Effective Generation of Closed-form Soliton Solutions of the Continuous Classical Heisenberg Ferromagnet Equation

The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and description.


Introduction
In this article we study the Heisenberg Ferromagnetic equation (HF), i.e., m t = m ∧ m zz , (1.1) where m(z, t) ∈ R 3 is a vector function satisfying m(z, t) → e 3 as z → ±∞ and m(z, t) = e 3 = 1 with e 3 = (0, 0, 1) T , and ∧ denotes the vector product. Here z denotes position and t time. Equation (1.1) describes the dynamics of the magnetization vector m of an isotropic ferromagnetic chain at the nanoscale in the absence of an external magnetic field.
We underline that this article is the preliminary version of a much longer paper currently in preparation, which will account for all the proofs of the outcomes reported herein, and discuss in a more detailed and extensive way the exact solutions arising from our main result, i.e. formula (3.10), along with their interactions and classification.
It is well known that (1.1) is integrable in the sense that it is possible to find a Lax pair associated with it (see [14]). For later (notational) convenience, let us briefly recall that if V is a 2 × 2 invertible matrix depending on position z ∈ R, time t ∈ R, and a spectral parameter λ, the Lax pair associated to (1.1) is given by: In (1.2) τ is defined as τ = m ∧ m z and σ is the column vector with entries the Pauli matrices Of course, the knowledge of the Lax pairs for (1.1) assures that the Inverse Scattering Tranform (IST) (see [1,2,13]) can be applied to solve the initialvalue problem for this equation, i.e., m t = m ∧ m zz , m(z, 0) known (1. 3) The first authors who applied the IST to the HF equation were Takhtajan and Zakharov [14,17]. The aim of this paper is twofold. The first goal is to present a (somehow novel) rigorous theory for the inverse scattering transform for the HF equation. In particular, the direct scattering problem is proved to be well posed for potentials such that: 1. m(z) · σ has an almost everywhere existing derivative with respect to z with entries in L 1 (R) 2. m 3 (z) > −1 for each z ∈ R.
Such conditions are less restrictive with respect to the usual Schwartz class hypotheses. For potentials satisfying 1. and 2. we establish the analyticity properties of eigenfunctions and scattering data. In order to obtain these results we derive a convenient set of Jost solutions (see (2.10) below) which allows to study their asymptotic behavior at large λ. The inverse scattering problem is formulated in terms of the Marchenko integral equations. These are obtained by using an appropriate triangular representation of the Jost solutions (see (2.13) in Section 2). The second objective of this paper is to find an explicit multi-soliton solution formula for (1.1), which allows a classification of all its localized solutions, along with a description of their interaction. In order to achieve this goal, we will apply the matrix triplet method, which has proved successful in solving exactly -in the reflectionless case -several integrable equations [6,7,8,9,10]. The idea of this method is to represent the Marchenko kernel as Ce −(y+z)A B (where (A, B, C) is a suitable matrix triplet) in such a way that the Marchenko integral equation can be solved explicitly via separation of variables. The solutions obtained in this way will contain nothing more complicated than matrix exponentials and solutions of Lyapunov or Sylvester matrix equations [12,3], hence can potentially be "unzipped" into lengthy expressions containing elementary functions. In the present preliminary paper we will show how to recover the famous HF one-soliton solution from (3.10).
The paper is organized as follows. In Section 2 we study the analyticity of the Jost solutions and scattering data, and determine their time evolution. Furthermore, we formulate the inverse scattering problem in terms of the Marchenko integral equation. In Section 3, combining the IST and the matrix triplet method, we get an explicit solution formula for (1.1). As a first example, we derive the explicit expression of the one-soliton solution.

