On the non-chiral intermediate long wave equation

We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) derive a B\"{a}cklund transformation, (d) use, separately, the B\"{a}cklund transformation and the Lax representation to obtain an infinite number of conservation laws. In the last section, we extend our results to the elliptic integrable system obtained by imposing periodic boundary conditions on the non-chiral intermediate long wave equation.


Introduction
The intermediate long wave (ILW) equation is a model for long gravity waves in a stratified, finite-depth fluid [1]. The model is integrable: it has a Lax pair [2,3], a Hirota bilinear form and N -soliton solutions [4], a Bäcklund transformation [5], and an infinite number of conservation laws [5].
In this paper we consider the following ILW-type equation which was recently introduced in [6]: where u(x, t)and v(x, t) are in general complex functions of x, t ∈ R and (1. 2) The parameter δ > 0 is assumed to be positive. We refer to (1.1) as the non-chiral ILW equation because it is invariant under the parity transformation [u(x, t), v(x, t)] → [v(−x, t), u(−x, t)] (the standard ILW equation and its degenerations are not parity-invariant). A quantum version of the non-chiral ILW equation arises from a second-quantization of the elliptic Calogero-Sutherland model [6], but here we consider only mathematical aspects of the classical model (1.1). More precisely, we show that the usual structures of integrability, listed above for the standard ILW equation, are present also in (1.1).
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. A Bäcklund transformation is constructed from the Hirota bilinear form in Section 4. Section 5 contains two independent, complementary derivations of an infinite sequence of conservation laws. In Section 6, we extend the above results to the periodic setting, where Weierstrass elliptic functions arise naturally and play a fundamental role in the analysis. Some properties of the operators T andT defined in (1.2) and of certain elliptic functions are collected in the two appendices.
In what follows we assume that the arguments of T andT are sufficiently regular and decay sufficiently rapidly to justify our arguments. We occasionally comment on specific necessary or sufficient conditions for clarity.

Lax pair
In this section we derive a Lax pair for (1.1). The ansatz used to obtain the Lax pair is a generalization of the ansatz used in [7] to find a Lax pair for the standard ILW equation; the operative difference here is that the underlying Riemann-Hilbert (RH) problem on the cylinder has a pair of jumps instead of a single jump. In Section 5, we will illustrate the utility of the Lax pair by using it to derive an infinite number of conservation laws for (1.1).
For fixed δ > 0, let C denote the cylinder C = C/2iδZ and let π : C → C be the natural projection. We can identify C with the strip C {z ∈ C : 0 ≤ Imz < 2δ} and a function f : C → C can be viewed as a function f : C → C which is periodic with period 2iδ, i.e., f (z + 2inδ) = f (z), n ∈ Z. Let C 0 and C 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 C \ (C 0 ∪ C δ ) → C with jumps across C 0 and C δ . The boundary values of the eigenfunction, ψ ± (z), are functions C 0 ∪ C δ → C defined by ψ ± (x, t; k) := lim ↓0 ψ(x ± i , t; k), ψ ± (x + iδ, t; k) := lim ↓0 ψ(x + iδ ± i , t; k). (2.1) where µ 1 = µ 1 (k), µ 2 = µ 2 (k), ν 1 = ν 1 (k), and ν 2 = ν 2 (k) are complex-valued functions of the spectral parameter k ∈ C, and A(z, t; k) and B(z, t; k) are, for each t ∈ R and k ∈ C, bounded analytic functions C \ (C 0 ∪ C δ ) → C 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 Condition (2.3a) shows that B has no jump across C 0 ∪ C δ . Hence B is a bounded analytic function C → C, so B(z, t) = B 0 must be a constant. Condition (2.3b) then shows that A is a solution of the following scalar RH problem on C: • A : C \ (C 0 ∪ C δ ) → C is an analytic function, • across C 0 ∪ C δ , A satisfies the jump condition • A(z) = O(1) as z → ∞. To solve this problem we use the following lemma.
Lemma 2.1 (RH problem on C with a jump across C 0 ∪C δ ). Let J 0 : C 0 → C and J 1 : C δ → C be continuous functions such that has the unique solution (2.4) where both C 0 and C δ are oriented from Re z = −∞ to Re z = ∞. Moreover, this solution satisfies coth π(z + 2inδ) 2δ = coth πz 2δ .
Hence A descends to an analytic function C \ (C 0 ∪ C δ ) → C. For x ∈ C 0 ∼ = R, the Plemelj formula gives Similarly, for x + iδ ∈ C δ ∼ = R + iδ, This proves the expressions for the boundary values and shows that A satisfies the correct jump condition. As x → ±∞, we have coth(x + iy) = 1 + 2e ∓2iy e −2|x| + O(e −4|x| ) uniformly for y ∈ R. Hence the assumption that R J 0 (x) dx = 0 and R J 1 (x) dx = 0 implies that A(z) → 0 as z ∈ C tends to ∞.
Substituting these expressions for A ± into (2.3e) and (2.3d) and using that T andT commute with ∂ x from Proposition A.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 (2.6) have an analytic extension to C \ (C 0 ∪ C δ ) and can be alternatively written as where A = A(z, t; k) is given by (2.5).

