Advertisement

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

  • F. Demontis
  • G. Ortenzi
  • M. Sommacal
  • C. van der Mee
Article

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

In this paper we show how to construct a formula containing all the reflectionless solutions of the classical, continuous Heisenberg ferromagnet chain equation [1, 2, 3, 4],
$$\begin{aligned} \varvec{m}_t=\varvec{m}\wedge \varvec{m}_{xx}, \end{aligned}$$
(1a)
to which we impose the in-plane asymptotic condition
$$\begin{aligned} \varvec{m}(x)\rightarrow \cos (\gamma )\varvec{e}_1-\sin (\gamma )\varvec{e}_2 \quad \text{ as }\quad x\rightarrow \pm \infty , \end{aligned}$$
(1b)
where \(\gamma \in [0,2\pi )\) is a constant angle, as discussed in [5]. Here
$$\begin{aligned} \varvec{m}:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {S}^2,\quad \varvec{m}(x,t)=\sum _{j=1}^{3}m_{j}(x,t)\,\varvec{e}_{j}, \end{aligned}$$
(2)
is the magnetization vector at position 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.

In the first part of this work [5], we have remarked that (1a) admits the following Lax pair representation
$$\begin{aligned} \left\{ \begin{array}{l} V_x=\varvec{A}\,V=[i\lambda (\varvec{m}\cdot \varvec{\sigma })]\,V\\ \\ V_t=\varvec{B}\,V=[-2i\lambda ^2(\varvec{m}\cdot \varvec{\sigma })-i\lambda (\varvec{m}\wedge \varvec{m}_x\cdot \varvec{\sigma })]\,V, \end{array}\right. \end{aligned}$$
(3)
It is well-known [28, 29, 30] that the knowledge of the Lax pair for (1a) assures that the inverse scattering transform (IST) can be applied to solve the initial-value problem [2, 3],
$$\begin{aligned} \left\{ \begin{array}{l} \varvec{m}_t=\varvec{m}\wedge \varvec{m}_{xx}\\ \varvec{m}(x,0)\quad \text {known}. \end{array}\right. \end{aligned}$$
(4)
After the association of (1a)–(3), the following classical diagram shows how the IST works:Let us recall that the initial datum \(\varvec{m}(x, 0)\) which appears in (4) (and in the first box in the diagram) has to be considered as a coefficient in the first equation of system (3). However, in the first part of this work [5], we have developed the direct scattering problem—which consists of the construction of the scattering data when \(\varvec{m}(x,0)\) is assigned—for the first of (3) (horizontal top arrow in the above diagram), we have discussed the evolution of the scattering data (vertical right arrow in the above diagram) and, finally, we have formulated the inverse scattering problem—which consists of the reconstruction of the potential \(\varvec{m}(x)\) corresponding to a set of a given scattering data—in terms of certain Marchenko integral equations (horizontal down arrow in the above diagram). So, in the first part of this work [5] we have treated the arrows of the IST scheme.
In the present second part, we are interested in solving explicitly the inverse scattering problem when the reflection coefficient is identically zero, aiming at an explicit soliton solution formula for (1a) under condition (1b). We will present this soliton solution formula in the next section. In order to derive the formula, we employ the matrix triplet method. Indeed, if the reflection coefficient vanishes identically, there exists a triplet of matrices \(\big (\varvec{A},\varvec{B},\varvec{C}\big )\), of sizes \(2\bar{n}\times 2\bar{n}\), \(2\bar{n}\times 2\), and \(2\times 2\bar{n}\), respectively, such that the Marchenko kernel is given by
$$\begin{aligned} \varvec{{\varOmega }}(x+y,t)=\varvec{C}e^{t\varvec{H}}e^{-(x+y)\varvec{A}}\varvec{B}, \end{aligned}$$
where the \(2\bar{n}\times 2\bar{n}\) matrices \(\varvec{A}\) and \(\varvec{H}\) commute and \(\varvec{A}\) has only eigenvalues with positive real parts. Typically, \(\varvec{H}\) is a function of \(\varvec{A}\). After solving the Marchenko equation by separation of variables, in the next section we arrive at the solution of the initial-value problem in terms of the matrix triplet \(\big (\varvec{A},\varvec{B},\varvec{C}\big )\) and the matrix \(\varvec{H}\), providing the time dependence via (8). The expression obtained can then be written in terms of elementary functions, and is particularly amenable to computer algebra.

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).

