On the non-chiral intermediate long wave equation II: periodic case

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a B\"{a}cklund transformation, (e) use the B\"{a}cklund transformation to obtain an infinite number of conservation laws.


Introduction
The introduction of the inverse scattering transform for the solution of equations such as the Korteweg-de Vries (KdV), nonlinear Schrödinger (NLS), and sine-Gordon equations was a major development in the field of nonlinear PDEs in the 20th century. This development, which began in the late 1960s [1], made it clear that certain nonlinear equations, called integrable, possess unique properties which allow them to be solved exactly, at least in appropriate circumstances. Two classes of solutions of integrable equations are particularly well-studied: (a) the class of solutions on the real line with decay at spatial infinity and (b) the class of (spatially) periodic solutions. These two classes are superficially similar, but they are very different when it comes to details. In fact, throughout the history of integrable PDEs, there has been a fruitful interplay between the theories for these two classes. For example, one of the main tools in the study of solutions on the line is the inverse scattering transform, which provides a way to solve the initial value problem via a sequence of linear operations [2]. The search for a generalization of this approach to the periodic setting led to the introduction of the so-called finite-gap integration method, a development which has in turn influenced the evolution of diverse branches of mathematics as well as theoretical physics; see [3] for a review.
In this paper, we consider the periodic version of an integrable equation introduced in [4]. This equation is referred to as the non-chiral intermediate long-wave (ncILW) equation, because it involves the same integral operator that appears in the standard intermediate longwave equation [5,6]. However, whereas the latter is chiral in the sense that it only allows for solitons moving in one direction, left or right, the ncILW equation supports solitons moving in both directions. While the ncILW equation was discovered in the context of a quantum field theory describing fractional quantum Hall effect systems [4], we expect that it will find applications also in the theory of nonlinear waves and other areas of theoretical physics; see [7, Sections 1.1-1.3] for a more detailed discussion of the physics behind the ncILW equation.
The periodic ncILW equation is given by where u = u(x, t) and v = v(x, t) are real-or complex-valued 1 functions of a space variable x ∈ R and a time variable t ∈ R. The integral operators T andT in (1.1) are defined by is equal, up to a term linear in z, to the Weierstrass ζ-function with periods L > 0 and 2iδ, δ > 0; see Appendix A for the precise relation. We are interested in L-periodic solutions of this equation, i.e., solutions such that u(x + L, t) = u(x, t) and v(x + L, t) = v(x, t). The non-chirality of the ncILW equation corresponds to the invariance of (1.1) under the parity transformation which maps x to −x and interchanges u and v [4]. The limiting case L → ∞ where ζ 1 (z|L/2, iδ) → π 2δ coth π 2δ z , ζ 1 (z + iδ|L/2, iδ) → π 2δ tanh π 2δ z corresponds to the ncILW equation on the real line, i.e., (1.1) with the integral operators ( T R f )(k) = i coth(kδ)f (k), ( T R f )(k) = i csch(kδ)f (k), (1.5) where csch(z) := 1/ sinh(z). We assume f (x) is a zero-mean L-periodic function with the Fourier transform pair: f (x) = n∈Z\{0}f n e 2iπnx/L ,f (k) = 2π n∈Z\{0}f n δ(k − 2πn/L). (1.6) It follows that (1.7) Comparing (1.7) with the Fourier series for the functions ζ 1 (z) and ζ 1 (z + iδ) [8,Eq. 23.8.2] appearing in (1.4), it is straightforward to show (at least formally) [9] that T R = T andT R =T on such functions (1.6) (the functions u xx , v xx appearing as arguments of T,T in (1.1) are in this class).
In a recent paper [7], we obtained a Lax pair, a Hirota form, a Bäcklund transformations and an infinite number of conservation laws for the ncILW equation on the real line. In this paper, we present corresponding results in the periodic case. We show that, even though several steps are significantly more complicated in the periodic setting, it is nevertheless possible to prove results which parallel those obtained in [7]. We also provide an alternative derivation based on the Hirota method of the multi-soliton solutions of the periodic ncILW equation; the multisolitons were previously obtained by a different method in [4]. More precisely, we show that the multi-solitons can be obtained via a pole ansatz in terms of a Weierstrass ζ-function, where the poles evolve according to the elliptic Calogero-Moser (CM) system; see Proposition 4.1 for the precise formulation.
In the limit L → ∞, the results we obtain here for the periodic problem reduce (at least formally) to analogous results for the problem on the line. However, we emphasize that only a subset of the results of [7] can be obtained in this way: proving the results directly on the real line is not only technically simpler but also leads to more general results.
It is important to note that the non-chiral ILW equation (1.1) is an elliptic integrable systems. This is clear already from the definition of the operators T andT in (1.2): while the ncILW equation on the line involves integral operators whose kernels are given in terms of the hyperbolic tangent, the kernels in (1.2) involve a Weierstrass ζ-function. The fact that (1.1) is an elliptic system is also evident from the relation to the elliptic CM system mentioned above. In fact, the periodic ncILW equation is related to the elliptic CM system in the same way as the ncILW equation on the line is related to the hyperbolic CM system [4].
The plan of this paper is as follows. In Section 2, we derive a Lax pair for (1.1). A Hirota bilinear form is presented in Section 3, where we additionally prove that the Hirota bilinear form is equivalent to (1.1) by constructing explicit transformations from (u, v) to the Hirota variables (F, G) and vice-versa. We use the Hirota bilinear form to construct N -periodic soliton solutions via a pole ansatz in Section 4. A Bäcklund transformation is constructed from the Hirota bilinear form in Section 5. Definitions and basic properties of certain elliptic functions are collected in Appendix A. Some properties of the operators T andT defined in (1.2) are established in Appendix B.
In what follows we assume that the arguments of T andT are sufficiently regular to justify our arguments. We occasionally comment on specific necessary or sufficient conditions for clarity.
A function f : Π → C can be viewed as a function f : C → C which is doubly periodic with periods L and 2iδ, i.e.
f (z + mL + 2inδ) = f (z), m, n ∈ Z. Let Π 0 and Π δ denote the images of the lines Imz = 0 and Imz = δ, respectively, under π. We consider an eigenfunction ψ(z, t; k); for each t ∈ R and k ∈ C, ψ(z) := ψ(z, t; k) is an analytic function Π \ (Π 0 ∪ Π δ ) with jumps across Π 0 and Π δ . The boundary values of the eigenfunction are functions Π 0 ∪ Π δ → C defined by We take the following ansatz for the Lax pair: Here, µ 1 , µ 2 , ν 1 , and ν 2 are complex-valued functions of the spectral parameter; A(z, t; k) and B(z, t; k) are analytic functions on Π \ (Π 0 ∪ Π δ ) to be determined. To obtain the compatibility conditions for (2.2), we write the boundary values of the t-part of the Lax pair: This equation and its x-derivative can be used to eliminate ψ ± t and ψ ± tx from the t-derivative of the x-part of (2.2), leading to Setting the coefficients of ψ ± x and ψ ± to zero, we find the equations

