + ) and v = − i ∂ x log( F + /G − ), where F ± ( x, t ) ≡ F ( x ± i δ / 2 , t ) a nd similarly for G , us ing standard Hir ota deriv atives. 51 The fir st three o f the conserv atio n laws w e found ar e I 1 = Z R ( u + v )d x, I 2 = 1 2 Z R ( u 2 − v 2 )d x, I 3 = Z R  u 3 3 + uT u x 2 + u ˜ T v x 2 + ( u ↔ v )  d x (43) with ( u ↔ v ) s hort for the same three terms but with u and v in terchanged. B¨ acklund tr ansformations , other conse r v ation laws, and detailed deriv ations are given elsewhere. 36 V. RESUL T S: ELLIPTIC CASE T o ge ne r alize (6) to the per io dic setting, w e use the W eierstr a ss functions ℘ ( z ) and ζ ( z ) with p erio ds (2 ω 1 , 2 ω 2 ) ≡ ( L, 2i δ ), 34 L > 0, and the rela ted functions ζ j ( z ) ≡ ζ ( z ) − η j z / ω j , η j ≡ ζ ( ω j ), j = 1 , 2. The function ζ 1 ( z ) is L -p erio dic, ζ 1 ( z + L ) = ζ 1 ( z ), whereas the func- tion ζ 2 ( z ) is 2i δ -p e rio dic, ζ 2 ( z + 2i δ ) = ζ 2 ( z ); recall that ζ ( z ) is neither L - nor 2i δ -p erio dic. W e note that ℘ 1 ( x ) in (12) equals − ζ ′ 1 ( x ) = ℘ ( x ) + η 1 /ω 1 . The perio dic nonchiral IL W equation is given by (6) with the integral o per ators T , ˜ T in (8)–(9). With that, T in (41) s till satisfies T 2 = − I , and the der iv ation of the m ultisoliton equation outlined a b ove g eneralizes str a ight- forwardly to the elliptic ca se pr ovided α ( z ) in (A9) is chosen as the 2i δ - p er io dic v a riant of ζ ( z ): The functions u ( x, t ) and v ( x, t ) gi ven in (37) , with α ( x ) = ζ 2 ( z ) , sat- isfy the p erio dic nonchir al IL W e quation pr ovide d that z j ( t ) satisfy Newton ’s e quations (1) with the el liptic CMS mo del p otential V ( r ) = ℘ ( r ) , and with initial c onditions z j (0) = a j and ˙ z j (0) in (38) , for arbitr ary c omplex a j satisfying δ / 2 < Im a < 3 δ / 2 and − L/ 2 ≤ Re a j < L/ 2 , j = 1 , . . . , N . It is impo r tant to note that the m ultisoli- ton s o lution is L -perio dic even though ζ 2 ( z ) is not. The int erested rea der ca n find further deta ils in App endix A 2. VI. OTHER APPLICA TIONS W e present arguments suggesting that the nonchiral IL W equation introduced in this paper will find other a p- plications in physics be yond the applicatio n to the FQHE describ ed ea rlier (Section VI A). As a sp ecific example, we discuss a p ossible application in the context of nonlin- ear w a ter w av es , and thereb y provide a complementary ph ysical in ter pretation of our ma thematical results (Sec- tion VI B). A. The wide appli cability of soliton equations Nonlinear evolution equations are typically more dif- ficult to solve than linear ones, and theo retical physics to ols are o ften not eq ually p ow erful when no nlinea r ef- fects are impo rtant. Soliton equations are an imp or- tant exce ptio n: these nonlinear equations are int egra ble , and it is therefor e p ossible to dev elop a na lytic 52 and nu meric 53 metho ds to solve them reliably . Thus, phe- nomena desc r ib e d by soliton equa tions can b e v ery well understo o d despite of the crucial imp o rtance of nonlin- ear effects. The cla ss of suc h phenomena is remark ably large, with many e xamples fro m different ar eas in physics such as hydrodyna mics, nonlinear optics, plasma ph ysics, dislo cation theory of crystals , etc. A well-known