# The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. II. IST and closed-form soliton solutions

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## Abstract

A new, general, closed-form soliton solution formula for the classical Heisenberg ferromagnet equation with in-plane asymptotic conditions is obtained by means of the inverse scattering transform technique and the matrix triplet method. This formula encompasses the soliton solutions already known in the literature as well as a new class of soliton solutions (the so-called multipole solutions), allowing their classification and description. Examples from all classes are provided and discussed.

## Keywords

Classical Heisenberg ferromagnet equation Soliton solutions Inverse scattering transform Magnetic droplet Ferromagnetic materials## Mathematics Subject Classification

35C08 35G25 35P25 35Q40 35Q51## 1 Introduction

*x*and time

*t*, where the vectors \(\varvec{e}_{j}\), \(j=1,2,3\), are the standard Cartesian basis vectors for \(\mathbb {R}^{3}\), \(\mathbb {S}^2\) is the unit sphere in \(\mathbb {R}^3\) and then \(\Vert \varvec{m}(x, t)\Vert =1\).

After the recent, first enucleation and experimental observation in a nano-contact spin-torque oscillator device of magnetic-droplet solitons [6, 7, 8, 9, 10, 11, 12, 13, 14], following their theoretical prediction [15, 16, 17, 18, 19, 20, 21, 22, 23], it has been theoretically shown in [24] how, as an extended magnetic thin film is reduced to a nano-wire with a nano-contact of fixed size at its center, the observed excited modes undergo transitions from a fully localized two-dimensional droplet into a pulsating one-dimensional droplet. This result has contributed to renew the interest in the study of low-dimensional magnetic solitons as a tool for better understanding the physics of ferromagnetic systems at the nano-meter length scale.

In this spirit, the present work aims at extending the analysis carried out in [25] for the classical, continuous Heisenberg ferromagnet equation with perpendicular (“easy-axis”) asymptotic conditions, \(\varvec{m}(x)\rightarrow \varvec{e}_3\) as \(x\rightarrow \pm \infty \), by constructing a new, general formula which generates all reflectionless solutions of (1a) under condition (1b), allowing their classification.

Special soliton solutions of (1a) with (1b) have been also recently constructed by means of the method of the Darboux transformation [26, 27].

In the present work, to reach our goal, that is, to find a general formula for the soliton solutions of (1a) satisfying condition (1b), we apply the inverse scattering transform (IST) [28, 29, 30] and the matrix triplet method [31, 32, 33, 34, 35] to (1a). For the sake of clarity let us briefly recall how the IST and the matrix triplet method work.

## 2 Soliton solutions formula

In this section we construct an explicit soliton solution formula for equation (1a) under the asymptotic condition (1b). To this aim, we apply the IST method (see, for instance [28, 29, 30] for more details on this method) combined with the matrix triplet technique, successfully used in [31, 32, 33, 34, 35] and more recently in [25] in the context of the classical Heisenberg ferromagnet equation.

### 2.1 Inverse scattering transform

Having presented in the first part of this work [5] the *direct scattering problem* (consisting in the construction of the scattering data when \(\varvec{m}(x, 0)\) is known), the *inverse scattering problem* (amounting to the construction of \(\varvec{m}(x)\) when the scattering data are given), and the *time evolution of the scattering data* associated to the first equation in system (3), we are now ready to discuss how the IST allows us to obtain the solution to the initial value problem for (1a).

### 2.2 Matrix triplet method

*n*is the number of discrete eigenvalues \(\{ia_j\}_{j=1}^{n}\), namely the poles of the transmission coefficient \(T(\lambda )\) in \(\mathbb {C}^+\) (thus, satisfying \(\mathrm {Re}(a_{j})>0\)); the quantities \(a_{j}\) are obtained by multiplying the discrete eigenvalues by \(-i\); \(n_j\) is the algebraic multiplicity of \(ia_j\); and \(\left\{ N_{jk}(t)\right\} _{k=0}^{n_{j}-1}\), for all \(j=1,2,\ldots ,n\), are the (time-dependent) norming constants corresponding to \(ia_{j}\). For algebraically simple eigenvalues \(ia_j\) we obtain the norming constants evolving in time according to (5).

