Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics

We present a class of periodic solutions of the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation, which was recently introduced by the authors together with Langmann as a classical, continuum limit of an Inozemtsev-type spin chain. These exact analytic solutions are constructed via a spin-pole ansatz written in terms of certain elliptic functions. The dynamical parameters in our solutions solve an elliptic spin Calogero-Moser (CM) system subject to certain constraints. In the course of our construction, we establish a novel B\"acklund transformation for this constrained elliptic spin CM system.


Introduction
A classic result in the theory of integrable systems [1,2] states that the soliton dynamics of the Korteweg-de Vries equation is governed by an (A-type) Calogero-Moser (CM) system. This relation between two of the best-known integrable systems is but one instance of a soliton-CM correspondence, whereby integrable PDEs are linked to many-body systems of CM type. For many such PDEs, including the Korteweg-de Vries [1,2], nonlinear Schrödinger [3], Benjamin-Ono [4], and intermediate long wave [5,Chapter 3] equations, this is accomplished by making an ansatz for the solution with time-dependent poles in the complex plane and showing that the locations of these poles evolve according to a (complexified) CM system. As such CM systems are exactly-solvable [6], this process provides classes of exact analytic solutions to the PDEs.
A complementary approach is to construct integrable systems with infinite degrees of free-dom by taking continuum limits of CM systems. The long-range character of the interactions in the CM system corresponds to nonlocal terms in the continuum description, resulting in partial integro-differential equations of Benjamin-Ono type [7,Chapter 4]. This concept was pioneered by Abanov, Bettelheim, and Wiegmann [8], who showed that the continuum dynamics of the rational CM system is described by Euler hydrodynamic equations that are equivalent to an integro-differential variant of the nonlinear Schrödinger equation [9]. Recent studies have applied this idea to CM-type systems with spin degrees of freedom, first introduced by Gibbons and Hermsen [10]; see also [11]. The half-wave maps (HWM) equation was derived in [12] and [13,14] as a continuum limit of a classical Haldane-Shastry spin chain [15,16], a limiting case of the trigonometric spin CM system [17]. Lax integrability and an infinite number of conservation laws were established for the HWM equation in [18] and multi-soliton solutions were constructed in [19,20]. Moreover, the HWM equation admits a family of periodic solutions governed by a trigonometric spin CM system [19]. Thus, the HWM equation is linked in two distinct ways to the trigonometric spin CM system. In the present paper, we show that this twofold relation can be lifted to the elliptic setting.
The non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation is a generalization of the HWM equation related to the elliptic spin CM system. Together with Langmann, we introduced the ncIHF equation in [21] as a continuum limit of a classical Inozemtsev spin chain [22]; the latter is simultaneously an elliptic generalization of the Haldane-Shastry spin chain and a limiting case of the elliptic spin CM system. It is important to note that the ncIHF equation comes in two related variants: (i) an equation with periodic boundary conditions and (ii) an equation posed on the real line, which may be obtained as an infinite-period limit of the first. In this paper, we study the former, which we call the periodic ncIHF equation. Basic integrability results for the ncIHF equation on the real line, where the analysis is technically simpler, have already been obtained in [21]: the (aperiodic) ncIHF equation admits a Lax pair, an infinite number of conservation laws, and multi-soliton solutions governed by the hyperbolic spin CM system. One major result of this paper is that the periodic ncIHF equation admits a family of solutions, analogous to the multi-solitons of the (aperiodic) ncIHF equation, governed by the elliptic spin CM system. As the elliptic spin CM system is exactly-solvable [23], this gives a new class of exact analytic solutions to the periodic ncIHF equation.
The periodic ncIHF equation generalizes known integrable systems (the HWM and real-line ncIHF equations) and originates from another (the elliptic spin CM system) but a Lax pair for it has not yet been established. While we regard the construction of such a Lax pair as an interesting question for future work, the class of exact solutions presented in this paper provides evidence for the integrability of the periodic ncIHF equation. The construction of these solutions is more involved than that of analogous solutions for the ncIHF equation on the real line [21] due to the presence of elliptic functions in both (1.2) and the spin-pole ansatz (1.16) given below. More specifically, a dynamical background vector is necessitated in the spin-pole ansatz and the resulting spin-pole dynamics must satisfy extra constraints versus the real-line case. To overcome these complications and link the spin-pole dynamics to an elliptic spin CM system, we prove a new Bäcklund transformation for the latter. This Bäcklund transformation is a key result of this paper; we believe it is also of independent interest for its striking difference from known Bäcklund transformations in degenerate cases [25,26,27,21]; namely, a new degree of freedom, corresponding to the background vector in the spin-pole ansatz, is required to mediate the transformation between two solutions of spin CM systems.
In the remainder of this section, we focus on stating and describing our two main results, a Bäcklund transformation for the elliptic spin CM system and a class of exact solutions of the periodic ncIHF equation, and describe the organization of the paper. Before proceeding, we introduce notation used in this section and throughout the paper.