Using the initial condition \(\varvec{m}(x,0)\) given in (4) as a potential in the system (3), we develop the direct scattering theory as shown in the first part [5] and build the scattering data. For the sake of clarity, let us recall that the scattering data (see Section 3 of [5] for more details) evolve in time as follows
$$\begin{aligned} R(\lambda ,t)=e^{-4i\lambda ^2t}\,R(\lambda ,0),\quad T(\lambda ,t)=T(\lambda ,0),\quad N_j(t)=e^{-4ia_j^2t}\,N_j(0), \end{aligned}$$
(5)
where \(R(\lambda ,t)\) is the reflection coefficient, \(T(\lambda ,t)\) the transmission coefficients, and \(N_j\) are the norming constants associated to the discrete eigenvalues \(ia_j\) (see Section 2 in [5] for a detailed description of the scattering data).
As we have seen in the first part of this work [5], the inverse scattering problem requires one to solve the following Marchenko equation
$$\begin{aligned} \varvec{L}(x,y)+\varvec{{\varOmega }}(x+y)+\int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi )\,\varvec{{\varOmega }}(\xi +y)=0_{2\times 2}, \end{aligned}$$
(6)
where the kernel \(\varvec{{\varOmega }}(x)\) of (6) is given by
$$\begin{aligned} \varvec{{\varOmega }}(x)=\begin{pmatrix} 0&{}{\varOmega }(x)\\ -{\varOmega }(x)^*&{}0\end{pmatrix}, \quad \text {with}\quad {\varOmega }(x)=\rho (x)+\sum _{j=1}^{n}\,N_j\,e^{-a_jx}. \end{aligned}$$
(7)
The solution of the Heisenberg equation (1a) under condition (1b) is then obtained by replacing \(\varvec{{\varOmega }}(x)\) with \(\varvec{{\varOmega }}(x;t)\) in the Marchenko equation (i.e. taking into account (5)) and using the relation
$$\begin{aligned} \varvec{m}(x)\cdot \varvec{\sigma }=U\left( I_2+\tilde{\varvec{L}}(x)^\dagger \right) \sigma _3\left( I_2+\tilde{\varvec{L}}(x)\right) U^{-1}, \end{aligned}$$
(8)
where
$$\begin{aligned} U= & {} \frac{1}{\sqrt{2}}\begin{pmatrix} 1&{}\quad -e^{i\gamma }\\ e^{-i\gamma }&{} \quad 1\end{pmatrix}, \end{aligned}$$
(9)
$$\begin{aligned} \tilde{\varvec{L}}(x)= & {} \int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi ), \end{aligned}$$
(10)
and \(\varvec{\sigma }\) is the column vector whose entries are the Pauli matrices
$$\begin{aligned} \sigma _1=\begin{pmatrix} 0&{}\quad 1\\ 1&{}\quad 0\end{pmatrix},\qquad \sigma _2=\begin{pmatrix} 0&{}\quad -i\\ i&{}\quad 0\end{pmatrix},\qquad \sigma _3=\begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad -1 \end{pmatrix}. \end{aligned}$$
Note that, in the expression of \({\varOmega }(x)\), with \(\rho (x)\) we have denoted the Fourier transform of the reflection coefficient.

2.2 Matrix triplet method

In the remaining part of this work we will focus on the reflectionless case, i.e. the case \(R(\lambda ,t)=0\). In this case the expression for \(\varvec{{\varOmega }}(x;0)\) is obtained from the expression of the kernel given above and setting \(\rho (x)=0\). In particular, we can treat the situation where the discrete eigenvalues are not necessarily algebraically simple [36] by generalizing formula (7) as follows:
$$\begin{aligned} {\varOmega }(x;t)=\sum _{j=1}^n\sum _{k=0}^{n_j-1}N_{jk}(t)\frac{x^k}{k!}\,e^{-a_jx}. \end{aligned}$$
(11)
In (11), 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).
To recover the solution of (4) we follow the three steps indicated here below.
  1. (a)
    Suppose that the scattering data, namely the discrete eigenvalues and the corresponding norming constants,
    $$\begin{aligned} \{i a_j\}_{j=1}^{n}\,\quad \text{ and } \quad \left\{ \{N_{jk}(t)\}_{k=0}^{n_j-1}\right\} _{j=1}^{n}, \end{aligned}$$
    are given. Then, we construct \(\varvec{{\varOmega }}(x)\) as in (7) and let it evolve in time using (11):
    $$\begin{aligned} \varvec{{\varOmega }}(x;t)=\begin{pmatrix} 0&{}{\varOmega }(x;t)\\ -{\varOmega }(x;t)^*&{}0\end{pmatrix}. \end{aligned}$$
    (12)
     
  2. (b)
    We solve the Marchenko integral equation (6):
    $$\begin{aligned} \varvec{L}(x,y;t)+\varvec{{\varOmega }}(x+y;t)+\int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi ;t)\,\varvec{{\varOmega }}(\xi +y;t)=0_{2\times 2}. \end{aligned}$$
    where \(\xi >x\) and the kernel \(\varvec{{\varOmega }}(x,y)\) is given in (12).
     