expla - nation of this wide applicability of soliton equa tio ns is by Calo g ero: 54 c ertain “universal” nonline ar PDEs 55 c an b e obtaine d, by a limiting pr o c e dur e involving r esc alings and an asymptotic exp ans ion, fr om very lar ge classes of nonline ar evolution e quations [. . . ]. Be c ause this limiting pr o c e dure is the c orr e ct one to evinc e nonline ar effe cts, the universal mo del e qu ations obtaine d in this manner [. . . ] ar e widely applic able. B e c ause this limiting pr o- c e dur e gener al ly pr eserves inte gr ability, these u niversal mo del e quations ar e likely to b e inte gr able [. . . ]. This sugg ests that the no nch iral IL W (6) will find other applications in ph ysics. B. Nonlinear water wa ves Consider the following cla ss o f soliton equations de- scribing, e.g ., nonlinear water wa ves in different situa- tions: u t + 2 uu x + D u xx = 0 (44) where D is one of the linear op e rators sp ecified b e- low and u = u ( x, t ), where x is a co or dina te on one- dimensional spa ce a nd t time. This c la ss includes the famous Kortewe g-de V ries (KdV) equatio n, 56 the BO equation, 57–59 the IL W equation interpola ting b etw e en the K dV and the BO equa tions, 30 and p erio dic v ari- ants of these three equations 33 depe nding o n a fur ther parameter, L > 0, corr esp onding to the s patial p erio d: u ( x + L, t ) = u ( x, t ). While the nonlinear term, 2 uu x , is the same in all cases, the dispersive term, D u xx , is differen t: it amount s to multiplication of u by functions iΩ( k ) in F ourier space: D u xx = iΩ ( − i ∂ x ) u , with the following disp ers io n rela- 10 tions in the different case s, 60 Ω( k ) =      k 3 δ / 3 (KdV) k 2 sgn( k ) (BO) k 2 coth( k δ ) (IL W) (45) where the wa ve num b er, k , is re stricted to in teg e r m ulti- ples of 2 π /L in the perio dic cases (it is real otherwise), and δ > 0 is a constant . Note that, in p osition space, the opera tor D is represented by a different ial oper a tor in the KdV case, D f = δ ∂ x f / 3 , whereas in the B O - and IL W cases it is given by an integral op era tor deno ted as H (Hilb ert transfor m) and T , resp e ctively; see (A3). It is impo rtant to note tha t, in ge ne r al, one should add a term cu x to the LHS in (44), with c some v elo city pa- rameter, to make manifest that (44) is a generalization of the chiral w av e equa tion u t + cu x = 0 ; how ever, since this ter m is triv ial in that it ca n b e remov ed by a trans- formation u → u − c/ 2, we ignore it in our discussion. The s oliton equations in (44)– (45) provide effective de- scriptions of nonlinea r w ater wav es taking into accoun t the most imp ortant nonlinear and dissipative terms. 30 It is imp or tant to note that, when deriving these equa- tions from fundamental hydrodynamic laws, par it y in- v ariance is broken and, for this rea s on, the equa tio ns in (4 4)–(45) are chir al : they can only describ e solitons moving to the right. Obviously , o ne can obta in a corre- sp onding equation descr ibing solito ns moving to the le ft by a parity tr ansformation: v ( x, t ) ≡ u ( − x, t ) satisfies v t − 2 vv x − D v xx = 0. Thu s, the chiral equation in (44)– (45) ac tually cor resp onds to a system of t wo equations for u and v describing solito ns moving