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

We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.

Introduction. One important feature of the Fractional Quantum Hall Effect (FQHE) is the strikingly high accuracy by which the Hall conductance, σ H , is measured in units of the inverse von Klitzing constant, e 2 /h. 1 Therefore, satisfactory explanations of these FQHE measurements, σ H h/e 2 = 1 3 , 2 5 , 3 7 , . . ., must be based on exact analytic arguments, and theories of the FQHE have close connections to integrable systems. Two important classes of integrable systems which are seemingly very different but which are both connected with the FQHE are (i) Calogero-Moser-Sutherland (CMS) 2 models describing FQHE edge states, [3][4][5][6] and (ii) soliton equations of Benjamin-Ono (BO) type describing the dynamics of nonlinear waves propagating along FQHE edges. [7][8][9] These systems are related by a fundamental correspondence between CMS systems and BO-type soliton equations, which provides the basis for a mathematically precise derivation of hydrodynamic descriptions of CMS systems. [10][11][12][13] It is worth noting that this subject has recently received considerable attention in the context of non-equilibrium physics. [14][15][16][17] While the CMS-BO correspondence has been successfully used to understand FQHE physics, it is incomplete. Indeed, CMS systems come in four types: (I) rational, (II) trigonometric, (III) hyperbolic, and (IV) elliptic 18,19 , and while the soliton equations related to the rational and trigonometric cases are well-understood since a long time [11][12][13] , soliton equations related to the hyperbolic and elliptic cases were only recently identified as the Intermediate Long Wave (ILW) equation and the periodic ILW equation, respectively [20][21][22] . However, as we will show in this paper, the latter two soliton equation are not unique: there are other equations which are more interesting in that they are of a different kind and describe new physics.
The correspondence between CMS and BO systems exists both at the classical 12,13 and at the quantum 7,11 level, and our discussion above applies to both cases. We first discovered the quantum elliptic version of the soliton equation presented in this letter from a second quantization of the quantum elliptic CMS model. 23 However, when presenting results in the following, we restrict ourselves to the classical case for simplicity; corresponding quantum results will be presented elsewhere. We first give and prove our results in the hyperbolic case; the generalization to the elliptic case is surprisingly easy, as will be shown later on.
Outline. We start with some physics motivation which suggests that there exists a parity-invariant soliton equation accounting for nonlinear waves propagating on two opposite edges of a FQHE system boundary and taking into account interactions between different edges. Next, the non-chiral ILW equation is presented, together with its N -soliton solutions governed by the hyperbolic CMS model. To show integrability, a Lax pair, a Hirota form, and conservation laws for this equation are presented. Finally, we generalize of our results to the elliptic case. Some technical details are given in two appendices.
Motivation. The CMS models can be defined by Newton's equations where the two-body interaction potential is V (r) = 4r −2 in the rational case and V (r) = 4(π/L) 2 sin −2 (πr/L), L > 0, in the trigonometric case 18 (the arguments in this paragraph apply to both cases). Eq. (1) describes an arbitrary number, N , of interacting particles with positions z j ≡ z j (t) at time t. While one often restricts to real positions when interpreting the CMS model as a dynamical system, one has to allow for complex z j when studying the relation to the BO equation; 12,13 this generalization preserves the integrability. 24 The CMS model is invariant under the parity transformation P : z j → −z j for all j. However, the corresponding BO equation is not parityinvariant: it is given by u t + 2uu x + Hu xx = 0, where u ≡ u(x, t) and H is the Hilbert transform (in the rational case, (Hf )( , and under the parity transformation P : This mismatch of symmetry is paradoxical at first sight, but the paradox is resolved by interpreting u as a wave propagating on one edge of a FQHE system and noting that, in general, there is another edge far away carrying another wave v. Thus, actually, the rational CMS model corresponds to two uncoupled BO equations for u and v. This system of equations is invariant under a parity transformation interchanging u and v, It is peculiar that these two BO equations are uncoupled, and it is for this reason that one can reduce the system to a single equation, ignoring the other. While this uncoupling is reasonable if the two edges are infinitely far apart, it is natural to ask what would happen if the two edges are parallel and close together; see Fig. 1. In this case, one would expect that