  3. (c)
    We construct the potential \(\varvec{m}(x;t)\) by using formula(8):
    $$\begin{aligned} \varvec{m}(x)\cdot \varvec{\sigma }=U_+ \left( I_2+\tilde{\varvec{L}}(x)^\dagger \right) \sigma _3 \left( I_2+\tilde{\varvec{L}}(x)\right) U_+^{-1}, \end{aligned}$$
    where \(\tilde{\varvec{L}}(x)=\int _x^\infty \mathrm {d}\xi \,\varvec{L}(x,\xi )\).
     
Let us follow the above procedure (an analogous procedure can be developed with the kernel \(\overline{\varvec{{\varOmega }}}\), as per in formula (3.10b) of [5], and solving the corresponding Marchenko equation for \(\overline{\varvec{L}}\), that is, equation (3.11) of [5]). We start by disregarding the time dependence (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.
It is well known [37, 38] that it is possible to factorize a matrix function which is in the form (12) with (11) by using a suitable triplet of matrices. More precisely, let \(\bar{n}=\sum _{j=1}^{n} n_j\), and suppose that \((\mathcal {A},\mathcal {B},\mathcal {C})\) be a matrix triplet such that all the eigenvalues of the \(2\bar{n}\times 2\bar{n}\) matrix \(\mathcal {A}\) have positive real parts, \(\mathcal {B}\) is a \(2\bar{n}\times 2\) matrix, and \(\mathcal {C}\) is a \(2\times 2\bar{n}\) matrix. We then set Alternatively, equation (13a) can be written by setting
$$\begin{aligned} {\varOmega }(x)=\sum _{j=1}^n\sum _{k=0}^{n_j-1}c_{jk}\frac{x^k}{k!}\,e^{-a_jx}=C\,e^{-xA}\,B, \end{aligned}$$
(13b)
with
$$\begin{aligned} \mathcal {A}=\begin{pmatrix} A&{}\quad 0_{\bar{n}\times \bar{n}}\\ 0_{\bar{n}\times \bar{n}}&{}\quad A^\dagger \end{pmatrix}, \quad \mathcal {B}=\begin{pmatrix} 0_{\bar{n}\times 1}&{}\quad B\\ -C^\dagger &{}\quad 0_{\bar{n}\times 1}\end{pmatrix}, \quad \mathcal {C}=\begin{pmatrix} C&{}\quad 0_{1\times \bar{n}}\\ 0_{1\times \bar{n}}&{}\quad B^\dagger \end{pmatrix}. \end{aligned}$$
(13c)
Here 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 (ABC) 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 (ABC) entails that the geometric multiplicity of the eigenvalues of A be one (see [31]).
We observe that it is not restrictive (in fact, it is the typical choice) to set the triplet (ABC) as follows [38]:
$$\begin{aligned} A_{\bar{n}\times \bar{n}}=\begin{pmatrix} A_1&{}\quad 0&{} \quad \cdots &{}\quad 0\\ 0&{}\quad A_2&{}\quad \cdots &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0&{}\quad 0&{}\quad \cdots &{}\quad A_n \end{pmatrix},\,\, B_{\bar{n}\times 1}=\begin{pmatrix} B_1\\ B_2\\ \vdots \\ B_n\end{pmatrix},\,\, C_{1\times \bar{n}}=\begin{pmatrix} C_1&C_2&\cdots&C_n\end{pmatrix}, \end{aligned}$$
(14a)
where 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 \(ia_j\),