in b oth directions . Clearly , this desc ription is simplistic with regar d to the following: solita r y w aves in nature moving in opp osite directions interact when they meet, but such interactions are ignor ed by this uncoupled system for u and v . This suggests to try to find integrable generalizations of these equations o f the fo r m u t + 2 uu x + D u xx + X ( v , u ) = 0 , v t − 2 v v x − D v xx − X ( u, v ) = 0 (46) with coupling terms, X ( v , u ) and X ( u, v ), such that the system (46) is inv aria n t under the parity transformation in (2). W e b elieve that neither the K dV equation nor the BO equation a llow for such a coupling; how ever, the IL W equa tion do es: it is given by the disp ers ive term I ( v , u ) = ˜ D v xx = i ˜ Ω( − i ∂ x ) v (indep endent of u ) with ˜ Ω( k ) = k 2 sinh( k δ ) ; (47) indeed, using (A3), one sees tha t (46) in this ca se is equiv- alent to the nonchiral IL W equation (6) ff . One can c heck that (6) does not hav e a w ell- defined limit δ → 0, a nd that ˜ T u xx → 0 in the limit δ → ∞ : the KdV-limit of the nonc hiral IL W equation do es not exis t, and its BO -limit is trivia l. Thus, to describe nonchiral ph ysics, one has to work in the regime 0 < δ < ∞ . This s uggests that it would b e int eresting to revisit the deriv ation of the K dV-equation fr om more fundamen tal parity in v ariant equations, and to see if this can be gen- eralized so as to obtain the nonchiral IL W equatio n. VII. FINAL REMARKS W e prese n ted the no vel soliton equation (6). W e call it the no nchiral IL W equation b eca use it is par ity inv a ri- ant a nd ca n describe interacting solitons mo v ing in both directions. W e obtained exact mult isoliton solutio ns de- termined b y p oles s a tisfying the equations of mo tio n of the hype r b o lic CMS mo del, and we gave a Lax pa ir, a Hirota form, and co nserv ation la ws. W e also presen ted a per io dic nonchiral IL W equa tio n and its soliton so lutio ns determined by the elliptic CMS mo del. W e prop os ed tha t the nonchiral IL W equation can mo del coupled nonlinear wav es in FQHE systems, and we gav e background information to make this propo sal precise. How ever, a s w e argued, our results are of wider int erest: Ma ny soliton equatio ns containing only first- order deriv atives in time are chiral, i.e., they can only describ e solitons mo ving in one direction, left or right, and thus ar e no t par ity in v ariant. Exa mples include the KdV equation, the BO eq ua tion and, more generally , the IL W equation. How ever, the fundamental equations in hydrodynamics from which these soliton equations ar e derived a re parity inv a r iant. This mismatc h of symme- tries is not fully satis fa ctory . Using the nonc hiral IL W equation instead of the standa rd IL W equation re c o n- ciles sy mmetr ies, and w e therefore b elieve that, in v ar- ious applications in physics, the former ca n be a better approximation to fundamen tal equations than the latter. W e hop e that our r esults op en up a route to generalize recent re s ults on a generalized h ydro dynamic description of the T o da chain 19,20 to the elliptic CMS mo del. This would b e interesting s ince, in the elliptic CMS mo del, o ne can change the qualita tive character of the interaction from long- range