Notation
We use the shorthand notation N k =j for sums N k=1,k =j , etc. The components of a three-vector s ∈ C 3 are denoted by (s 1 , s 2 , s 3 ) and the dot and cross products of two vectors s, t ∈ C 3 are defined as s · t = 3 a=1 s a t a and s ∧ t = (s 2 t 3 − s 3 t 2 , s 3 t 1 − s 1 t 3 , s 1 t 2 − s 2 t 1 ), respectively. The set of real vectors s ∈ R 3 satisfying s · s = 1, i.e., the two-sphere, is denoted by S 2 . We write the zero vector as 0 = (0, 0, 0). Dots above a variable indicate differentiation with respect to time while primes indicate differentiation with respect to the argument of a function. Complex conjugation and matrix transposition are denoted by * and ⊤, respectively.

Bäcklund transformation for an elliptic spin CM system
(Complexified) spin CM systems describe the time evolution of a system of N ∈ Z ≥1 particles with internal degrees of freedom moving in the complex plane. We consider the case where the internal degrees of freedom can be represented by complex three vectors, which is a special case of more general systems introduced by Gibbons and Hermsen [10] and Wojciechowski [11]; see [27] for the precise relation. Each particle is represented by a position a j = a j (t) ∈ C and a spin vector s j = s j (t) ∈ C 3 . We define the elliptic spin CM system to be the following system of equations, where ℘ 2 (z) is, up to an additive constant, the Weierstrass ℘-function with half-periods ℓ and iδ, Our definition of the elliptic spin CM system (1.5) differs from others in the literature, e.g. [23,26]. More specifically, we use the potential ℘ 2 (z) in place of either ℘(z) or ℘ 1 (z) := ℘(z) + ζ(ℓ)/ℓ, which differ from ℘ 2 (z) by additive constants. However, by multiplying each s j in (1.5) by an appropriate time-dependent complex rotation R = R(t) ∈ SO(3; C), the potential ℘ 2 (z) can be shifted to ℘ 2 (z) + c for any constant c ∈ C. A proof of this claim can be found in Appendix B.
Elliptic spin CM systems are known to be exactly-solvable [23], which gives, in principle, exact analytic solutions of the periodic ncIHF equation via our main result, Theorem 2, presented below. From our current perspective, the most important property of (1.5) is the existence of a Bäcklund transformation relating certain distinct solutions of the elliptic spin CM system; later we will employ this Bäcklund transformation to link the periodic ncIHF equation to the elliptic spin CM system. A Bäcklund transformation valid for the rational, trigonometric, and hyperbolic Gibbons-Hermsen spin CM systems [10] was presented in [25]; see [26] for a detailed proof. We now describe a Bäcklund transformation for the elliptic spin CM system (1.5) subjected to certain constraints which arise in our analysis of the periodic ncIHF equation (1.1).
Consider a second elliptic spin CM system for M ∈ Z ≥1 particles described by positions b j = b j (t) and spin vectors t j = t j (t) ∈ C 3 ; the equations of motion read Under appropriate circumstances, solutions of (1.5) and (1.7) may be related via a system of first-order differential equations involving also a vector φ = φ(t) ∈ C 3 , and The precise statement is as follows.
A proof of Theorem 1 is given in Section 4. Each of the constraints (1.12)-(1.14) corresponds to a conserved quantity; if the constraints are satisfied at t = 0, they also hold at future times when the first-order equations (1.9), (1.8), (1.5b), and (1.7b) are satisfied; this fact is proven for (1.12) and (1.13) in Proposition 3.1 and for (1.14) in Lemma 2.3. , and (1.14) unchanged. We will use Theorem 1 in the proof of Theorem 2 below; for this application, it is convenient to have the terms iδ in place.