Hirota bilinear form
In this section, we show that the bilinear system where D t and D x are the usual Hirota derivatives: and is equivalent to (1.1) in the sense of the following theorem.
A. Let F (z, t) and G(z, t) be 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, and

4)
for some constants f 0 , g 0 , f 1 ∈ C. Then F, G satisfy the bilinear system (3.1) if and only if the functions B. Suppose u(x, t), v(x, t) are solutions of (1.1) and the transforms T u, T v,T u,T v exist. Then F (x, t), G(x, t), defined up to inessential multiplicative constants by where and (3.4), are analytic for −δ/2 < Im z < δ/2, continuous for −δ/2 ≤ Im z ≤ δ/2, and satisfy the Hirota equations (3.1).
3.1. Proof of Theorem 2A. Suppose (F, G) and (u, v) are related as in (3.5). We write where u ± and v ± are defined by with z the complex extension of x. 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 and u − and v + are analytic in the strip −δ < Im z < 0. Additionally, we observe that Lemma 3.1. If g + (z) is analytic in the strip 0 < Im z < δ, continuous in the strip 0 ≤ Im z ≤ δ, and Similarly, if g − (z) is analytic in the strip −δ < Im z < 0, continuous in the strip −δ ≤ Im z ≤ 0, and Proof. Suppose g + (z) is a function which is analytic in 0 < Im z < δ, continuous in 0 ≤ Im z ≤ δ, and which satisfies (3.10). Using the definition ofT and then changing variables to z = x + iδ, we find We next use the identity tanh(z + iπ/2) = coth(z) and 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 where E(x) is defined by As R → ∞, the function coth( π(z −x) 2δ ) tends to 1 (resp. −1) uniformly for z ∈ [R, R + iδ] (resp. z ∈ [−R, −R + iδ]). Hence, thanks to the assumption (3.10), we have E = 0. Thus, using that equation (3.14) reduces to which is (3.11). The proof of (3.13) is similar.
We are now in a position to prove Theorem 2A.
Proof of Theorem 2A. According to Lemma 3.2, the non-chiral ILW equation (1.1) can be written as Since u t = i log F − G + xt and v t = i log G − F + xt , integration of (3.16) with respect to x gives where λ 1 (t) and λ 2 (t) are complex-valued functions. By considering the limit x → ∞ and using (3.4) and (3.8), we conclude that λ 1 (t) = λ 2 (t) = 0. Rewriting the system in terms of F and G, we obtain Simplification shows that the first equation can be rewritten as i.e., In the same way, the second equation can be written as Multiplying (3.17a) and (3.17b) by F − G + and F + G − , respectively, we conclude that (1.1) is equivalent to the bilinear system (3.1). This completes the proof.

3.2.
Proof of Theorem 2B. We decompose the solution u, v of (1.1) as . We view (3.5) as a pair of differential-difference equations for F , G and seek solutions satisfying We note that, using the hyperbolic identity for csch z := 1/ sinh z csch 2z = 1 2 (coth z − tanh z), the functions u ± + v ± may be written as As x → ±∞, we have csch x → ±2e −|x| + O(e −3|x| ) uniformly, so the second term decays rapidly. Hence the existence of T (u ± + v ± ) andT (u ± + v ± ) follows from the existence of T u, T v,T u,T v. LetC denote the cylinder C/iδZ,π the natural projection C →C, andC 0 the image of Im z = 0 underπ. Then (3.19) defines a pair of RH problems for the functions ∂ z log F and ∂ z log G onC. The following lemma can be proved similarly to Lemma 2.1. Then the scalar RH problem • A :C \ C 0 is analytic.
• AcrossC 0 , A satisfies the jump condition Moreover, this solution satisfies We are now in a position to prove Theorem 2B.
Remark 3.4. We note that the ansatz for (3.1), together with Theorem 2A, leads to an alternative proof of the following result in [6]. 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