$$\begin{aligned} C_{j} = \begin{pmatrix}c_{j,0}&c_{j,1}&\cdots&c_{j,n_j-1}\end{pmatrix}, \end{aligned}$$
(14b)
so that the elements of C are chosen to be the \(\bar{n}\) norming constants \(\left\{ \{c_{jk}\}_{k=0}^{n_j-1}\right\} _{j=1}^{n}\).
For later convenience, we also introduce the matrix \(\mathcal {P}\), which is—under the conditions satisfied by our triplet—the unique solution of the Sylvester equation
$$\begin{aligned} \mathcal {A}\,\mathcal {P}+\mathcal {P}\,\mathcal {A}=\mathcal {B}\,\mathcal {C}, \end{aligned}$$
(15a)
namely
$$\begin{aligned} \mathcal {P}=\int _0^\infty \mathrm {d}\xi \,e^{-\xi \mathcal {A}}\,\mathcal {B}\,\mathcal {C}\,e^{-\xi \mathcal {A}}. \end{aligned}$$
(15b)
Note that it is also possible to write \(\mathcal {P}\) as
$$\begin{aligned} \mathcal {P}=\begin{pmatrix} 0_{\bar{n}\times \bar{n}}&{}\quad N\\ -Q&{}\quad 0_{\bar{n}\times \bar{n}} \end{pmatrix}, \end{aligned}$$
(15c)
where N and Q solve the Lyapunov matrix equations
$$\begin{aligned} A^{\dagger }\,Q+Q\,A&=C^{\dagger }\,C, \end{aligned}$$
(15d)
$$\begin{aligned} A\,N+N\,A^{\dagger }&=B\,B^{\dagger }, \end{aligned}$$
(15e)
that is
$$\begin{aligned} N=\int _0^\infty \mathrm {d}\xi \,e^{-\xi \,A}\,B\,B^\dagger \,e^{-\xi \,A^\dagger },\qquad Q=\int _0^\infty \mathrm {d}\xi \,e^{-\xi \,A^\dagger }\,C^\dagger \,C\,e^{-\xi \,A}. \end{aligned}$$
(15f)
By the minimality of the triplet (ABC) [38, Sect. 4.1], we see that N and Q are positive Hermitian matrices and then \(\mathcal {P}\) is invertible and
$$\begin{aligned} \mathcal {P}^{-1}=\begin{pmatrix} 0_{\bar{n}\times \bar{n}}&{}\quad -Q^{-1} \\ N^{-1}&{}\quad 0_{\bar{n}\times \bar{n}}\end{pmatrix}. \end{aligned}$$
(15g)
Now we are ready to express the solution \(\varvec{L}(x,y)\) of the Marchenko integral equation (6) in terms of the triplet \((\mathcal {A},\mathcal {B},\mathcal {C})\) and of the matrix \(\mathcal {P}\). Indeed, by substituting the expression of the kernel (13) into (6), we arrive at the following Marchenko equation
$$\begin{aligned} \varvec{L}(x,y)+\mathcal {C}\,e^{-(x+y)\,\mathcal {A}}\,\mathcal {B}+\int _x^\infty \mathrm {d}\xi \, \varvec{L}(x,\xi )\,\mathcal {C}\,e^{-(\xi +y)\,\mathcal {A}}\,\mathcal {B}=0_{2\times 2}. \end{aligned}$$
(16)
Equation (16) can be solved explicitly via separation of variables and we obtain [see [31, 35] for more details on the resolution of (16)]
$$\begin{aligned} \varvec{L}(x,y)=-\mathcal {C}e^{-x\mathcal {A}}[I_{2\bar{n}}+e^{-x\mathcal {A}}\mathcal {P}e^{-x\mathcal {A}}]^{-1}e^{-y\mathcal {A}}\mathcal {B}, \end{aligned}$$
(17)
provided the inverse matrix exists for all \(x\in \mathbb {R}\).