A class of elliptic solutions of the periodic ncIHF equation
We construct solutions of the periodic ncIHF equation with dynamics governed by a pair of elliptic spin CM systems, which are related to each other through the Bäcklund transformation of Theorem 1. More specifically, we make the ansatz for solutions of the periodic ncIHF equation, where φ(t), s j (t), t j (t) ∈ C 3 and a j (t), b j (t) ∈ C and show that these parameters must satisfy the assumptions of Theorem 1. In this case, the ansatz (1.16) will satisfy u(x, t) 2 = v(x, t) 2 = ρ 2 , for some constant ρ ∈ C, provided certain constraints on the initial values of the parameters are fulfilled. Theorem 1 and standard results concerning the existence and uniqueness of solutions to systems of ODEs allow us to formulate our result as a relation between (i) certain solutions of the elliptic spin CM systems (1.5), (1.7) and background dynamics (1.9) and (ii) a class of solutions of the periodic ncIHF equation satisfying u(x, t) 2 = v(x, t) 2 = ρ 2 . The precise statement is now given.
for some constant ρ ∈ C at t = 0. Moreover, suppose that the conditions  and v(x+2ℓ)−v(x) are proportional to N j=1 s j − M j=1 t j , i.e., the left hand side of (1.14). We later show that (1.14) corresponds to a conserved quantity of the elliptic spin CM system: if it is satisfied at t = 0, as required in Theorem 2, then it holds for t ∈ [0, T ), see Lemma 2.3. The constraint (1.14) is also required for u(x, t) 2 = v(x, t) 2 = ρ 2 , see Proposition 2.1. Remark 1.4. We emphasize that the solutions in Theorem 2 are generically complex-valued, i.e., u(x, t), v(x, t) ∈ C 3 and satisfy u(x, t) 2 = v(x, t) 2 = ρ 2 for some constant ρ ∈ C. Real-valued solutions of unit length are described by the consistent reduction M = N , ρ = 1, φ * = φ, b j = a * j , and t j = s * j of the theorem, which is given as Corollary 2.1 in Section 5, where examples of such solutions are presented. We have chosen our approach because (i) the proofs in the generic, complex case are no more difficult than in the real case and (ii) at least one interesting class of solutions, considered in Section 5.1, is necessarily complex: in the case N = M = 1 of Theorem 2, which contains one-soliton, traveling wave solutions in the analogous real-line case [21, Section 6.1], there is no solution of the constraints (1.12) and (1.14) satisfying s * 1 = t 1 ; all solutions obtained under these conditions from Theorem 2 are complex.

Plan of the paper
We prove Theorem 2 by establishing a sequence of intermediate results including Theorem 1. In Section 2, we derive constraints on the parameters in (1.16) and we show that the parameters satisfy the first-order system of ODEs of Theorem 1. We show that this system of ODEs preserves the constraints (1.12)-(1.14) and (1.17) in Section 3. In Section 4, we prove the Bäcklund transformation, Theorem 1, in the course of proving Theorem 2. Examples of solutions of the ncIHF equation from Theorem 2 are constructed in Section 5. Appendix A contains identities for the special functions used in the paper. Appendix B contains a formal statement and proof of the claim in Remark 1.1.