Bäcklund transformation
Our derivation of a Bäcklund transformation for (1.1) is inspired by the analogous derivation for the Benjamin-Ono equation presented in [8,Chapter 3].
Suppose (u, v) and (ũ,ṽ) are two solutions of (1.1) with associated Hirota bilinear forms (3.1) and Then, in terms of the variables F , G,F ,G, the Bäcklund transformation of (1.1) is given by where α, β, γ ∈ C are arbitrary constants. Proof. Suppose the relations in (4.2) hold and that (F, G) satisfy (3.1). We will show that (F ,G) satisfies (4.1). The reverse implication then follows by symmetry. Consider the quantity To simplify this, we need a pair of identities [8, Appendix I]: Using (4.4a), we have We now use (4.2a) and (4.2b) to write Using both identities in (4.4), we find Finally, using (4.2a), we see that Q = 0. Thus, recalling (4.3), we see that (3.1a) is satisfied if and only if (4.1a) is satisfied. The proof for (3.1b) and (4.1b) is similar.

4.1.
Bäcklund transformation in terms of u, v,ũ,ṽ. To transform (4.2) into a form written in the original variables, we introduce potential functions U , V ,Ũ ,Ṽ by Theorem 3 (Bäcklund transformation for the non-chiral ILW equation). Suppose the following relations hold: Then (u, v) satisfy the non-chiral ILW equation (1.1) if and only if (ũ,ṽ) do.
Lemma 4.2. The following identities hold: (4.10) Proof. By Lemma 3.2, T u +T v + iu = 2iu + and the expression for u + follows after simplification. The expression for u − then follows because u = u + + u − . The expressions forũ ± follow in the same way.
Utilizing Lemma 4.2, equation (4.9) can be rewritten as Using (4.7) and setting and which is (4.6a). We next rewrite the t-parts (4.2a)-(4.2b) of the Bäcklund transformation as Subtracting the second of these equations from the first gives Multiplying by i and using the definitions (3.8) and (4.5) of u ± ,ũ ± and U,Ũ , this becomes Recalling (4.7) and using Lemma 3.2, we find Equation (4.11) can be written as Using this relation to eliminate (U +Ũ ) xx , we arrive at That is, which is (4.6b).
We next rewrite the x-part (4.2f). Dividing (4.2f) by G −F + yields (4.14) Lemma 4.3. The following identities hold: Proof. By Lemma 3.2, T v +T u − iv = −2iv + and the expression for v + follows after simplification. The expression for v − then follows The expressions forṽ ± follow in the same way.
Utilizing Lemma 4.3, equation (4.14) can be rewritten as With (4.7) and (4.12) this becomes which is (4.6c). We next rewrite the t-parts (4.2d)-(4.2e) of the Bäcklund transformation. As before, we find that (4.13) 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 (3.8) and (4.5) of v ± ,ṽ ± and V,Ṽ , this becomes Recalling (4.7) and using Lemma 3.2, we find Equation (4.16) can be written as Using this relation to eliminate (V +Ṽ ) xx , we arrive at That is, which is (4.6d). This completes the proof of Theorem 3.