3b)
From (2.3b), we see that B(z) is an analytic function on Π and so must be constant: B(z) = B 0 . Then (2.3c-2.3d) shows that A is a solution of the following RH problem on Π: • across Π 0 ∪ Π δ , A satisfies the jump condition Lemma 2.1 (RH problem on Π with a jump across Π 0 ∪Π δ ). Let J 0 : Π 0 → C and J 1 : Π δ → C be continuous functions satisfying Then the scalar RH problem: has the general solution where A 0 is an arbitrary complex constant and both Π 0 and Π δ are oriented from Im z = −L/2 to Im z = L/2. Moreover, this solution satisfies (2.5) Proof. If A 1 and A 2 are two different solutions, then A 1 − A 2 is analytic on Π and hence constant. Let A be given by (2.4). Using periodicity properties of ζ 1 , we observe that A(z+L) = A(z) and where the integral vanishes by assumption. Hence A descends to a well-defined function For x ∈ Π 0 , the Plemelj formula gives (we suppress the second and third arguments of ζ 1 ) Similarly, for x + iδ ∈ Π δ , This proves the expressions for the boundary values and shows that A satisfies the correct jump condition.
Using (2.4), we see that Substituting these expressions for A ± into (2.3e) and (2.3d) and using that T andT commute with ∂ x from Proposition B.1, we arrive at the two-component equation Choosing µ 1 = µ 2 = µ and B 0 = −2iµ, this becomes the non-chiral ILW equation (1.1). We summarize the results above in a theorem.
Remark 2.2. The t-parts of (2.9) have an analytic continuation to Π \ (Π 0 ∪ Π δ ) and can be alternatively written as where A = A(z, t; k) is given by (2.7).