Direct Scattering Theory
In this section we study the direct and inverse scattering theory associated to the first of equation (1.2). In particular, we analyze the analytic properties and the asymptotic behavior at large λ of the Jost solutions and the scattering data, and formulate the inverse scattering problem in terms of the Marchenko integral equations.
The proofs of the main theorems given in this section, i.e, Propositions 2.3, 2.4 and Theorems 2.5, 2.11, can be found in [16,11].
In the sequel, we also use the following notations: Then the differential equations Ψ z = AΨ and Φ z = AΦ (cf. with (1.2)) can be written as It is then easily verified 1 that Ψ(z, λ) and Φ(z, λ) belong to the group SU (2). As a result, Since the two Jost matrices are both solutions to the same first order linear homogeneous differential system, there exists a so-called transition matrix T (λ), depending on λ and belonging to SU (2), such that For λ ∈ R, we have where |a(λ)| 2 + |b(λ)| 2 = 1. We assume that a(λ) = 0 for each λ ∈ R, i.e., we assume that no spectral singularities exist. In order to formulate the Riemann-Hilbert problem we need to establish the properties of analyticity as well as the asymptotic behavior of the Jost solutions and the coefficients a(λ) and b(λ) at large λ. To get these results, let us put m 0 = m − e 3 . We can convert the differential systems (2.2) with corresponding asymptotic conditions (2.1) into the Volterra integral equations As a result of Gronwall's inequality (see Appendix of [4]) we get for (z, λ) ∈ R 2 where we assume that m 0 (z) = m(z) − e 3 has its entries in L 1 (R). We can easily prove the following Proposition 2.1 If m 0 (z) = m(z) − e 3 has its entries in L 1 (R), then the Faddeev functions e −iλz ψ up (z, λ), e −iλz ψ dn (z, λ), e iλz φ up (z, λ), and e iλz φ dn (z, λ) are analytic in λ ∈ C + and continuous in λ ∈ C + ∪R, while the Faddeev functions e iλz ψ up (z, λ), e iλz ψ dn (z, λ), e −iλz φ up (z, λ) and e −iλz φ dn (z, λ) are analytic in λ ∈ C − and continuous in λ ∈ C − ∪ R.
Proof. We give the proof only for the the Faddeev functions e −iλz ψ up (z, λ), e −iλz ψ dn (z, λ), e iλz ψ up (z, λ), e iλz ψ dn (z, λ) because the proof for the other Faddeev functions is very similar. The Volterra integral equations can be written in the form Using Gronwall's inequality, uniformly in (λ, z) for λ ∈ C ± and z ≥ z 0 > −∞, we obtain continuity in λ ∈ C ± and analyticity in C ± .
From (2.8) and (2.9) it is clear that no information is available on their asymptotics as λ → ∞. In order to get such information, let us derive a different set of Volterra integral equations. To do so we need to require that a. m(z) · σ has an almost everywhere existing derivative m (z) · σ with respect to z which has its entries in Under the above hypothesis, after straightforward calculations we get which is a matrix of determinant 1 2 (1 + m 3 (z)). Since we have required that m 3 (z) > −1 for each z ∈ R this matrix D(z) is invertible and its inverse 3 is bounded in z ∈ R. We may therefore apply Gronwall's inequality to (2.10) and find that Remark 2.2 In the same way and under the assumptions a and b, adapting the procedure above presented to the Jost matrix Φ(z, λ), we get We may therefore apply Gronwall's inequality to (2.12) to obtain 2 Under the first condition, m(z) is absolutely continuous in z ∈ R. Hence its pointwise values make sense and hence it makes mathematical sense to assume that, in addition, Equations (2.10) and (2.12) allow us to prove that the analyticity and continuity properties of the Jost solutions extend to the closed upper and lower half-planes. In other words, the Jost solutions and the coefficient a(λ) have a finite limit as λ → ∞ from within the closure of its half-plane of analyticity, while b(λ) vanishes as λ → ±∞. In order to prove these results we need to find a "suitable" triangular representation for the Jost solutions. We have the following: There exist an auxiliary matrix function K(x, y) such that The proof of Proposition 2.3 can be found in [16,11]. Analogously we have the following