the nonlinear waves propagating on the two edges interact. We now give a simple heuristic argument to suggest that the hyperbolic CMS model can describe this situation. The hyperbolic CMS model can be defined by Newton's equations (1) with the interaction potential where δ > 0 is an arbitrary length parameter. Dividing the particle positions z j into two groups and shifting the ones in the second group by the imaginary half-period, w k ≡ z k−N1 + iδ for k = 1, . . . , N 2 ≡ N − N 1 , with 1 < N 1 < N , we can write these Newton's equations as withṼ (r) ≡ V (r − iδ) = −4(π/2δ) 2 cosh −2 (πr/2δ), for j = 1, . . . , N 1 and k = 1, . . . , N 2 . This can be interpreted as a model of two kinds of particles, z j and w k , in which particles of the same kind interact via the singular repulsive two-body potential V , whereas particles of different kinds interact via the weakly attractive non-singular po-tentialṼ . We interpret δ as the distance between the two edges of the FQHE system. In the rational limit δ → ∞, we haveṼ → 0, so particles of different types do not interact and the two corresponding soliton equations for u and v decouple; for finite δ, the system is coupled. Non-chiral ILW equation. In the hyperbolic case, the two-component generalization of the BO equation we propose in this letter is given by for u = u(x, t) and v = v(x, t), with The ILW equation is given by u t + 2uu x + T u xx = 0; it reduces to the BO equation in the limit δ → ∞. [25][26][27] Thus, if one drops theT -terms, (5) corresponds to a system of uncoupled ILW equations generalizing the system of uncoupled BO equations discussed above. However, due to the presence of theT -terms, the nonlinear waves u and v interact. For this reason, and since equation (5) is invariant under the parity transformation (2), we call it the non-chiral ILW equation. 28 As shown below, it provides an integrable model of nonlinear waves on two edges of a FQHE system taking into account interaction effects between the edges.
N -soliton solutions. The following fundamental result shows that (5) admits N -soliton solutions whose dynamics is described by the hyperbolic CMS model, thus generalizing a famous result for the rational case: 29 For arbitrary integers N ≥ 1 and complex parameters a j with imaginary parts in the range δ/2 < Ima j < 3δ/2 for j = 1, . . . , N , the following is an exact solution of the non-chiral ILW equation (5): where α(x) ≡ (π/2δ) coth(πx/2δ) and the poles z j (t) are determined by Newton's equations (1) with V (r) given by (3) and with initial conditions z j (0) = a j anḋ (the bar denotes complex conjugation, c.c.). Thus, to obtain an exact solution of (5), one chooses complex parameters a j satisfying δ/2 < Ima j < 3δ/2; next, the time-evolution of z j (t) is obtained by solving the hyperbolic CMS model with initial conditions determined by the a j ; finally, the solution of (5) is obtained from (7). Using the exact analytic solution of the hyperbolic CMS model obtained by the projection method, 18 the numerical effort to compute such an N -soliton solution at an arbitrary time, t, is reduced to diagonalizing an explicitly known N × N matrix. As elaborated in Appendix B, we tested this result by comparing with a numeric solution of (5).  (5) with a u-channel dominated soliton (big blue and small red humps) colliding with a v-channel dominated soliton (big red and small blue humps), as explained in the main text. The plots show u(x, t) (blue line) and v(x, t) (red line) at successive times t = (n − 1)t0, n = 1, . . . , 5; the parameters are δ = π, a1 = −4 + 1.2iδ, a2 = 3 + 0.85iδ, and t0 = 2.25. Examples. The 1-soliton solution of (5) is given by where z(t) = a +ż(0)t,ż(0) = −2iα(a −ā + iδ), with a ∈ C such that δ/2 < Ima ≤ 3δ/2. It is important to note thatż(0) is real, and therefore, Imz(t) = Ima independent of t. Thus, the functions u(x, t) and v(x, t) both describe humps whose shapes do not change with time. These humps are centered at the same point and move with constant velocity, Rez(t) = Rea +ż(0)t, and their heights, max u > 0 and max v > 0, are determined by Ima. For Ima close to 3δ/2, max u max v, and the solitons move to the right,ż(0) > 0. As Ima decreases, max u andż(0) decrease while max v increases until, at Ima = δ, max u = max v andż(0) = 0. Thus, if Ima lies in the range δ < Ima < 3δ/2, then the 1-soliton is mainly in the u-channel and moves to the right; it is therefore similar to the 1-soliton solution of the standard ILW equation u t + 2uu x + T u xx = 0. Similarly, when δ/2 < Ima < δ, the 1-soliton is mainly in the v-channel and moves to the left, similar to a 1-soliton solution of For parameters a j such that Re(a j − a k ) δ for all j = k, the N -soliton solution of (5) is well-approximated by a sum of N 1-solitons of the form (9) whereż j (t) ≈ −2iα(a j −ā j + iδ) is time-independent for times such that Re(z j (t) − z k (t)) δ; see Fig. 2 for a 2-soliton solution, with the corresponding motion of poles in Fig. 3(a).