Hirota bilinear form
In the periodic setting, the Hirota bilinear form of (1.1) is whereū andv are the spatial means of u(x, t) and v(x, t), respectively and λ 1 (t) and λ 2 (t) are complex functions. By the following lemma, we may takeū andv to be constants. Proof. Starting from the definition of the mean A bilinear form similar to (3.1) was used in [10] to construct periodic solutions of the standard ILW equation. We show that (3.1) is equivalent to (1.1) in the sense of the following theorem.
A. Let F (z, t) and G(z, t) be L-periodic functions of z ∈ C and t ∈ R such that log F (z, t) and log G(z, t) are analytic for −δ/2 < Im z < δ/2 and continuous for −δ/2 ≤ Im z ≤ δ/2. Then F, G satisfy the bilinear system (3.1) for someū,v ∈ C and complex-valued functions λ 1 (t) and λ 2 (t) if and only if , defined up to multiplication by an arbitrary function of t by are analytic for −δ/2 < Im z < δ/2, continuous for −δ/2 ≤ Im z ≤ δ/2, and satisfy the Hirota equations (3.1) with Each of these functions has zero mean by L-periodicity of F and G. By our assumptions on the analyticity of log F and log G, we see that u + and v − are analytic in the strip 0 < Im z < δ, u − and v + are analytic in the strip −δ < Im z < 0. Additionally, we observe that Lemma 3.2. If g + (z) is L-periodic, analytic in the strip 0 < Im z < δ, and continuous in the strip 0 ≤ Im z ≤ δ, then, Similarly, if g − (z) is L-periodic, analytic in the strip −δ < Im z < 0, and continuous in the strip −δ ≤ Im z ≤ 0, then Proof. Suppose g + (z) is an L-periodic function which is analytic in 0 < Im z < δ, and continuous in 0 ≤ Im z ≤ δ. Using the definition ofT (1.2) and then changing variables to z = x + iδ, we find We deform the contour down towards the real axis. Utilizing the Plemelj formula to evaluate the contribution from the simple pole at z = x, we obtain Because the integrand of E is L-periodic, we have E = 0. Thus, using that Res The proof of (3.9) is similar, but there is a correction term due to the non-2iδ-periodicity of ζ 1 . Suppose g − (z) is an L-periodic function which is analytic in −δ < Im z < 0 and continuous in −δ ≤ Im z ≤ 0. Using the definition ofT , changing variables to z = x + iδ, and using the identity We deform the contour up towards the real axis. Utilizing the Plemelj formula to evaluate the contribution from the simple pole at z = x, we obtain Because the integrand of E is L-periodic, we have E = 0. Thus, using that Res Lemma 3.3. The functions u and v obey the identities Proof. By (3.7) and the identities T We see from (3.6) that u + is L-periodic, analytic for 0 < Im z < δ, and continuous for 0 ≤ Im z ≤ δ, so that T u + −T [u + (· + iδ)] = iu + by Lemma 3.2. Similarly, u − is L-periodic, analytic for −δ < Im z < 0, continuous for −δ ≤ Im z ≤ 0, and has zero mean, so that By our analyticity assumptions on F, G, we see from (3.6) that v − is analytic for 0 < Im z < δ, and continuous for 0 ≤ Im z ≤ δ, so that T v − −T [v − (· + iδ)] = iv − by Lemma 3.2. Similarly, v + is analytic for −δ < Im z < 0, continuous for −δ ≤ Im z ≤ 0, and has zero mean, so that (3.12) and Lemma 3.2.
Rewriting the system in terms of F and G, we obtain Simplification shows that the first equation can be rewritten as In the same way, the second equation can be written as Multiplying (3.15a) and (3.15b) by F − G + and F + G − , respectively, we conclude that (1.1) is equivalent to the bilinear system (3.1). This completes the proof.