Scattering Data
In this subsection, for the sake of completeness, we introduce the scattering matrix and the scattering coefficients. From now on, we assume that the coefficients a(λ) introduced in the preceding section is such that a(λ) = 0 for each λ ∈ R, i.e., there are no spectral singularities. We can write the identity (2.4) as the following Riemann-Hilbert problems: (2.16b) Putting F − (z, λ) = φ(z, λ) ψ(z, λ) and F + (z, λ) = ψ(z, λ) φ(z, λ) , we obtain the Riemann-Hilbert problem where the scattering matrix S(λ) is defined by We write where T , R, and L are called the transmission coefficient, the reflection coefficient from the right, and the reflection coefficient from the left, respectively. Equations (2.16) then imply that Thus S(λ) is σ 3 -unitary and has determinant a(λ) * /a(λ). Also, S(λ) → e −iα I 2 as λ → ±∞ for a suitable complex number e −iα of modulus 1. We easily derive the Fourier representations converges uniformly in z ∈ R.
The scattering data associated with the first of equation (1.2) are: 1. one of the reflection coefficients, 2. the poles of the transmission coefficient T (λ) (or of T (λ * ) * ). We call such poles the discrete eigenvalues in the upper half-plane C + (or in the lower half-plane) and denote them by ia j (or by −ia * j ) for j = 1, . . . , N .
3. a set of constants N j (N j ) for j = 1, . . . , N associated to the discrete eigenvalues ia j (−ia * j ) j = 1, . . . , N in the upper half-plane (lower halfplane). These constants are called the norming constants.
By using elementary arguments of complex analysis it is possible to prove that if there are no spectral singularities then the number of discrete eigenvalues is finite. We present, for the sake of simplicity, the scattering theory assuming that each pole of the transmission coefficient has multiplicity equal to one (the general case can be treated as shown in [5]). The way to construct the norming constants is standard (see [1,2,13]). However, for the sake of completeness, we prefer to insert their construction. Let us assume that there are finitely many poles ia 1 , . . . , ia N of the transmission coefficient T (λ) in the upper half-plane C + , all of which are assumed to be simple. We let τ s stand for the residue of T (λ) at λ = ia s , i.e., where the dot indicates differentiation with respect to λ. We then introduce the norming constants N s by By the same token, T (λ * ) * has the simple poles −ia * 1 , . . . , −ia * N in C − , all of them simple. The corresponding norming constants N s are defined by 20b) The next proposition shows how the norming constants introduced in the upper half-plane are related to those defined in the lower half-plane.
Taking the complex conjugate of the first equation and premultiplying the result by ( 0 1 −1 0 ), we obtain the second equation, provided N s = −(N s ) * .

Marchenko Equations
In this subsection we formulate the Marchenko integral equations and establish the connection between the solutions of these equations and the solution of the HF equation. We refer the reader to [16,11,5] for the details on the derivation of (2.24) below. In order to derive the Marchenko equations we need the following We refer to [16,11] for the proof.
We have the following where and ρ(w) is the Fourier transform of the reflection coefficient (see (2.21)).
Recall that H(z) ∈ SU (2). Writing K(z, y) = H(z)L(z, y) we can convert (2.22) into the traditional Marchenko integral equation. By following the same proof as in the focusing AKNS case [5,3], we find that the integral equation is uniquely solvable on the space L 1 (z, +∞) 2×2 . We observe that analogous Marchenko equations are satisfied by the auxiliary function N (x, y) which appear in (2.14). More precisely, we have the following Theorem 2.9 The auxiliary function N (x, y) which appears in (2.14) satisfies the following integral Marchenko equations wherẽ where K(z, y) and L(z, y) satisfy the Marchenko integral equations (2.22) and (2.24), respectively. Using the Volterra equation (2.5a) and the asymptotic relation Ψ(z, λ)e −iλzσ 3 → I 2 as z → +∞, we get from the triangular representation (2.13) (2.30) We refer the reader to [16,11] for the proof.

Time Evolution of the Scattering Data
In this subsection we derive the time evolution of the scattering data. We shall arrive at the same time evolution as for the NLS equation.
Recall the AKNS pair is given by (1.2). Suppose V (z, t; λ) is a nonsingular 2 × 2 matrix function satisfying where V does not need to be one of the Jost matrices. Then there exist invertible matrices C Ψ and C Φ , depending on (t, λ) but not on z, such that Here the left-hand side does not depend on z, whereas the right-hand side only seemingly depends on z. We may therefore allow z to tend to +∞ without losing the validity of (2.31). Since B −2iλ 2 σ 3 and Ψ e iλzσ 3 as z → +∞, we obtain The same result can be obtained for the other Jost matrix Φ(z, λ). Using that Ψ = ΦT , we get Hence, a(λ) and T (λ) do not depend on t, whereas R(λ, t) = e −4iλ 2 t R(λ, 0), L(λ, t) = e 4iλ 2 t L(λ, 0). (2.34) Differentiating (2.20a) with respect to t, we obtain Using (2.31) for Φ and Ψ and using (2.32), we get