Finally, in order to reconstruct the solution of (4) we have to integrate (17) with respect to y, obtaining the explicit formula
$$\begin{aligned} \tilde{\varvec{L}}(x)=-\mathcal {C}e^{-x\mathcal {A}}[I_{2\bar{n}}+e^{-x\mathcal {A}}\mathcal {P}e^{-x\mathcal {A}}]^{-1}e^{-x\mathcal {A}}\,\mathcal {A}^{-1}\,\mathcal {B}. \end{aligned}$$
(18)
The right-hand side of (18) is now explicit and we can use such formula to recover the components \(m_j(x)\), \(j=1,2,3\), of the vector \(\varvec{m}(x)\).
Let us now introduce the dependence on the time 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:
$$\begin{aligned} {\varOmega }(x;t)&=\sum _{j=1}^n\sum _{k=0}^{n_j-1}c_{jk}(t)\frac{x^k}{k!}\,e^{-a_jx} =C\,e^{-4itA^2}\,e^{-xA}\,B, \end{aligned}$$
(19a)
$$\begin{aligned} {\varOmega }(x;t)^*&=\sum _{j=1}^n\sum _{k=0}^{n_j-1}c_{jk}^*(t)\frac{x^k}{k!}\,e^{-a_j^*x} =B^\dagger \,e^{-xA^\dagger }\,e^{4it{A^\dagger }^2t}\,C^\dagger . \end{aligned}$$
(19b)
In other words, we may replace the matrix triplet (ABC) for the triplet (AB\(Ce^{-4itA^2})\) 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:
$$\begin{aligned} \tilde{\varvec{L}}(x;t)&=-\mathcal {C}(t)\,e^{-x\mathcal {A}}\,\left[ I_{2\bar{n}}+e^{-x\mathcal {A}}\mathcal {P}(t) e^{-x\mathcal {A}}\right] ^{-1}\,e^{-x\mathcal {A}}\,\mathcal {A}^{-1}\,\mathcal {B}(t) \nonumber \\&=-\mathcal {C}(t)\,\left[ e^{2\,x\mathcal {A}}+\mathcal {P}(t)\right] ^{-1}\,\mathcal {A}^{-1}\,\mathcal {B}(t), \end{aligned}$$
(20a)
where
$$\begin{aligned} \mathcal {B}(t)=\begin{pmatrix} 0_{\bar{n}\times 1}&{}\quad B\\ -\left( Ce^{-4itA^2} \right) ^\dagger &{}\quad 0_{\bar{n}\times 1} \end{pmatrix}, \quad \mathcal {C}(t)=\begin{pmatrix} Ce^{-4itA^2}&{}\quad 0_{1\times \bar{n}} \\ 0_{1\times \bar{n}}&{}\quad B^\dagger \end{pmatrix}, \end{aligned}$$
(20b)
and
$$\begin{aligned} \mathcal {P}(t)=\begin{pmatrix} 0_{\bar{n}\times \bar{n}}&{}\quad N \\ -Q(t)&{}\quad 0_{\bar{n}\times \bar{n}}\end{pmatrix}, \end{aligned}$$
(20c)
with
$$\begin{aligned} Q(t)=\int _0^\infty dx\,e^{-xA^\dagger } \left( Ce^{-4itA^2}\right) ^\dagger Ce^{-4itA^2}e^{-xA}, \end{aligned}$$
(20d)
satisfying
$$\begin{aligned} A^{\dagger }Q(t)+Q(t)A=(Ce^{-4itA^2})^{\dagger }(Ce^{-4itA^2}). \end{aligned}$$
(20e)
Finally, after some algebra, using (8) with (20) and taking into account that \(\tilde{\varvec{L}}(x,y)\) belongs to SU2, so that
$$\begin{aligned} \tilde{\varvec{L}}=\begin{pmatrix} {{\tilde{L}}_1}&{}\quad {{-\tilde{L}_2}}^*\\ {{\tilde{L}}_2}&{}\quad {{\tilde{L}_1}}^*\end{pmatrix}, \quad \text{ and }\quad \left[ I_2+\tilde{\varvec{L}}\right] ^{-1}=\left[ I_2+\tilde{\varvec{L}}\right] ^{\dagger }= \left[ I_2+{\tilde{\varvec{L}}^{\dagger }}\right] , \end{aligned}$$