in the trig onometric case, to short-r ange in the h yp erb olic case , to nearest-neig hbor in the T o da limit. ACKNO WLEDGMENTS W e thank L. Bystricky , M. Noumi, and in par ticula r J. Shira ishi for very helpful and inspir ing discussions. B.K.B. a ckno wledg e s supp ort from the G¨ or an Gustafss on F oundation and the E urop ean Research Council, Gra nt Agreement No. 682 537. E.L . ac k nowledges support by the Swedish Research Council, Gran t No. 2 016-0 5167, and by the Stiftelse Olle Engkvist Byggm¨ astare, Contract 184-0 573. J.L. acknowledges supp ort from the G¨ ora n Gustafsson F oundation, the Ruth and Nils-Er ik Stenb¨ ack F oundation, the Swedish Res earch Council, Gra n t No. 2015- 05430 , and the E ur op ean Resear ch Co uncil, Gr ant Agreement No. 682537 . 11 App endix A: De riv ation of soliton solutions W e giv e details on the deriv atio n of the N -soliton solu- tions presented in the ma in text, both in the hyperb olic and elliptic cases . 1. Hyp erbol ic case W e construct solutions of (41) with T , ˜ T defined in (7) by genera lizing a known metho d for the BO equation. 15,47 a. Inte gr al op er ators in F ourier sp ac e W e compute the F ourier s pace repr esentation of the matrix op er ator T in (41). W e start by transforming the op era tors T , ˜ T in (7) to F ourier space, using the following exact integral, Z R π 2 δ coth  π 2 δ ( x ∓ i a )  e − i kx dx = − π i e ± ( ak − kδ ) sinh( k δ ) (A1) for real parameter s a, k s uch that 0 < a < 2 δ and k 6 = 0 (a de r iv ation of this r esult ca n b e found at the end of this section). This implies − Z R 1 2 δ coth  π 2 δ x  e − i kx dx = − i coth( kδ ) , Z R 1 2 δ tanh  π 2 δ x  e − i kx dx = − i 1 sinh( k δ ) (A2) for rea l k 6 = 0. Indeed, the fir st of these identities is equiv alent to the av erag e o f the tw o int egra ls in (A1) in the limit a ↓ 0, and the second is obtained from (A1) in the sp ecia l cas e a = δ . Obser ve that the integrals in (7) are co n volutions. Using the fo llowing conv entions for F ourier transforma tion, ˆ u ( k ) = R R u ( x )e − i kx dx , the op erators defined in (7) can there fo re b e express ed in F ourier space as follows, [ ( T u )( k ) = i coth( k δ ) ˆ u ( k ) , [ ( ˜ T u )( k ) = i 1 sinh( k δ ) ˆ u ( k ) . (A3) Thu s, for the matrix op era to r T defined in (41), d T u ( k ) = ˆ T ( k ) ˆ u ( k ) with ˆ T ( k ) = i  coth( k δ ) 1 / sinh( kδ ) − 1 / sinh ( k δ ) − coth( kδ )  (A4) and ˆ u ( k ) = ( ˆ u ( k ) , ˆ v ( k )) t for u ( x ) = ( u ( x ) , v ( x )) t . Us- ing this, it is easy to c heck that ˆ T ( k ) 2 = − I , whic h is equiv alent to T 2 = − I . Derivation of (A1) . Supp ose 0 < a < 2 δ and define the function h ( x ) b y h ( x ) = π 2 δ coth  π 2 δ ( x − i a )  . Even though h ( x ) do es no t decay as x → ± ∞ , the F ourier transform ˆ h of h is w ell-defined as a temp ered distribu- tion. Indeed, the deriv a tive h ′ ( x ) = −  π 2 δ sinh( π ( x − i a ) 2 δ )  2 has exp onential decay as x → ± ∞ and has a double p ole at x = i a + 2i δ n for e a ch integer n . Its F ourier transform d ( h ′ ) can b e computed by a residue co mputation. The F ourier transform ˆ h can then be o btained for k 6 = 0 by ˆ h ( k ) = d ( h ′ )( k ) / (i k ). A simila r c o mputation applies if − 2 δ < a < 0, and we arrive at (A1). b. Eigenfunctions