Constraints and first-order dynamics
We derive conditions under which the ansatz (1.16) satisfies (i) u(x, t) 2 = v(x, t) 2 = ρ 2 and (ii) solves (1.1). The first requirement yields a number of nonlinear constraints on the parameters appearing in (1.16), which are obtained in Section 2.1. In Section 2.2, we show that when the ansatz (1.16) is subjected to one of these constraints and inserted into (1.1), the latter is reduced to a system of first-order ODEs.
To prove results in this section, we employ certain notation developed in [21]. Given C-valued functions F j , G j , j = 1, 2, we form two-vectors and define the following product, Similarly, we can combine pairs of three-vectors a j , b j ∈ C 3 , j = 1, 2, and define analogs of the dot and wedge products, and using (2.2) we may write the periodic ncIHF equation (1.1) as with T andT as defined in (1.2).
It is also useful to write the ansatz (1.16) using this two-vector notation. We define so that (1.16) can be written as using also the shorthand notation

Constraints
The following proposition establishes the conditions required for the functions in ansatz (1.16) to have constant length.
Proof. Using (2.7) and (2.2), we compute To proceed, we need the identities and and The identities (2.10) and (2.11) follow from the elliptic identities (A.1) and (A.2), respectively together with the definitions of E, A ± (z) (2.6) and F ± (z) (2.13). We evaluate the double sum in (2.9) using (2.10) for j = k and (2.11) for j = k: (2.14) Next, we recall that the functions ζ 2 (z) and f 2 (z) appearing in A ± (z) and F ± (z) are odd and even, respectively (A.5). Using this symmetry to rewrite the double sums in the second and third lines of (2.14) and collecting terms, we find We set (2.15) equal to ρ 2 E and note that E, are linearly independent as a consequence of (1. and and by inserting (2.18) into the sum proportional to E in (2.15), we obtain The constraints (2.16)- (2.19) are seen to be equivalent to (1.12-1.14) and (1.17) after recalling the notation (2.8) and (2.12).

First-order dynamics
The following proposition describes conditions, in the form of a system of ODEs, when the ansatz (1.16) solves the periodic ncIHF equation Proof. We compute both terms in the periodic ncIHF equation in the form (2.4), again making use of the form (2.7) of the pole ansatz (1.16) and the shorthand notation (2.8) and (2.12). First, we have using thatȧ j =ȧ j . We compute the remaining term in (2.4) in several steps. To compute T U x , we use the fact that the A ′ r j (x − a j ) are eigenfunctions of T when (1.18) holds, The identity (2.21) is established by verifying that the functions ℘ 2 (z − a j ) appearing in A ′ ± (z) (2.6) satisfy the conditions of the following result proved in [19, Appendix A]: 2 for a 2ℓ-periodic function g(z) analytic in a strip −d < Im z < d with d > δ/2 and satisfying ℓ −ℓ g(x) dx = 0, the functions G ± (x) := (g(x ± iδ/2), g(x ∓ iδ/2)) ⊤ are eigenfunctions of the operator T with eigenvalues ∓i.
and hence (2.21) implies We compute U T U x by combining (2.7) with (2.23): Differentiating (2.11) with respect toã k gives inserting this identity into (2.24), we have Next, since ∧ is antisymmetric and ℘ 2 (z) is an even function (A.5), we can rewrite the double sum in the first line of (2.26) according to Hence, inserting this and swapping some indices j ↔ k (using the antisymmetry of ∧ and the fact that ζ 2 (z) is an odd function (A.5)) in (2.26), we obtain which may be rearranged to where we have used s k ∧ s k = 0 to rewrite the final sum.
Inserting (2.20) and (2.29) into (2.4) and using the linear independence of E,  Proof. We differentiate S in (2.33) with respect to t and insert (1.5b) to find The sum vanishes because ℘ 2 (z) is an even function (A.5) and hence the summand is antisymmetric under the interchange of j and k. The proof for T is similar.
Because S and T are conserved quantities, so is their difference and hence, (1.14) holds on [0, T ) if it is satisfied at t = 0.

Conserved quantities
This section is devoted to proving that the constraints in Proposition 2.1 correspond to conserved quantities of the ODE system of Proposition 2.2, i.e., if the constraints are satisfied at t = 0, as required by Theorem 2, they hold at all future times. We note that the constancy of the total spins appearing in the constraint (1.14) was already proved in Lemma 2.3.
We prove the following.
Proposition 3.1. Under the assumptions of Proposition 2.2, the following quantities are conserved:

Proof of Proposition 3.1
We prove the proposition in three parts corresponding to the quantities (3.1), (3.2), and (3.3).