However, when two solitons meet, they interact in a non-trivial way, and after the interaction they re-emerge with the same shape but with phase-shifts; see Fig. 3 Non-trivial such interactions between solitons can also be modeled by the system of decoupled ILW equations obtained from (5) by dropping theT -terms. A qualitatively new effect stemming from theT -terms is that u-channel dominated solitons (u-solitons) interact nontrivially with v-solitons, as clearly seen in our example in Figs. 2 and 3. It is interesting to note that the poles corresponding to the u-and v-solitons interchange their imaginary parts and directions during the collision and thus, in this sense, exchange their identities: while the first pole corresponds to the u-soliton and the second to the v-soliton before the collision, it is the other way round after the collision; see Figs. 3(a) and (b). We note that such an identity change of poles during soliton collisions is known for the BO equation, 30 but only for solitons moving in one direction.
Derivation of N -soliton solutions. We explain the key difference between the derivation of solitons for (5) and the corresponding derivation in the rational case; 29 further details can be found in Appendix A 1.
The Hilbert transform, H, satisfies H 2 = −I, and this property is crucial for the existence of eigenfunctions of H needed in the derivation of the CMS-related soliton solutions of the BO equation u t + 2uu x + Hu xx = 0. 29 However, while the trigonometric generalization of H also has this property, the hyperbolic generalization of H is the operator T in (6), and T 2 = −I. This is the reason why the soliton solution of the BO equation straightforwardly generalizes to the trigonometric case, 29 but the naive generalization to the hyperbolic case fails. However, the non-chiral ILW equation (5) can be written in vector form as where the matrix operator, T , satisfies T 2 = −I. Moreover, (α(x+z±iδ/2), −α(x+z∓iδ/2)) t are eigenfunctions of T with eigenvalues ±i. The latter are the eigenfunctions needed to be able to use the method developed for the rational case: 29 using well-known identities for the function α(x), 31 as well as a Bäcklund transformation for the hyperbolic CMS model, 32 it is straightforward to adapt a known derivation of N -soliton solutions of the BO equation 29 to the hyperbolic case. Integrability. We found a Lax pair, a Hirota bilinear form, a Bäcklund transformation, and an infinite number of conservation laws for (5). Thus, the non-chiral ILW equation is a soliton equation that is integrable in the same strong sense as the standard ILW equation. 26 Below we present some of these results that can be checked by straightforward computations.
The Lax pair we found is as follows: Let ψ(z; t, k) be an analytic function on the union of the strips 0 < Imz < δ and δ < Imz < 2δ and extended to C by 2iδ-periodicity, ψ ± 0 (x; t, k) and ψ ± δ (x; t, k) the boundary values of this function on R and R + iδ, respectively, and µ 1 , µ 2 , ν 1 , and ν 2 arbitrary functions of the spectral parameter k. Then the compatibility of the following linear equations yields (5): Inspired by known results for the BO equation, 13 we obtained the following Hirota bilinear form of (5), with u = i∂ x log(F − /G + ) and v = i∂ x log(G − /F + ), where F ± (x, t) = F (x ± iδ/2, t) and similarly for G, using standard Hirota derivatives. 33 The first three of the conservation laws we found are with (u ↔ v) short for the same three terms but with u and v interchanged. Bäcklund transformations, other conservation laws, and detailed derivations will be given elsewhere.
Final remarks. We presented the novel soliton equation (5). We call it the non-chiral ILW equation because it is parity invariant and can describe interacting solitons moving in both directions. We obtained exact N -soliton solutions determined by poles satisfying the equations of motion of the hyperbolic CMS model, and we gave a Lax pair, a Hirota form, and conservation laws. We also presented a periodic non-chiral ILW equation and its soliton solutions determined by the elliptic CMS model.
Many soliton equations containing only first-order derivatives in time are chiral, i.e., they can only describe solitons moving in one direction, left or right, and thus are not parity invariant. Examples include the Kortewegde Vries equation, the BO equation and, more generally, the ILW equation. However, the fundamental equations in hydrodynamics from which these soliton equations are derived are parity invariant. This mismatch of symmetries is similar to the one discussed in this paper. Using the non-chiral ILW equation instead of the standard ILW equation would reconcile symmetries, and it therefore is tempting to speculate that the former is a better approximation to the fundamental equations than the latter.