Since T 2 = − I , the eigenv a lue s of T are ± i. W e no w construct the corre spo nding eigenfunctions. By stra ightf orward computatio ns we o btain the follow- ing eigenvectors o f the matrix ˆ T ( k ) in (A4), ˆ g ( k )  e ± kδ / 2 − e ∓ kδ / 2  (A5) with cor resp onding eigenv a lues ± i, for an arbitr a ry func- tion ˆ g ( k ) of k . T o get eig e nfunctions of T with appropri- ate analyticity prop erties, we r estrict ourselves to func- tions ˆ g ( k ) s uch tha t ˆ g ( k )e kα has a well-defined inverse F ourier trans form g ( x − i α ) in a strip − A < α < A with A > δ / 2. F or such functions, Z R dk 2 π ˆ g ( k )e ± kδ / 2 e i kx = g ( x ∓ i δ / 2) , (A6) and the eigenfunctions o f the op erato r T are therefore as follows: F or arbitr ary c omplex value d functions g ( z ) of z ∈ C analytic in a strip − A < Im ( z ) < A with A > δ/ 2 , the ve ctor value d functions v ± ( x ) ≡  g ( x ∓ i δ / 2) − g ( x ± i δ / 2)  (A7) satisfy T v ± ( x ) = ± i v ± ( x ) . (A8) c. Pole ansatz Inspired by the CMS-related soliton solutio ns known for the BO equation, 15,47 we ma ke the following ansatz 12 to solve (41), u ( x, t ) = i N X j =1  α ( x − z j ( t ) − i δ / 2) − α ( x − z j ( t ) + i δ / 2)  − i M X j =1  α ( x − w j ( t ) + i δ / 2) − α ( x − w j ( t ) − i δ / 2)  (A9) where α ( x ) = ( π / 2 δ ) coth( π x/ 2 δ ), N , M are ar bitrary int egers ≥ 0, a nd with po le s z j ( t ) and w j ( t ) to be de- termined. W e note that, to obtain r e al-v alued s o lutions, one must restrict this ansatz to (37), i.e., M = N and w j ( t ) = ¯ z j ( t ) fo r all j , but we find it co nv enient to deriv e a more g e ne r al result. In the following, we sometimes write z j as sho r thand for z j ( t ), etc. The function α ( z ) is meromorphic with p oles a t z = 2i δ n , n integer. Thus, if we restrict the imagina r y parts of z j and w j as follows, Im ( z j ± i δ / 2 ) 6 = 2 δ n , Im ( w j ± i δ / 2 ) 6 = 2 δn (A10) for all in tegers n , then the result in (A7)–(A8) implies T u xx = − N X j =1  α ′′ ( x − z j − i δ / 2 ) − α ′′ ( x − z j + i δ / 2 )  − M X j =1  α ′′ ( x − w j + i δ / 2 ) − α ′′ ( x − w j − i δ / 2 )  (A11) with α ′ ( z ) ≡ ∂ z α ( z ) etc. W e now use α ( − z ) = − α ( z ) and the w ell-known identit ies 49 α ′ ( z ) = − V ( z ) , ∂ z  α ( z ) 2  = V ′ ( z ) , α ( z + 2i δ ) = α ( z ) , ∂ z  α ( z − a ) α ( z − b )  = ∂ z  α ( z − a ) − α ( z − b )  α ( a − b ) , (A12) with V in (3), and for ar bitrary z , a, b ∈ C . Using this we compute u t + ( u . u ) x + T u xx = N X j =1  V ( x − z j − i δ / 2 ) − V ( x − z j + i δ / 2 )  ×   i ˙ z j + 2 N X k 6 = j α ( z j − z k ) − 2 M X k =1 α ( z j − w k + i δ )   + M X j =1  V ( x − w j + i δ / 2 ) − V ( x − w j − i δ / 2 )  ×   − i ˙ w j + 2 M X k 6 = j α ( w j − w k ) − 2 N X k =1 α ( w j − z k + i δ )   (the computations leading to this res ult ar e near ly the same as in the BO case 15 and th us omitted). This implies the following r esult: The funct ion in (A9) satisfies the nonchir al IL W e quation in (41) pr ovide d t he fol lowing system of e qu ations is satisfie d, ˙ z j = 2i N X k 6 = j α ( z j − z k ) − 2i M X k =1 α ( z j − w k + i δ ) , ˙ w j = − 2i M X k 6 = j α ( w j − w k ) + 2i N X k =1 α ( w j − z k + i δ ) , (A13) and the c onditions in (A10) hold true. The system in (A13) is known as a B¨ acklund tr ansfor- mation for the hyperb olic CMS system. 