Conservation of P j
Using the notation (2.8), we write (3.1) as Differentiating this with respect to time and inserting (2.31), we havė where we have used the invariance of the vector triple product under cyclic permutations, with the shorthand notation (2.8) and (2.12).
Differentiating (3.7) with respect to time and inserting (2.31)-(2.32), we finḋ We use (3.6) again to reorder triple products, To proceed, we rewrite the quantity (b j −b k )℘ 2 (ã j −ã k ) in a convenient way. By the definition of b j (3.8), (3.11) where we have used the fact that ζ 2 (z) is an odd function (A.5) in the second step. To proceed, we use the identities and The first identity (3.12) is obtained by differentiating (A.1) with respect to z and the second identity (3.13) is obtained by differentiating (A.2) with respect to a and setting z =ã j , a =ã k , and b =ã l .
Inserting (3.12) and (3.13) into (3.11) and simplifying gives where we have used Lemma 2.3 in the final step to replace N l=1 r l s l by 0.
Inserting (3.14) into (3.10) giveṡ The sum in the first line of (3.15) vanishes as a consequence of (3.6). The double sum in the second line of (3.15) may be symmetrized, (using (3.6), the antisymmetry of ∧, and the fact that ℘(z) is an even function (A.5)) and the final sum in (3.15) may be rewritten as where we have used s j · (s k ∧ s j ) = 0 and, similarly as in (3.16), symmetrized the double sum in the final step. Hence, using (3.6), we see that (3.16) and (3.17) are equal, leading to cancellation in (3.15). We are left witḣ where we have symmetrized the double sum (using (3.6), the antisymmetry of ∧, and the fact that f ′ 2 (z) is an odd function (A.5)) and inserted (1.9) in the second step. Noting that all terms proportional to s j · (s k ∧ s l ) with j = k and j = l are zero in the second double sum in (3.18) and using (3.6), we see thatQ j = 0.

Conservation of R
We write where By differentiating R (1) with respect to t and inserting (2.30) and which follows from (3.8), we computė where we have used (3.6) in the last step.

Bäcklund transformation
We prove the Bäcklund transformation between the elliptic spin CM systems (1.5) and (1.7) stated in Theorem 1 in Section 4.1. Building on this result and using the results of Sections 2 and 3, we prove Theorem 2 in Section 4.2.

Proof of Theorem 1
This proof consists in deriving the deriving second-order equations Differentiating (4.2) with respect to t and rearranging gives We compute the terms on the right hand side of (4.3). Using (2.31) and then (2.32), To simplify, we use r 2 j = 1, the standard vector identities and (1.13) in the form b j · s j = 0. Hence, To compute the remaining term in (4.3), we first differentiate (3.8) with respect to t to finḋ where we have used thatȧ j =ȧ j . Taking the cross product with −r j s j and using (2.31) gives The double sum in (4.8) can be rewritten as using r 2 j = 1 and the parity properties of ℘ 2 (z) and ζ 2 (z) (A.5) in the second step. Then, the second identity in (4.5) and the constraint (1.12) (4.10) We simplify the remaining sum in (4.8) using (2.32), r 2 j = r 2 k = 1, and (4.5): By using (4.10) and (4.11) in (4.8), we arrive at Then inserting (4.6) and (4.12) into (4.3), we geẗ (4.13) using (1.13) in the form s j · b j = s k · b k = 0 in the second step.
We insert (3.14) (which was derived under the assumption (1.14)) into (4.13) to obtain, after combining some terms, (4.14) Since ℘(z) is an even function (A.5), the double sum in the third line of (4.14) is antisymmetric under the interchange of k and l and hence vanishes. The first double sum in the fourth line of (4.14) similarly vanishes by symmetry, because f ′ 2 (z) is an odd function (A.5). We again use the identity (3.12), leading to, after some rearrangement, The second identity in (4.5) and lead tö (4.17) Symmetrizing the double sum (using the antisymmetry of ∧ and the fact that f ′ 2 (z) is an odd function (A.5)) and inserting (2.30) gives the result (4.1) after recalling thatã j −ã k = a j − a k for r j = r k .