We hope that our results open up a route to generalize recent results on a generalized hydrodynamic description of the Toda chain 16,17 to the elliptic CMS model. This would be interesting since, in the elliptic CMS model, one can change the qualitative character of the interaction from long-range in the trigonometric case, to short-range in the hyperbolic case, to near-neighbor in the Toda limit.

Hyperbolic case
We construct solutions of (10) with T,T defined in (6) by generalizing a known method for the BO equation. 12,29 a. Integral operators in Fourier space We compute the Fourier space representation of the matrix operator T in (10).
We start by transforming the operators T,T in (6) to Fourier space, using the following exact integral, for real parameters a, k such that 0 < a < 2δ and k = 0 (a derivation of this result can be found at the end of this section). This implies for real k = 0. Indeed, the first of these identities is equivalent to the average of the two integrals in (A1) in the limit a ↓ 0, and the second is obtained from (A1) in the special case a = δ. Observe that the integrals in (6) are convolutions. Using the following conventions for Fourier transformation,û(k) = R u(x)e −ikx dx, the operators defined in (6) can therefore be expressed in Fourier space as follows, Thus, for the matrix operator T defined in (10), T u(k) = T (k)û(k) witĥ andû(k) = (û(k),v(k)) t for u(x) = (u(x), v(x)) t . Using this, it is easy to check thatT (k) 2 = −I, which is equivalent to T 2 = −I.
Derivation of (A1). Suppose 0 < a < 2δ and define the function h(x) by Even though h(x) does not decay as x → ±∞, the Fourier transformĥ of h is well-defined as a tempered distribution. Indeed, the derivative has exponential decay as x → ±∞ and has a double pole at x = ia + 2iδn for each integer n. Its Fourier transform (h ) can be computed by a residue computation. The Fourier transformĥ can then be obtained for k = 0 bŷ h(k) = (h )(k)/(ik). A similar computation applies if −2δ < a < 0, and we arrive at (A1).

b. Eigenfunctions
Since T 2 = −I, the eigenvalues of T are ±i. We now construct the corresponding eigenfunctions.
By straightforward computations we obtain the following eigenvectors of the matrixT (k) in (A4), with corresponding eigenvalues ±i, for an arbitrary functionĝ(k) of k. To get eigenfunctions of T with appropriate analyticity properties, we restrict ourselves to functionsĝ(k) such thatĝ(k)e kα has a well-defined inverse Fourier transform g(x − iα) in a strip −A < α < A with A > δ/2. For such functions, and the eigenfunctions of the operator T are therefore as follows: For arbitrary complex valued functions g(z) of z ∈ C analytic in a strip −A < Im(z) < A with A > δ/2, the vector valued functions c. Pole ansatz Inspired by the CMS-related solition solutions known for the BO equation, 12,29 we make the following ansatz to solve (10), where α(x) = (π/2δ) coth(πx/2δ), N, M are arbitrary integers ≥ 0, and with poles z j (t) and w j (t) to be determined. We note that, to obtain real-valued solutions, one must restrict this ansatz to (7), i.e., M = N and w j (t) =z j (t) for all j, but we find it convenient to derive a more general result. In the following, we sometimes write z j as shorthand for z j (t), etc. The function α(z) is meromorphic with poles at z = 2iδn, n integer. Thus, if we restrict the imaginary parts of z j and w j as follows, Im(z j ± iδ/2) = 2δn, Im(w j ± iδ/2) = 2δn (A10) for all integers n, then the result in (A7)-(A8) implies with α (z) ≡ ∂ z α(z) etc. We now use α(−z) = −α(z) and the well-known identities 31 with V in (3), and for arbitrary a, b ∈ C. Using this we compute (the computations leading to this result are nearly the same as in the BO case 12 and thus omitted). This implies the following result: The function in (A9) satisfies the non-chiral ILW equation in (10) provided the following system of equations is satisfied, and the conditions in (A10) hold true.