50 It implies tw o decoupled systems of Newton’s equations, ¨ z j = − N X k 6 = j 4 V ′ ( z j − z k ) ( j = 1 , . . . , N ) , (A14a) ¨ w j = − M X k 6 = j 4 V ′ ( w j − w k ) ( j = 1 , . . . , M ) (A14b) with V a s in (3); see Ref. [61] for a recent a lternative deriv ation of this result. W e thus o btain the follo wing generaliza tion of the r e s ult stated in the main text: F or arbitr ary nonne gative inte gers N , M and c omplex p ar am- eters a j , j = 1 , . . . , N , and b j , j = 1 , . . . , M , satisfying Im ( a j ± i δ / 2 ) 6 = 2 δn , Im ( b j ± i δ / 2 ) 6 = 2 δ n (A15) for al l inte gers n , t he function u ( x, t ) in (A9) is a so- lution of the n onchir al IL W e quation (41) pr ovide d the p oles z j ( t ) and w j ( t ) satisfy Newton ’s e qu ations for the hyp erb olic CMS mo del in (A14) with initial c onditions z j (0) = a j , w j (0) = b j , ˙ z j (0) = 2i N X k 6 = j α ( a j − a k ) − 2i M X k =1 α ( a j − b k + i δ ) , ˙ w j (0) = − 2i M X k 6 = j α ( b j − b k ) + 2i N X k =1 α ( b j − a k + i δ ) . Restricting to M = N and b j = ¯ a j for all j , we obtain the result stated in the main text (note that, in this special case, the initial conditions imply w j ( t ) = ¯ z j ( t ) for all t ). A tec hnical remark is in order. Strictly sp eaking, we prov e d the result above only for times, t , wher e the con- ditions in (A10) hold true. W e did not p o int o ut this restriction befor e since we believe that, if the conditions in (A10) and (A13) hold true a t time t = 0, then the solutions z j ( t ) and w j ( t ) of (A14) satisfy the conditions in (A10) for a ll t > 0. W e chec ked this in several sp ecial cases by int egra ting (A14) n umer ically . W e expect that this can be prov ed in g eneral using the known explicit solution of the hyper bo lic CMS mo del obtained with the pro jection metho d; 21 this is left fo r future work. 13 2. Ell iptic case W e give details on how the deriv ation in Appendix A 1 generalizes to the L -pe rio dic cas e . a. Perio dic nonchir al IL W e quation T o see that (6) with T , ˜ T in (8)–(9) is the correct L - per io dic g eneralization of the no nch iral IL W equation, one can c heck that (A3) still holds true but with F ourie r mo des, k , r estricted to in teger m ultiples of (2 π / L ), and for L -p erio dic functions f ( x ) tha t hav e z e ro mean, ˆ f (0) ≡ R L/ 2 − L/ 2 f ( x ) dx = 0 . Thus, T 2 = − I , and the r esult in (A7)–(A8) holds true a s it stands provided the function f ( z ) is L -p erio dic, has zer o mean, and is ana lytic in a strip − A < Im ( z ) < A for A > δ / 2. In pa rticular, T ∂ 2 x  ζ 2 ( x − z ∓ i δ / 2) − ζ 2 ( x − z ± i δ / 2)  = ∓ i  ℘ ′ ( x − z ∓ i δ / 2) − ℘ ′ ( x − z ± i δ / 2)  (A16) using ζ ′′ 2 ( z ) = − ℘ ′ ( z ). W e can use this to constr uct soli- ton solutions related to the elliptic CMS model defined by Newton’s equations (1) with the p otential V ( x ) = ℘ ( x ) . (A17) b. Pole ansatz The dis c us sion ab ove suggests to use the p ole ans atz in (A9) with α ( x ) equal to ζ 1 ( x ). How ever, this choice do e s not work since the third identit y in (A12) is not satisfie d. The choice that works is α ( x ) = ζ 2 ( x ) (A18) since ζ 2 ( z ) is 2i δ -p e rio dic. How