Proof of Theorem 2
We first show that the assumptions of the theorem imply those of Proposition 2.2.
By assumption, φ, {a j , s j } N j=1 , and {b j , t j } M j=1 in the statement of the theorem is a solution of the following initial value problem (IVP) for some choice of initial conditions. Initial value problem 1. Note that when s j = 0 and t j = 0, (1.8) may be written aṡ (4.20) We consider the following IVP. We denote the maximal solution of IVP 2 byφ, {â j ,ŝ j } N j=1 , and {b j ,t j } M j=1 . Theorem 1 shows that this solution of IVP 2 is also a solution of IVP 1 on [0, T ′ ). Suppose T ′ < T . We now consider two cases.
In the second case, suppose that either (4.21) is violated or least one of the quantitiesŝ j ·ŝ * j andt k ·t * We conclude that T ′ = T and so IVP 2 admits a unique maximal solution on [0, T ). By Theorem 1 and the uniqueness of the known solution φ, {a j , s j } N j=1 , and {b j , t j } M j=1 to IVP 1, we see that this known solution solves IVP 2 on [0, T ). It follows that assumptions of Proposition 2.

Examples of solutions
We construct examples of solutions of the periodic ncIHF equation (1.1) using Theorem 2. Analogs of one-soliton traveling wave solutions known for the ncIHF equation on the real line [27] are given in Section 5.1. We use an elliptic parameterization of S 2 to construct a class of real initial data for the periodic ncIHF equation satisfying the constraints of Theorem 2 in Section 5.2. The results of Section 5.2 are used to obtain a breather-type solution of the periodic ncIHF equation in Section 5.3.

Sections 5.2-5.3 are concerned with real-valued solutions
of the periodic ncIHF equation satisfying u(x, t) = v(x, t) 2 = 1. Such solutions are characterized by the following consistent reduction of Theorem 2 where Corollary 2.1. For N ∈ Z ≥1 and T > 0, let φ and {a j , s j } N j=1 be a solution of the equations (1.5) andφ on the interval [0, T ) with initial conditions that satisfy the following equations at t = 0,

Solutions that are sums of traveling waves
As mentioned in Remark 1.4, Theorem 2 does not include real traveling wave solutions when N = M = 1. In fact, as we will show, the class of solutions with N = M = 1 does not contain any traveling waves but instead consists of solutions that are the sums of two traveling waves moving in opposite directions.
Remark 5.1. The absence of nontrivial traveling wave solutions for the periodic ncIHF equation in Theorem 2 is surprising in view of the the rich structure of analogous solutions, obtainable via pole ansatz [27], for the half wave maps equation [12,13]. We regard the classification of traveling wave solutions of the periodic ncIHF equation as an interesting open problem.

Initial data from an elliptic parameterization of the two-sphere
One way to find initial data satisfying (1.12)-(1.14) and (1.17) is by considering the following parameterization of the two-sphere, defined by a map from R 2 to S 2 , (x 1 , x 2 ) → sn(x 1 |m)cn(x 2 |m), sn(x 1 |m)sn(x 2 |m), cn(x 1 |m) , (5.15) where sn(·|m) and cn(·|m) are the Jacobi sine and cosine functions with elliptic modulus m. The S 2 -valuedness of (5.15) can be shown using the identity (A.6). Requisite details on the functions sn(z|m) and cn(z|m) and the elliptic integrals 3 K = K(m) and K ′ = K ′ (m), which determine the periods of Jacobi elliptic functions, can be found in Appendix A. The functions sn(z|m) and cn(z|m) are elliptic functions of z with half-periods (2K, iK ′ ) and (2K, K + iK ′ ), respectively. Both functions have simple poles at with corresponding residues Proposition 5.1. Let m ∈ (0, 1), p, q ∈ Z ≥1 , and x 0 ∈ (0, 4K(m)) such that the sets and are disjoint. Then, (5.1) with N = 2(p 2 + q 2 ), ℓ = 2K(m), iδ = 2iK ′ (m), and (where c.c. denotes the complex conjugate of the terms within the parentheses) provides initial data for the periodic ncIHF equation satisfying the conditions of Theorem 2.

Proof of Proposition 5.1
We begin by writing the function r(x) in (5.18) in terms of the function ζ 2 (z) (1.10).