Inverse Scattering Transform.
Having presented the direct scattering problem (consisting of the construction of the scattering data when m(z, 0) is known), the inverse scattering problem (amounting to the construction of m(z) when the scattering data are given), and the time evolution of the scattering data associated to the first of equation (1.2), we can discuss how the IST allows us to obtain the solution to the initial value problem for (1.1).
Using the initial condition m(z, 0) as a potential in system (1.3), we develop the direct scattering theory as shown above and build the scattering data. Successively, let the initial scattering data evolve in time in agreement with equation (2.33)-(2.35). The solution of the Heisenberg equation is then obtained by solving the Marchenko equation (2.24) where the kernel Ω(w) is replaced by Ω(w; t) (i.e., taking into account (2.33), (2.34), and (2.35)), and then using relation (2.30).
Gauge transformation. In this paragraph we show how the gauge transformation (see [14,17]) between the solutions of the Heisenberg equation and the solutions of the Nonlinear Schrödinger equation is determined by the matrix H(z) (H(z)) introduced in the triangular representation (2.13) ((2.14)).
Then we have the following: The solution of the initial value problem (1.3) are expressed in terms of the Jost solutions of the Zakharov-Shabat system as: The proof can be found in [16,11].

Matrix Triplet Method
In this section we construct an explicit soliton solution formula for equation (1.1). We apply the same technique successfully used in [6,7,8,9,10] to solve the NLS, the cmKdV, the sine-Gordon, the discrete NLS, and the Hirota equations, respectively. Furthermore, we use the triplet method to get explicit expressions for the Jost solutions in the reflectionless case when the corresponding scattering data are specified.

Explicit soliton solutions for equations (1.1).
We want to restrict ourselves to the case R(λ) = 0. In this case the expression for Ω l (w; 0) is given by (2.23), with ρ = 0. In particular, we can treat the situation where the discrete eigenvalues are not necessarily simple [5] by introducing the function are the corresponding norming constants, and n j is the geometric multiplicity of i a j .
To recover the solution of (1.3) we follow the three steps indicated below: a. Assume that the discrete eigenvalues {ia j } N j=1 and the norming constants {N js (t)} n j −1 s=0 N j=1 are given, with N denoting the number of discrete eigenvalues in C + , and n j denoting the geometric multiplicity of i a j . By using the scattering data we introduce the matrix function where u > z, and the kernel Ω(z, y) is given by (3.2).
c. Construct the potential m(z; t) by using formula(2.30): An analogous procedure can be followed by using the Marchenko equation (2.27).
Let us follow the above procedure, momentarily disregarding the time dependence. We will then show how to take into account also the time dependence.
It is well known [12,3] that it is possible to factorize a matrix function in the form (3.2) by using a suitable triplet of matrices. More precisely, suppose (A, B, C) is a triplet of matrices such that all the eigenvalues of the q × q matrix A have positive real parts, B is 2q × 2, and C is 2 × 2q. Let Ω(w) be defined as where the scalar function ω(w) is defined as ω(w) = Ce −wA B, with A a q × q matrix having only eigenvalues with positive real part, B a q × 1 matrix, and C a 1 × q matrix. Furthermore, let us assume that the triplet (A, B, C) is a minimal triplet in the sense that the matrix order of A is minimal among all triplets representing the same Marchenko kernel by means of (3.3) [12,3]. We then define Then provided the inverse matrix exists for each z ∈ R. Actually, the inverse matrix exists thanks to the unique solvability of (3.6). Finally, in order to reconstruct (1.3) we have to integrate (3.7) with respect to y, obtaining the explicit formulaL By means of this latter expression we can write the right-hand side of (2.30) in explicit form, and recover the components m j (z) of the vector m(z).
In order to introduce the time dependence we have to take into account the time evolution of the scattering data expressed by (2.33)-(2.35). Defining the (reflectionless) Marchenko kernels as follows: we may replace the matrix triplet (A, B, C) by (A, B, Ce −4itA 2 ) in such a way that (2.33), (2.34) and (2.35) are satisfied (A contains the discrete eigenvalues which are time independent, and C the norming constants). As a consequence, the explicit right-hand side of (2.30) is given by: where We remark that formula (3.10) allows us to compute explicitly the functions m i (z, t) satisfying the HF equation (1.1). The following example illustrates this fact in the most elementary case: the one-soliton solution.  Consequently, Ψ(z, λ)e −iλzσ 3 = H(z) I 2q − iC P + e 2zA −1 (λI 2q − iσ 3 A) −1 Bσ 3 = H(z) I 2q − iCe −zA I 2q + e −zA Pe −zA −1 e −zA (λI 2q − iσ 3 A) −1 Bσ 3 .