we arrive at the following soliton solution formula of (1a) with asymptotic boundary conditions (1b)
$$\begin{aligned} m_1(x,t)&=\sin ^2(\gamma )\,\tilde{m}_1(x,t)+\cos (\gamma )\sin (\gamma )\,\tilde{m}_2(x,t)+\cos (\gamma )\,\tilde{m}_3(x,t), \end{aligned}$$
(21a)
$$\begin{aligned} m_2(x,t)&=\cos (\gamma )\sin (\gamma )\,\tilde{m}_1(x,t)+\cos ^2(\gamma )\,\tilde{m}_2(x,t)-\sin (\gamma )\,\tilde{m}_3(x,t), \end{aligned}$$
(21b)
$$\begin{aligned} m_3(x,t)&=-\cos (\gamma )\,\tilde{m}_1(x,t)+\sin (\gamma )\,\tilde{m}_2(x,t), \end{aligned}$$
(21c)
where \(\left( \tilde{m}_1(x,t), \tilde{m}_2(x,t), \tilde{m}_3(x,t)\right) \) have the following explicit expression in terms of the elements of the matrix \(\tilde{\varvec{L}}(x;t)\)
$$\begin{aligned} \tilde{m}_1(x,t)&=-2\,\mathrm {Re}\Big ((1+\tilde{L}_1(x,t)) \,\tilde{L}_2(x,t)\Big ). \end{aligned}$$
(22a)
$$\begin{aligned} \tilde{m}_2(x,t)&=-2\,\mathrm {Im}\Big ((1+\tilde{L}_1(x,t)) \,\tilde{L}_2(x,t)\Big ), \end{aligned}$$
(22b)
$$\begin{aligned} \tilde{m}_3(x,t)&=2\,\left| 1+\tilde{L}_1(x,t)\right| ^2-1. \end{aligned}$$
(22c)
We note that formulae (22) give the soliton solutions of (1a) with the so-called easy-axis conditions (i.e., \(\varvec{m}(x)\rightarrow \varvec{e}_3\) as \(x\rightarrow \pm \infty \), see [25]). Thus, Eq. (21) allow one to generate the soliton solutions of (1a) with boundary conditions (1b) when a soliton solutions of (1a) with easy-axis conditions is known.
We observe that the matrix
$$\begin{aligned} V=U\,\text {diag}\left( e^{i\delta }, e^{-i\delta }\right) =\frac{1}{\sqrt{2}} \begin{pmatrix} e^{i\delta }&{}\quad -e^{i(\gamma -\delta )}\\ e^{i(\delta -\gamma )}&{}\quad e^{-i\delta }\end{pmatrix}, \end{aligned}$$
(23)
where \(\delta \) is a constant and the expression of 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
$$\begin{aligned} \varvec{m}(x)\cdot \varvec{\sigma }=V\left( I_2+\tilde{\varvec{L}}(x)^\dagger \right) \sigma _3 \left( I_2+\tilde{\varvec{L}}(x)\right) V^{-1}, \end{aligned}$$
(24)
where V is given by (23). By using (24) and after some straightforward computations, we arrive at the the more general soliton solution formula
$$\begin{aligned} m_1(x,t)&=\sin (\gamma )\sin (\hat{\delta })\,\tilde{m}_1(x,t)+\sin (\gamma )\cos (\hat{\delta })\,\tilde{m}_2(x,t)+\cos (\gamma )\,\tilde{m}_3(x,t), \end{aligned}$$
(25a)
$$\begin{aligned} m_2(x,t)&=\cos (\gamma )\sin (\hat{\delta })\,\tilde{m}_1(x,t)+\cos (\gamma )\cos (\hat{\delta })\,\tilde{m}_2(x,t)-\sin (\gamma )\,\tilde{m}_3(x,t), \end{aligned}$$
(25b)
$$\begin{aligned} m_3(x,t)&=-\cos (\hat{\delta })\,\tilde{m}_1(x,t)+\sin (\hat{\delta })\,\tilde{m}_2(x,t), \end{aligned}$$
(25c)
where \(\hat{\delta }=\gamma -2\delta \).