Proof of Theorem 2B.
Proof. We decompose the L-periodic solution u, v of (1.1) as u =ū + u + + u − , v =v + v + + v − , with u ± , v ± as in (3.4). We view (3.2) as a pair of differential-difference equations for F , G and seek solutions satisfying LetΠ denote the torus C/Λ, whereΛ := LZ + iδZ,π the natural projection C →Π, andΠ 0 the image of Imz = 0 underπ. Then (3.17) defines a pair of RH problems for the functions ∂ z log F and ∂ z log G onΠ. The following lemma can be proved similarly to Lemma 2.1. Then the scalar RH problem: • A :Π \Π 0 is analytic, • acrossΠ 0 , A satisfies the jump condition has the general solution where A 0 is an arbitrary complex constant. Moreover, this solution satisfies It follows from (3.4) and the anti-self-adjointness of T andT from Proposition B.1, that the functions u ± , v ± and hence u ± + v ± have zero mean. Then, Lemma 3.4 shows that the general solution to the scalar RH problem: is given by with boundary values Hence we find with corresponding boundary values where F 0 and G 0 are arbitrary complex functions of t.

Periodic solitons
We construct the N -periodic soliton solutions of (1.1) via an ansatz for the Hirota form (3.1). The ansatz for (3.1), together with Theorem 2A, leads to an alternative proof of the following result in [4]. The naïve ansatz fails to satisfy the conditions of Theorem 2A; the ansatz in (4.1) is a minor modification of the latter one which satisfies those conditions. Im (a j ± iδ/2) = 2δn, Im (b j ± iδ/2) = 2δn, for all integers n, the functions provide a solution of the non-chiral ILW equation (1.1) provided the poles z j (t) and w j (t) satisfy with initial conditions ζ 2 (w j − z k + iδ|L/2, iδ).

(4.5)
Our result is obtained by the observation that the equations in (4.5) are a Bäcklund transformations for the elliptic CM system, i.e., if (4.5) is fulfilled, then (4.3) is implied; to keep this paper self-contained, we also give the proof of this known fact [11].
Proof. We differentiate (4.14) with respect to t and insert (4.13): Using the identity which follows from Definition A.2 and the identity η 1 ω 2 − η 2 ω 1 = iπ/2 [8, Eq. 23.2.14], we can writeż We setū so that the equations of motion (4.13) become (4.17) Hence, after inserting (4.16) with (4.14) and (4.8) into (3.2), we have which is the first equation in (4.2). The corresponding result for v(x, t) in (4.2) is established similarly. By Theorem 2A, (4.2) provides a solution to (1.1) when (4.17) is satisfied. It remains to show that (4.17) with initial conditions (4.4a) is equivalent to (4.3) with the initial conditions (4.4). We write (4.12) as and claim that Indeed, by direct computation, alternatively, by differentiating (4.18) with respect to t and inserting (4.12), we obtain which is the last line in (4.20).
To show that (4.19) implies (4.3), we write Then, a lengthy but straightforward computation using the identities (A.12-A.13) shows that which, upon comparison with the first line of (4.19), implies (4.3) after recalling the notation (4.10).