ever, ζ 2 ( z ) is not L - per io dic: ζ 2 ( z + L ) = ζ 2 ( z ) + c for some nonzer o constant c . Thu s, u ( x + L, t ) = u ( x, t ) + i( N − M )( c, − c ) t , and, to get a L -p erio dic function u ( x, t ), we must restrict to M = N . W e use (A16 ) to obtain T u xx = N X j =1  V ′ ( x − z j − i δ / 2 ) − V ′ ( x − z j + i δ / 2 )  + N X j =1  V ′ ( x − w j + i δ / 2 ) − V ′ ( x − w j − i δ / 2 )  (A19) with V in (A17). W e define f 2 ( z ) ≡ ∂ z [ ζ 2 ( z ) 2 − ℘ ( z )] and observe that the genera lizations of the second and four th ident ities in (A12) a re ∂ z α ( z ) 2 = V ′ ( z ) + f 2 ( z ) (A20) and ∂ x  α ( x − a ) α ( x − b )  = ∂ x  α ( x − a ) − α ( x − b )] × α ( a − b ) + 1 2  f 2 ( x − a ) + f 2 ( x − b )  , (A21) resp ectively (the latter follows from the following well- known functional equatio n satisfied by the W eierstra ss functions, 34 [ ζ ( x ) + ζ ( y ) + ζ ( z )] 2 = ℘ ( x ) + ℘ ( y ) + ℘ ( z ) provided x + y + z = 0 ). The first and third ident ities in (A12) hold true as they sta nd. While f 2 ( z ) = 0 in the hyperb olic ca se, it is a nontrivial function in the elliptic case. How ever, going thro ugh the computations des crib ed in Appendix A 1 c, one finds that they generalize stra ightf orwardly to the elliptic case pro- vided M = N (that (A13) for M = N implies (A14) even in the elliptic case has been known for a lo ng time 50 ). One thus obtains the same result as in the hyperb olic case but with the restriction M = N . App endix B: Nume rical metho d W e v er ified o ur solito n so lutions n umerically by ada pt- ing a metho d develop ed for so lving the standa rd IL W equation 62 to the nonc hiral IL W equation (6). The n u- merical method applies to the p erio dic problem o n the int erv al [ − L / 2 , L/ 2]; fo r initial conditions and for times, t , suc h that u ( x, t ) and v ( x, t ) a re sig nificantly differ en t from zero o nly in an interv al [ − ℓ/ 2 , ℓ/ 2] with 0 < ℓ ≪ L , this is a n excellent a pproximation for the nonp erio dic problem on R . W e thus chec ked n umer ically v ario us 2- and 3 -soliton solutio ns b oth for the p erio dic and no npe- rio dic problem, and w e found excellent agreement. F or example, the tw o -soliton solutio n in Fig. 3 computed with our numerical metho d cannot b e distinguished with bare eyes from the one obta ined with our a nalytic res ult. W e men tion in pa s sing that our numerical method is m uch more stable for initial conditions which give rise to soli- ton solutions than for g eneric initial conditions. In what follows, w e describ e o ur numeric metho d in more detail. W e employ the disc r ete F ourier transform u ( x, t ) ≈ N − 1 X n = − N ˆ u n ( t )e i k n x , k n ≡ n 2 π L ˆ u n ( t ) = 1 2 N N − 1 X j = − N u ( x j )e − i k n x j , x j ≡ j L 2 N (B1) and the F ourier multiplier repres ent ations (A3) of the singular integral op erator s (7), [ ( T u ) n ( t ) = i coth( k n δ ) ˆ u n ( t ) , [ ( ˜ T u ) n ( t ) = i 1 sinh( k n δ ) ˆ u n ( t ) , (B2) 14 to obtain a system of ordinary differential equations for the time ev olution of the F o ur ier co efficient s via a semi- discrete co llo cation approximation 62 (note that ˆ u n ( t ) /L can b e identified with the F ourier transform ˆ u ( k n , t )). 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