- (a)Suppose that the scattering data, namely the discrete eigenvalues and the corresponding norming constants,are given. Then, we construct \(\varvec{{\varOmega }}(x)\) as in (7) and let it evolve in time using (11):$$\begin{aligned} \{i a_j\}_{j=1}^{n}\,\quad \text{ and } \quad \left\{ \{N_{jk}(t)\}_{k=0}^{n_j-1}\right\} _{j=1}^{n}, \end{aligned}$$$$\begin{aligned} \varvec{{\varOmega }}(x;t)=\begin{pmatrix} 0&{}{\varOmega }(x;t)\\ -{\varOmega }(x;t)^*&{}0\end{pmatrix}. \end{aligned}$$(12)
- (b)We solve the Marchenko integral equation (6):where \(\xi >x\) and the kernel \(\varvec{{\varOmega }}(x,y)\) is given in (12).$$\begin{aligned} \varvec{L}(x,y;t)+\varvec{{\varOmega }}(x+y;t)+\int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi ;t)\,\varvec{{\varOmega }}(\xi +y;t)=0_{2\times 2}. \end{aligned}$$
- (c)We construct the potential \(\varvec{m}(x;t)\) by using formula(8):where \(\tilde{\varvec{L}}(x)=\int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi )\).$$\begin{aligned} \varvec{m}(x)\cdot \varvec{\sigma }=U_+ \left( I_2+\tilde{\varvec{L}}(x)^\dagger \right) \sigma _3 \left( I_2+\tilde{\varvec{L}}(x)\right) U_+^{-1}, \end{aligned}$$

*e.g.*, we construct \(\varvec{{\varOmega }}(x)\) assuming no dependence on the time). We will subsequently show how to take the time dependence into account.

*A*is a \(\bar{n}\times \bar{n}\) matrix whose

*n*eigenvalues \(\left\{ a_{j}\right\} _{j=1}^{n}\) are obtained from the poles \(\left\{ i a_{j}\right\} _{j=1}^{n}\) of the transmission coefficient \(T(\lambda )\) (namely the discrete eigenvalues) by multiplication by a factor \(-i\) (a proof of this fact can be found in [25]);

*B*is a \(\bar{n}\times 1\) matrix; and

*C*is a \(1\times \bar{n}\) matrix. Furthermore, we 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 (13) [37, 38]. As the discrete eigenvalues \(\left\{ i a_{j}\right\} _{j=1}^{n}\) belong to the upper half-plane \(\mathbb {C}^{+}\), we have \(\mathrm {Re}(a_{j})>0\) for all

*j*, namely all the eigenvalues of the matrix

*A*have positive real parts: this fact is necessary in order to assure the convergence of the integrals in (15f). Moreover, we recall that the minimality of the triplet (

*A*,

*B*,

*C*) entails that the geometric multiplicity of the eigenvalues of

*A*be one (see [31]).

*A*,

*B*,

*C*) as follows [38]:

*A*is in Jordan canonical form, with \(A_j\) being the Jordan block of dimension \(n_j\times n_j\) corresponding to the discrete eigenvalue \(ia_j\), \(B_j\) is a column vector of dimension \(n_j\), typically chosen to be a vector of ones; and \(C_j\) is a row vector of dimension \(n_j\), typically chosen to be the vector of the norming constants corresponding to the discrete eigenvalue \(ia_j\),

*C*are chosen to be the \(\bar{n}\) norming constants \(\left\{ \{c_{jk}\}_{k=0}^{n_j-1}\right\} _{j=1}^{n}\).

*N*and

*Q*solve the Lyapunov matrix equations

*A*,

*B*,

*C*) [38, Sect. 4.1], we see that

*N*and

*Q*are positive Hermitian matrices and then \(\mathcal {P}\) is invertible and

*y*, obtaining the explicit formula

*t*. In order to recover it, we have to take into account the time evolution of the scattering data expressed by (5). Then the (reflectionless) Marchenko kernels become:

*A*,

*B*,

*C*) for the triplet (

*A*,

*B*, \(Ce^{-4itA^2})\) in such a way that, for algebraically simple eigenvalues

*iaj*, (5) are satisfied (A contains the discrete eigenvalues which are time independent and

*C*the norming constants). Consequently, the explicit right-hand side of (18) can be written as follows:

*U*is given by (9), is such that its columns form an orthonormal basis of eigenvectors of \(\cos (\gamma )\sigma _1-\sin (\gamma )\sigma _2\), corresponding to the eigenvalues 1 and \(-1\), respectively. Consequently, we can replace the matrix

*U*by the matrix

*V*in formula (8), obtaining a more general formula (indeed, a formula featuring the additional phase \(\delta \)). Indeed, the more general reconstruction formula for the soliton solutions is

*V*is given by (23). By using (24) and after some straightforward computations, we arrive at the the more general soliton solution formula

We conclude this section with two remarks.