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 \(m_2\).

Remark 2

It is worth observing that formulae (21) and (25), which have been obtained in this paper by means of the IST technique, can be obtained also (and straightforwardly) by employing the symmetries of the classical Heisenberg ferromagnet Eq. (1) (see, for instance [39, 40]).

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 (ABC) 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

The one-soliton solution corresponds to the choice \(n=1\), \(n_{1}=1\) in (13), so that \(\bar{n}=1\). If we set the matrix triplet (ABC) as
$$\begin{aligned} A=\left( a\right) ,\quad B=\left( 1\right) , \text{ and } C=\left( c\right) , \end{aligned}$$
we get
$$\begin{aligned} N = \left( \frac{1}{2\,\mathrm {Re}(a)}\right) ,\quad Q = \left( \frac{|c|^{2}}{2\,\mathrm {Re}(a)}\right) ,\quad Q(t) = \left( \frac{|c|^{2}\,e^{-4\,i\,\mathrm {Re}(a^{2})\,t}}{2\,\mathrm {Re}(a)}\right) , \end{aligned}$$
and from (20) we have
$$\begin{aligned} \tilde{L}_{1} =&-\frac{2\,|c|^{2}\,\mathrm {Re}(a)}{a^*\,\Big (|c|^{2}+4\,\mathrm {Re}(a)^{2}\,e^{4\,\mathrm {Re}(a)\,(x-4\,\mathrm {Im}(a)\,t)}\Big )},\\ \tilde{L}_{2} =&\frac{2\,c^*\,\mathrm {Re}(a)\,e^{2\,a\,(x+2\,i\,a\,t)}}{a^*\,\Big (|c|^{2}+4\,\mathrm {Re}(a)^{2}\,e^{4\,\mathrm {Re}(a)\,(x-4\,\mathrm {Im}(a)\,t)}\Big )}. \end{aligned}$$
Then we set, without any loss of generality (see [25]),
$$\begin{aligned} a = p+i\,q,\quad p>0, \end{aligned}$$
(26a)
and
$$\begin{aligned} c\equiv c(p,q,x_{0},\varphi _{0})=\left\{ \begin{array}{ll} 2\,i\,p\,\mathrm {sign}(q)\,\left( \frac{p+i\,q}{p-i\,q}\right) \,e^{2\,(p+i\,q)\,x_{0}-i\,\varphi _{0}}&{} \text{ if } q\ne 0\\ &{}\\ 2\,p\,e^{2\,p\,x_{0}-i\,\varphi _{0}}&{} \text{ if } q=0, \end{array} \right. \end{aligned}$$
(26b)
for some \(x_{0},\varphi _{0}\in \mathbb {R}\). From (22), after some simple algebra, we obtain the in-plane one soliton solution \(\left( m_{1},m_{2},m_{3}\right) \), via (21), or alternatively via (25), with \(\left( \tilde{m}_{1},\tilde{m}_{2},\tilde{m}_{3}\right) \) given by:
$$\begin{aligned} \tilde{m}_{+}=\,&2\, \frac{(p+i\,q)-(p-i\,q)\,e^{4\,p\,(x-4\,q\,t-x_{0})}}{(p-i\,q)^{2}\,\left[ 1+e^{4\,p\,(x-4\,q\,t-x_{0})}\right] ^{2}}\, \nonumber \\&\times e^{2\,(p+i\,q)\,(x-4\,q\,t)-2\,p\,x_0-i\,\mathrm {arg}(c)}\,e^{4\,i\,(p^{2}+q^{2})\,t}, \end{aligned}$$
(27a)
$$\begin{aligned} \tilde{m}_{1}=\,&\mathrm {Re}\left( \tilde{m}_{+}\right) ,\,\,\, \tilde{m}_{2}=\mathrm {Im}\left( \tilde{m}_{+}\right) ,\,\,\, \tilde{m}_{3}= 1-\frac{2\,p^{2}\,\mathrm {sech}^{2}\Big (2\,p\,(x-4\,q\,t-x_{0})\Big )}{p^{2}+q^{2}}. \end{aligned}$$
(27b)
This solution describes a localized, coherent magnetic configuration travelling at the constant speed
$$\begin{aligned} v = 4\,\mathrm {Im}(a) = 4\,q. \end{aligned}$$
(28a)
Furthermore, the exponent of the last exponential term in the right-hand side of (27a) is a phase factor depending only on the time 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
$$\begin{aligned} \omega = 4\,|a|^{2} = 4\,(p^{2}+q^{2}), \end{aligned}$$
(28b)
which, in turn, depend only on the real and imaginary parts of the eigenvalue a. By inverting (28a) and (28b),
$$\begin{aligned} p = \frac{1}{2}\,\sqrt{\omega -\frac{v^{2}}{4}},\quad q = \frac{v}{4}, \end{aligned}$$
(28c)
we immediately obtain the well-known condition for localization (see [16]),
$$\begin{aligned} \omega \ge 0,\quad |v|\le 2\,\sqrt{\omega }. \end{aligned}$$
(28d)
On the other hand, via (26b), the norming constant 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].
In Fig. 1 we illustrate an in-plane, propagating, one-soliton solution for the choice \(v=1\), \(\omega =2\), \(x_0=-4\), and \(\varphi _0=0\), entailing \(p = \tfrac{\sqrt{7}}{4}\), \(q = \tfrac{1}{4}\), and \(\displaystyle c=\tfrac{-7+i\,3\,\sqrt{7}}{8}\,e^{-2\,(i+\sqrt{7})}\), for several choices of the asymptotic angle \(\gamma \).
Fig. 1

Propagating, one-soliton solution, for different values of the asymptotic angle \(\gamma \)

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]).