The system in (A13) is known as a Bäcklund transformation for the hyperbolic CMS system. 32 It implies two decoupled systems of Newton's equations, with V as in (3); see Ref. [35] for a recent alternative derivation of this result. We thus obtain the following generalization of the result stated in the main text: For arbitrary non-negative integers N, M and complex parameters a j , j = 1, . . . , N , and b j , j = 1, . . . , M , satisfying Im(a j ± iδ/2) = 2δn, Im(b j ± iδ/2) = 2δn (A15) for all integers n, the function u(x, t) in (A9) is a solution of the non-chiral ILW equation (10) provided the poles z j (t) and w j (t) satisfy Newton's equations for the hyperbolic CMS model in (A14) with initial conditions Restricting to M = N and b j =ā 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 technical remark is in order. Strictly speaking, we proved the result above only for times, t, where the conditions in (A10) hold true. We did not point out this restriction before since we believe that, if the conditions in (A10) and (A13) hold true at time t = 0, then the solutions z j (t) and w j (t) of (A14) satisfy the conditions in (A10) for all t > 0. We checked this in several special cases by integrating (A14) numerically. We expect that this can be proved in general using the known explicit solution of the hyperbolic CMS model obtained with the projection method; 18 this is left for future work.

Elliptic case
We give details on how the derivation in Appendix A 1 generalizes to the L-periodic case.

a. Periodic non-chiral IWL equation
We recall that the function ζ 1 (z) defined in the main text has the following representation, 34 and it therefore is the natural L-periodic analogue of This suggests that the L-periodic periodic version of the non-chiral ILW equation is as in (5) but with To see that this is the correct generalization, one can check that (A3) still holds true but with Fourier modes, k, restricted to integer multiples of (2π/L), and for Lperiodic functions f (x) that have zero mean,f (0) ≡ L/2 −L/2 f (x)dx = 0. Thus, T 2 = −I, and the result in (A7)-(A8) holds true as it stands provided the function f (z) is L-periodic, has zero mean, and is analytic in a strip −A < Im(z) < A for A > δ/2. In particular, using ζ 2 (z) = −℘ (z). We can use this to construct soliton solutions related to the elliptic CMS model defined by Newton's equations (1) with the potential The discussion above suggests to use the pole ansatz in (A9) with α(x) equal to ζ 1 (x). However, this choice does not work since the third identity in (A12) is not satisfied. The choice that works is since ζ 2 (z) is 2iδ-periodic. However, ζ 2 (z) is not Lperiodic: ζ 2 (z +L) = ζ 2 (z)+c for some non-zero constant c. Thus, u(x + L, t) = u(x, t) + i(N − M )(c, −c) t , and, to get a L-periodic function u(x, t), we must restrict to M = N . We use (A18) to obtain and observe that the generalizations of the second and fourth identities in (A12) are and respectively (the latter follows from the following wellknown functional equation satisfied by the Weierstrass functions, 34 [ζ(x) + ζ(y) + ζ(z)] 2 = ℘(x) + ℘(y) + ℘(z) provided x + y + z = 0). The first and third identities in (A12) hold true as they stand. While f 2 (z) = 0 in the hyperbolic case, it is a nontrivial function in the elliptic case. However, going through the computations described in Appendix A 1 c, one finds that they generalize straightforwardly to the elliptic case provided M = N (that (A13) for M = N implies (A14) even in the elliptic case has been known for a long time 32 ). One thus obtains the same result as in the hyperbolic case but with the restriction M = N .

Appendix B: Numerical method
We verified our soliton solutions numerically by adapting a method developed for solving the standard ILW equation 36 to the non-chiral ILW equation (5). The numerical method applies to the periodic problem on the interval [−L/2, L/2]; for initial conditions and for times, t, such that u(x, t) and v(x, t) are significantly different from zero only in an interval [− /2, /2] with 0 < L, this is an excellent approximation for the non-periodic problem on R. We thus checked numerically various 2and 3-soliton solutions both for the periodic and nonperiodic problem, and we found excellent agreement. For example, the 2-soliton solution in Fig. 2 computed with our numerical method cannot be distinguished with bare eyes from the one obtained with our analytic result. We mention in passing that our numerical method is much more stable for initial conditions which give rise to soliton solutions than for generic initial conditions. In what follows, we describe our numeric method in more detail.
We employ the discrete Fourier transform u(x, t) ≈ to obtain a system of ordinary differential equations for the time evolution of the Fourier coefficients via a semidiscrete collocation approximation 36 (note thatû n (t)/L can be identified with the Fourier transformû(k n , t)). The numerical approximation for the nonlinear terms is (2uu x ) n (t) = ik n (u 2 ) n (t) with (u 2 ) n (t) ≈