### Remark 1

By following a procedure analogous to the one that has led to formulae (25), a similar generalization of the soliton solution formula can be obtained also in the easy-axis case studied in [25]. In that case what one gets is a rotation of the angle \(2\delta \) around the *z*-axis for the components \(m_1\) and \(m_2\).

## 3 Classes of soliton solutions

In the present section we discuss classes of soliton solutions of (1), as resulting from the explicit formula (21) with (22) and (20). Moreover, similarly to [25], we provide several numerical examples, obtained by computing (on *MATLAB R2017a*) the terms \(\tilde{L}_{1}\) and \(\tilde{L}_{2}\) appearing in (22) using formulae (C.2a) and (C.2d) in [25] when *x* is large and negative, and formulae (C.2b) and (C.2e) in [25] when *x* is large and positive.

An immediate classification of the soliton solutions of (1) can be had by considering the algebraic multiplicity of the eigenvalues of the matrix *A* in the matrix triplet (*A*, *B*, *C*) in (13b). Propagating and stationary soliton solutions (the so-called *magnetic-droplet solitons*, see [16]) are associated to algebraically simple eigenvalues of *A*. Multiple-pole (or, more simply, *multipole*) soliton solutions are instead associated to eigenvalues of *A* having algebraic multiplicity larger than one (i.e., *degenerate* eigenvalues). In the following, we choose *A* to be in Jordan canonical form as in (14): single eigenvalues on the main diagonal are associated to individual (stationary or propagating) solitons, whereas Jordan blocks of algebraic multiplicity \(n_{j}>1\) are associated to multipole solutions. No blocks are repeated, as the geometric multiplicity of each eigenvalue is one due to the minimality of the triplet [31, 38].

### 3.1 One-soliton solution

*A*,

*B*,

*C*) as

*t*. Consequently, the space and time evolution of the magnetic configuration is entirely described in terms of the constant speed

*v*and the constant frequency

*a*. By inverting (28a) and (28b),

*c*can be used to give the initial (\(t=0\)) position \(x_{0}\) of the minimum of \(\tilde{m}_{3}\) and the initial phase \(\varphi _{0}\), see [25].

### 3.2 Multi-soliton and breather-like solutions

By combining two or more one-soliton solutions, namely choosing \(n>1\), and \(n_{j}=1\) for all *j*, \(\bar{n}=n\) in (13), one can easily construct multi-soliton solutions. In this respect, we point out once more that formulae (21) as well as (25) are notably amenable to computer algebra, and allow to obtain explicit expressions (see [25]).

*A*(see [25]).

### 3.3 Multipole solutions

If \(n_{j}>1\) for some *j*, then *A* features a Jordan block of order \(n_{j}\), and one has multipole soliton solutions. Multipole solutions of (1) with (1a) are presented here for the first time. Their analysis can be achieved in analogy to the study of the multipole solutions of the nonlinear Schrödinger equation [41, 42], and is postponed to future investigation.

*A*is real (\(a=p\)), so that \(A=\left( {\begin{matrix}p&{}1\\ 0&{}p\end{matrix}}\right) \), if

*B*is chosen as a vector of ones, and if the norming constants in

*C*are chosen as follows

*A*is real (\(a=p\)), so that \(A=\left( {\begin{matrix}p&{}1&{}0\\ 0&{}p&{}1\\ 0&{}0&{}p\end{matrix}}\right) \), if

*B*is chosen as a vector of ones, and if the norming constants in

*C*are chosen as follows

## Notes

### Acknowledgements

The results contained in the present paper have been partially presented in Wascom 2017. The research leading to this article was supported in part by INdAM-GNFM and the London Mathematical Society Scheme 4 (Research in Pairs) Grant on “Propagating, localised waves in ferromagnetic nanowires” (Ref No. 41622). We also wish to thank an anonymous referee for his/her valuable comments.

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