Proposition 5.1 (Bäcklund transformation in terms of bilinear variables). Suppose (F, G)
and (F ,G) satisfy the relations in (5.2). Then (F, G) is a solution of (3.1) if and only (F ,G) is a solution of (5.1).
To transform (5.2) into a form written in the original variables, we introduce potential functions U , V ,Ũ ,Ṽ by Lemma 5.2. The functions λ 1 and λ 2 in (3.1) satisfy

4)
where is a constant.
Proof. We add the two equations in (3.14) and integrate over [−L/2, L/2] to obtain The second term vanishes by periodicity. Using (5.3) we obtain (5.4). It remains to show that I 2 is a conservation law. This is verified by a calculation analogous to the direct verification of I 2 in [7, Section 5.3], using the anti-self-adjointness of T andT from Proposition B.1.
Lemma 5.3. The following identities hold: Proof. By Lemma 3.3, and the expression for u + follows after simplification. The expression for u − then follows because u =ū + u + + u − . The expressions forũ ± follow in the same way.
Utilizing Lemma 5.3, equation (5.10) can be rewritten as Recalling (5.8) and setting which is (5.7a). We next rewrite the t-parts (5.2a)-(5.2b) of the Bäcklund transformation as Subtracting the second of these equations from the first gives Multiplying by i and using the definitions (3.6), (5.3), and (5.6a) of u ± ,ũ ± , U,Ũ , and Λ 1 ,Λ 1 , this becomes Recalling (5.8) and using Lemma 3.3, we find Equation (5.12) can be written as Using this relation to eliminate (U +Ũ ) xx , we arrive at That is, which is (5.7b). We next rewrite the x-part (5.2f). Dividing (5.2f) by G −F + yields (5.14) Lemma 5.4. The following identities hold: Proof. By Lemma 3.3, T v +T u − iv + i(v −ū) = −2iv + and the expression for v + follows after simplification. The expression for v − then follows because v =v + v + + v − . The expressions forṽ ± follow in the same way.
Utilizing Lemma 5.4, equation (5.14) can be rewritten as With (5.8) and setting which is (5.7c). We next rewrite the t-parts (5.2d)-(5.2e) of the Bäcklund transformation. As before, we find that (5.13) holds except that F,F and G,G are now evaluated at x + iδ/2 and x − iδ/2, respectively, i.e., Multiplying by i and using the definitions (3.6), (5.3), and (5.6b) of v ± ,ṽ ± ,V,Ṽ , and Λ 2 ,Λ 2 this becomes Recalling (5.8) and using Lemma 3.3, we find Equation (5.16) can be written as Using this relation to eliminate (V +Ṽ ) xx , we arrive at That is, which is (5.7d). This completes the proof of Theorem 3.

Conservation laws
Theorem 4 (Conservation laws of the periodic non-chiral ILW equation). The periodic nonchiral ILW equation (1.1) with (1.2) has an infinite number of conservation laws where W n and Z n can be computed recursively from the formal power series in with P ± as in (5.9). The first four conservation laws are 3a) 3b) (6.3c) Proof. Adding equations (5.7b) and (5.7d), we find Thus, d dt Using the anti-self-adjointness (B.2) of the operators T andT , the integral on the right-hand side vanishes. The remainder of the proof is identical to that of [7, Theorem 4] and hence omitted.

Appendix B. Properties of the T andT operators
In this section we collect and prove several identities for the T andT operators (1.2).
(B.7). We consider the integral Γ ζ 1 (z + iδ) dz, where Γ is a rectangular contour with vertices at ±L/2 and ±L/2 + i(Im a − δ), oriented so the integral along the real axis is positively-oriented. When 0 < Im a < 2δ, the contour encloses no poles, so we have, after cancelling vertical contributions by periodicity, (B.9) The first case in (B.7) follows from the real translation invariance of (B.9). The proof of the second case in (B.7) is similar after accounting for the pole enclosed by Γ.