Conservation laws
In this section, we provide two complementary proofs of the following theorem.
Theorem 4 (Conservation laws of the non-chiral ILW equation). The non-chiral ILW equation (1.1) has an infinite number of conservation laws where W n and Z n can be computed recursively from the following formal power series in , with P ± as in (4.8). The first four conservation laws are The first proof uses the Bäcklund transformation (4.6) while the second proof uses the Lax pair (2.6) and illustrates its utility. In both proofs we construct a conservation law with dependence on an auxiliary parameter; an infinite number of conservation laws are obtained by an appropriate expansion in this parameter. At the end of this section, we verify by direct computation that the three first three quantities in (5.3) are conserved. 5.1. Proof of Theorem 4 using the Bäcklund transformation. Adding equations (4.6b) and (4.6d), we find Using the anti-self-adjointness (A.2) of the operators T andT , the right-hand side vanishes. Hence R (W + Z) dx is a conserved quantity. If we expand W and Z formally in powers of as we find that ∞ n=1 dI n dt n = 0, (5.5) where The identity (5.5) must hold for arbitrary , therefore The I n defined in (5.6) is the nth conserved quantity of (1.1). To derive the explicit functional forms of W n and Z n , we substitute (5.4) into (4.6a) and (4.6c), which gives (5.2) Note that The terms of O(1) yield which gives the conservation law In fact the quantities I 1,u and I 2,v , defined in (5.3a), are individually conserved. We verify this by direct computation in Section 5.3 below. The terms of O( ) give that is, leading to the expression (5.3b) for I 2 . The terms of O( 2 ) give i.e., (5.10) After some simplifications, this gives the expression (5.3c) for I 3 : i.e., Then, a lengthy calculation using (5.7-5.10) and the identities (A.2-A.3) in (5.6) gives (5.3d).

5.2.
Proof of Theorem 4 using the Lax pair. We generalize the approach of [7]. It is first necessary to transform the Lax pair (2.6) into a more convenient form and define particular eigenfunctions. We define the functions We view W 1 (z) and W 2 (z) as analytic functions on the strip 0 < Im z < δ and we use the notation W ± for the continous boundary values of these functions as z approaches the lines C 0 (from above) and C δ (from below), respectively. In terms of (5.12), the x-part of the Lax pair (2.6) is written as We define a solution (W 1 , W 2 ) = (M 1 , M 2 ) to (5.16) by the asymptotic behavior 14) The existence of this solution implies the conditions We assume these conditions hold and define ζ(k) = k − µ(k), so that the full Lax pair in terms of the particular eigenfunctions M 1 and M 2 is Then σ 1 and σ 2 decay to zero as |x| → ∞, and, for large enough ζ, M + 1 and M + 2 are nonzero. Thus the function Lemma 2.1 therefore gives, for ζ sufficiently large, We next use these expressions to express the Lax pair (5.16) in terms of σ 1 and σ 2 . Dividing by M − 2 and M + 2 in the first and second equation, respectively, we find Using (5.19) and (4.8), this can be written as The next lemma provides the time evolution of σ 1 and σ 2 .
Lemma 5.1. The functions σ 1 and σ 2 satisfy Proof. Using that ζ = k − µ, the t-part of (5.16) can be written as Subtracting the first from the fourth equation and using that Similarly, subtracting the third from the second equation and using that we obtain Next note that and, by (5.19), Thus, the stated equations for σ 1,t and σ 2,t follow from (5.23) and (5.24). Equation (5.21) follows because T andT are anti-self-adjoint, see (A.2).
We can use the fact that R (σ 1 +σ 2 ) dx is conserved for all ζ to determine an infinite sequence of conservation laws for the non-chiral ILW equation (1.1).
As ζ → ∞, we have the expansions It follows from (5.21) that forms an infinite sequence of conserved quantities. Substituting (5.25) into (5.20) gives Using P * ± = −P ∓ , we see that (5.26) is precisely the complex conjugate of (5.2) with the identifications ζ = 1/ * , σ 1 = W * , σ 2 = Z * . Thus, the remainder of the proof is similiar to the proof of the first four conservation laws in Subsection 5.1 and hence omitted.
The verification for I 1,v is similar.
To verify the second conservation law (5.3b), we also need the anti-self-adjointness (A.2) of the operators T andT from Proposition A.1: To verify the third conservation law (5.3c), we need the identity (T T f )(x) = (TT f )(x) from Proposition A.1:

Periodic non-chiral intermediate long wave equation
In this section we extend the results obtained in Sections 2-5 to the spatially periodic setting. The corresponding periodic ILW equation has been studied in [7,9,10].
Let L > 0 and consider equation (1.1) where u(x, t) and v(x, t) are L-periodic functions of x ∈ R and the integral transforms are Here ζ 1 denotes the minor modification of the Weierstrass ζ-function defined in (B.2).
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.
Let Π 0 and Π δ denote the images on 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 defined by (2.1). We take the natural modification of (2.2) as our ansatz for the Lax pair: As before, 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 now analytic functions on Π \ (Π 0 ∪ Π δ ) to be determined. Demanding compatibility, using the same procedure as before, we obtain the equations These are precisely the same equations as in (2.3) except they now hold on Π 0 ∪ Π δ . From (6.3b), we see that B(z) is an analytic function on Π and so must be constant: B(z) = B 0 . Then (6.3c-6.3d) shows that A is a solution of the following RH problem on Π: Lemma 6.1 (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:

4)
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

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 (6.4). 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 (6.4), we see that A(z, t; k) = 1 π Π 0 ζ 1 (z − z|L, 2iδ)J 0 (z )dz (6.7) The remaining steps are identical to those in the non-periodic case and lead to the Lax pair for the L-periodic problem (1.1).

Hirota bilinear form.
In the periodic setting, the Hirota bilinear form of (1.1) is whereū andv are the 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 take theū andv to be constants. u(x, t) dx,

The proof forv is similar.
A bilinear form similar to (6.10) was used in [10] to construct periodic solutions of the standard ILW equation. We show that (6.10) is equivalent to (1.1) with (6.1) in the sense of the following theorem.

Theorem 5 (Hirota bilinear form of periodic non-chiral ILW).
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 (6.10) for someū,v ∈ C and complex-valued functions λ 1 (t) and λ 2 (t) if and only if satisfy (1.1). B. Suppose u(x, t), v(x, t) are L-periodic solutions of (1.1) with meansū andv, respectively. Then F (x, t), G(x, t), defined up to multiplication by an arbitrary function of t by x, t ∈ R, (6.13) are analytic for −δ/2 < Im z < δ/2, continuous for −δ/2 ≤ Im z ≤ δ/2, and satisfy the Hirota equations (6.10) with (6.14) 6.2.1. Proof of Theorem 5A. Suppose (F, G) and (u, v) are related as in (6.11). We write where u ± and v ± are defined by 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 (6.16) Lemma 6.3. 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 (6.1) 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 which is (6.17). The proof of (6.18) 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 ζ 1 (z − 2ω 2 ) = ζ 1 (z) + iπ/ω 1 from Proposition A.1, we find 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 which is (6.18).
Lemma 6.4. The functions u and v obey the identities x, t ∈ R.
Proof of Theorem 5A. According to Lemma 6.4, the non-chiral ILW equation (1.1) can be written as where λ 1 (t) and λ 2 (t) are arbitrary complex functions.
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 (3.17a) and (3.17b) by F − G + and F + G − , respectively, we conclude that (1.1) is equivalent to the bilinear system (6.10). This completes the proof.

Proof of Theorem 5B.
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 (6.13). We view (6.11) as a pair of differential-difference equations for F , G and seek solutions satisfying LetΠ denote the torus C/Λ, whereΛ is the lattice generated by {L, iδ},π the natural projection C →Π, andΠ 0 the image of Imz = 0 underπ. Then (6.26) 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 6.1. Lemma 6.5 (RH problem onΠ with a jump acrossΠ 0 ). Let J :Π 0 → C be a continuous function such that 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 Remark 6.6. Note that T δ 2 is the operator T but with δ replaced by δ 2 : It follows from (6.13) and the anti-self-adjointness of T andT from Proposition A.2, that the functions u ± , v ± and hence u ± + v ± have zero mean. Then, Lemma 6.5 shows that the general solution to the scalar RH problem: • A :Π \Π 0 is analytic, • acrossΠ 0 , A satisfies the jump condition is given by with boundary values i∂ z log G(z, t) = 1 2πi with corresponding boundary values where F 0 and G 0 are arbitrary complex functions of t.
Proof. We add the two equations in (6.23) and integrate over [−L/2, L/2] to obtain The second term vanishes by periodicity. Using (6.37) we obtain (6.38). It remains to show that I 2 is a conservation law. This is verified by a calculation identical to the direct verification of I 2 in Section 5.3, using the anti-self-adjointness of T andT from Proposition A.2.
Lemma 6.12. The following identities hold: (6.45) Proof. By Lemma 6.4, T u +T v + iu − i(ū −v) = 2iu + 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 6.12, equation (6.44) can be rewritten as Recalling (6.42) and setting which is (6.41a). We next rewrite the t-parts (6.36a)-(6.36b) of the Bäcklund transformation as Subtracting the second of these equations from the first gives Multiplying by i and using the definitions (6.15), (6.37), and (6.40a) of u ± ,ũ ± , U,Ũ , and Λ 1 ,Λ 1 , this becomes Recalling (6.42) and using Lemma 6.4, we find Equation (6.46) can be written as Using this relation to eliminate (U +Ũ ) xx , we arrive at That is, which is (6.41b). We next rewrite the x-part (6.36f). Dividing (6.36f) by G −F + yields Lemma 6.13. The following identities hold: Proof. By Lemma 3.2, 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 6.13, equation (6.48) can be rewritten as With (6.42) and setting which is (6.41c). We next rewrite the t-parts (6.36d)-(6.36e) of the Bäcklund transformation. As before, we find that (6.47) holds except that F,F and G,G are now evaluated at x + iδ/2 and x − iδ/2, respectively, i.e., (6.51) Multiplying by i and using the definitions (6.15), (6.37), and (6.40b) of v ± ,ṽ ± ,V,Ṽ , and Λ 2 ,Λ 2 this becomes Recalling (6.42) and using Lemma 6.4, we find Equation (6.50) can be written as Using this relation to eliminate (V +Ṽ ) xx , we arrive at That is, which is (6.41d). This completes the proof of Theorem 6. where W n and Z n can be computed recursively from the formal power series in with P ± as in (6.43). The first four conservation laws are 54a) 54b) (6.54c) Proof. Adding equations (6.41b) and (6.41d), we find Thus, d dt Using the anti-self-adjointness (A.8) of the operators T andT , the integral on the righthand side vanishes. The remainder of the proof is identical to that for Theorem 4 and hence omitted.