Numerical implementation
In the source file of our submission, we have included a Mathematica notebook to visualize solutions of the periodic ncIHF equation with initial data in the form (5.18). Using Proposition 5.1, such data may be transformed into the form (5.1) (a special case of (1.16)) to which Theorem 2 applies. For chosen p, q, m, and x 0 , our Mathematica notebook performs the transformation of Proposition 5.1 and uses the resulting parameters a j,0 , s j,0 , and φ 0 as initial conditions for the reduction (5.2) of the ODE system in Theorem 2. By numerically solving these ODEs, we obtain numerical solutions of the periodic ncIHF equation in the form (5.1). Visualizations of a particular solution obtained using this method are presented in Section 5.3.

A breather solution
We study a particular instance of the solution of the periodic ncIHF equation with initial data constructed using Proposition 5.1. This solution exhibits energy oscillations reminiscent of wellknown breather solutions of the nonlinear Schrödinger [30] and sine-Gordon equations [31]. To be more specific, we will present numerical evidence of a solution of the ncIHF equation where the To avoid misunderstanding, we emphasize that the results presented in this subsection are primarily numerical: a particular exact solution of the constraints (1.12)-(1.14) and (1.17) given in Section 5.2 provides admissible initial data for Theorem 2; we numerically solve the equations of motion of the spin CM system (1.5) and background dynamics (1.9) to evolve the solution (1.16) in time, using the method described in Section 5.2.2.
We set p = q = 1 and x 0 = K in (5.18) to obtain the following map from R to S 2 .  In accordance with Corollary 2.1, we solve (1.5) and (5.3) subject to the initial conditions (5.30) and with initial velocities computed from (5.6) (at t = 0). The resulting dynamics for the poles a j are time-periodic with period T ≈ 11.83. A visualization of the dynamics of the poles is shown in Fig. 1.
However, the dynamics of the spins s j and of the background vector φ are not time-periodic, and correspondingly, the solution (5.1) of the ncIHF equation is not time-periodic. This solution is shown in Fig. 2. We observe that at times t = T /4 + nT /2 (n ∈ Z ≥0 ), the solution has two points of non-differentiability. At these times, Corollary 2.1 does not apply but rather guarantees a solution of the ncIHF equation on intervals with the times {T /4 + nT /2 : n ∈ Z ≥0 } subtracted.
The energy density associated with this solution oscillates periodically in time, see Fig. 3. An explicit formula for the energy density is presented below in Section 5.3.1. At each time t, the total energy density ǫ(x, t) = ǫ u (x, t) + ǫ v (x, t) and the individual energy densities ǫ u (x, t) (red) and ǫ v (x, t) (blue) (5.32) of the u-and the v-channels are shown. The plots illustrate that the total energy density ǫ(x, t) is periodic with period T /2 ≈ 5.916, but the u-and v-channel energy densities are periodic with period T ≈ 11.83 only. At t = T the energy densities are exactly the same as at t = 0. s j · s * k ℘ 2 (a j − a * k + iδ)ζ 2 (x − a j + iδ/2) + (5.32)

A Special functions
We collect identities for the special functions needed in the main text.

A.2 Jacobi elliptic functions and elliptic integrals
We refer to [32,Chapter 16] for definitions of the Jacobi functions sn(z|m) and cn(z|m). These functions are elliptic in z; when the elliptic parameter m satisfies 0 < m < 1, the functions are real-valued for z ∈ R. The functions satisfy the identity sn 2 (z|m) + cn 2 (z|m) = 1. where S ∈ so(3; C) is defined to be Then, {Rs j , a j } N j=1 is a solution of (1.5) with ℘ 2 (z) → ℘ 2 (z) + c.

(B.3b)
We show that (B.3) holds if and only if (1.5) holds. Using the invariance of the dot product under orthogonal transformations, (Rs j ) · (Rs k ) = s j · s k , we see that (1.5a) and (B.3a) are equivalent. Cross products transform under orthogonal transformations as (Rs j ) ∧ (Rs k ) = R(s j ∧ s k ); using this fact and multiplying by R −1 in (B.3b) giveṡ We observe that R satisfies the differential equation where I is the 3 × 3 identity matrix and the choice of branch in √ S · S is immaterial. This is equivalent to Rodrigues' formula for the exponential map from so(3; C) to SO(3; C) [33].