Discussion
In this paper we have presented a Lax pair, Hirota bilinear form, Bäcklund transformation, and an infinite sequence of conservation laws for the non-chiral ILW equation (1.1) and its periodic counterpart. While our results are generalizations of those for the standard ILW equation [2,4,5], we emphasize that the non-chiral ILW equation does not contain the standard ILW equation as a limiting case and exhibits features not present in the single-component case [6]. We conclude by mentioning three directions for future research.
(1) The derivation of the Lax pair in Section 2 suggests that an N -component generalization of the non-chiral ILW equation (1.1) could be obtained starting from a RH problem with N jumps on the cylinder. It would be particularly interesting to investigate the N → ∞ limit of this construction. Integrable equations with nonlocalities in the form u(−x, t) have recently attracted considerable attention [11]. We note that (7.1) can be viewed as a doubly nonlocal (both from the T andT transforms and space reversal) variant of the standard ILW equation. (3) The Szegö equation [12] and the half-waves map [13,14] are two recently-introduced equations that, like the non-chiral ILW equation (1.1), have nonlocalities given by a Fourier multiplier. Both of these equations are integrable by virtue of Lax representations and possess an infinite number of conservation laws. It would be interesting to investigate these equations from the perspective taken in this paper by constructing their Hirota forms and Bäcklund transformations and studying the pole dynamics of their solitons.
Appendix A. Properties of the T andT operators In this section we collect and prove several identities for the T andT operators, both on the line and in the periodic case.
We note that a necessary condition for the existence of any of the products of transforms T T f ,TT f , TT f ,T T f on the line is R f (x) dx = 0. Proposition A.1 (Properties of T andT on R). The operators T andT have the following properties Proof. (A.1). By the definition of T , The proof of the corresponding formula forT is similar.
(A.2). By the definition of T , Similarly, by the definition ofT , . By the definitions of T andT , Deforming the x -contour downward, we find In fact, E vanishes because Thus we find after the change of variables y = x + iδ that Thus, (A.4). By the definition ofT Deforming the x -contour downward, we find In fact, E vanishes because Thus we find after the change of variables y = x + iδ that Thus, Deforming the x -contour downwards, we find csch π(z − y) δ f (z) dz dy Using (A.6) again, we find The first integral in (A.11) can be computed using the definition of the principal value integral and the standard elliptic identities [15] σ(−z) = −σ(z), σ(z + 2ω 1 ) = −e −2η 1 (z+ω 1 ) σ(z): The second integral in (A.14) is found to be Hence, the right-hand side of (A.11) vanishes. The function f (z) = 1 satisfies the conditions of Lemma 6.3, hence (A.12) follows from (A.11) and (6.17).