In particular, similarly to the easy-axis case [25], breather-like soliton solutions can be constructed out of two-soliton solutions, by creating two stationary, or two same-speed, propagating solitons close to each other. In the case of two stationary solitons (\(v^{(1)}=v^{(2)}=0\)), namely, in the case of two real eigenvalues \(a^{1} = p_{1}\) and \(a_{2} = p_{2}\), \(p_1\ne p_2\), it is possible to show that, if the norming constants are chosen as follows
$$\begin{aligned} C = \left( c_1,\,c_{2}\right) = 2\,\sqrt{\frac{(p_1+p_2)^2+(q_1-q_2)^2}{(p_1-p_2)^2+(q_1-q_2)^2}}\,\left( p_1,\,p_{2}\right) \end{aligned}$$
(29)
with \(q_1=0\) and \(q_2=0\), then a single, symmetrical, breather-like soliton solution is created, with \(m_{1}\) and \(m_{2}\) characterized by two identical, localized extrema oscillating in time around the origin with period
$$\begin{aligned} \nu = \frac{2\,\pi }{4\,(p_1+p_2)\,(p_1-p_2)}. \end{aligned}$$
(30)
Figure 2(a) shows an example of such a breather-like soliton for \(\gamma =\frac{\pi }{4}\), obtained via (28c) and (29) with
$$\begin{aligned} v^{(1)} = 0,\quad \omega ^{(1)} = 0.8, \quad \text{ and }\quad v^{(2)} = 0,\quad \omega ^{(2)} = 0.4, \end{aligned}$$
thus entailing an oscillation in time with period \(\nu \simeq 15.71\). Similarly to the easy-axis case [25], propagating, breather-like solitons can be constructed in the same way as above, but assigning the same non-zero imaginary part to both of the discrete eigenvalues. More generally, propagating or stationary, breather-like solitons can be had by creating two stationary, or two same-speed, propagating solitons close to one another: in other words, a single, breather-like soliton should always be regarded as a stable, periodic tangle of two interacting, but individual solitons, associated to two different eigenvalues in the matrix A (see [25]).
Fig. 2

Examples of breather-like and multipole solutions for \(\gamma =\frac{\pi }{4}\)

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.

If \(n=1\), \(n_{1}=2\), \(\bar{n}=2\) in (13), then we have a single two-pole soliton solution. In this case, it is possible to show [25] that, if the associated eigenvalue of 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
$$\begin{aligned} C = \left( c_1,\,c_{2}\right) =\left( 4\,p^{2},\,4\,p\,[1+p\,(2\,x_0-1)]\right) \,e^{2\,p\,x_0-i\,\varphi _0}, \end{aligned}$$
(31)
then a single, symmetrical, two-pole soliton solution is created, with \(m_{1}\)and \(m_{2}\) characterized by two minima, constituting two separated branches, that are expected to propagate in space at a velocity that varies logarithmically in time.
The same technique can be generalized to any value of the algebraic multiplicity \(n_{j}\). For instance, if \(n=1\), \(n_{1}=3\), \(\bar{n}=3\) in (13), then we have a single three-pole soliton solution: if the associated eigenvalue of 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
$$\begin{aligned} C^{T} = \left( \begin{array}{c} c_1\\ c_{2}\\ c_{3}\end{array}\right) = \left( \begin{array}{c} 8\,p^{3}\\ 4\,p^{2}\,[3+p\,(4\,x_0-2)]\\ 8\,p^{2}\,x_0\,(x_0-1)+6\,p\,(2\,x_0-1)+3\end{array} \right) \,e^{2\,p\,x_0-i\,\varphi _0}, \end{aligned}$$
(32)
then a single, symmetrical, three-pole soliton solution is created, with \(m_{1}\) and \(m_{2}\) characterized by three minima, constituting three separated branches, propagating in space at a velocity that varies logarithmically in time, and interacting in \(x=x_0\) at \(t=0\). Figure 2(b) shows an example of such a solution with \(\gamma =\frac{\pi }{4}\), obtained via (32) with \(p=1\), \(x_{0}=0\), and \(\varphi _{0}=0\).

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.

References

  1. 1.
    Lakshmanan, M.: Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 61, 53–54 (1977)CrossRefGoogle Scholar
  2. 2.
    Takhtajan, L.A.: Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 64, 235–237 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zakharov, V.E., Takhtajan, L.A.: Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys. 38, 17–23 (1979)CrossRefGoogle Scholar
  4. 4.
    Fogedby, H.C.: Solitons and magnons in the classical Heisenberg chain. J. Phys. A: Math. Gen. 13, 1467–1499 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Demontis, F., Ortenzi, G., Sommacal, M., van der Mee, C.: The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory. Ricerche Mat. (2018).  https://doi.org/10.1007/s11587-018-0394-8
  6. 6.
    Mohseni, S.M., Sani, S.R., Persson, J., Anh Nguyen, T.N., Chung, S., Pogoryelov, Y., Muduli, P.K., Iacocca, E., Eklund, A., Dumas, R.K., Bonetti, S., Deac, A., Hoefer, M.A., Åkerman, J.: Spin torque-generated magnetic droplet solitons. Science 339, 1295–1298 (2013)CrossRefGoogle Scholar
  7. 7.
    Macià, F., Backes, D., Kent, A.D.: Stable magnetic droplet solitons in spin-transfer nanocontacts. Nat. Nanotechnol. 9, 992–996 (2014)CrossRefGoogle Scholar
  8. 8.
    Mohseni, S.M., Sani, S.R., Dumas, R.K., Persson, J., Anh Nguyen, T.N., Chung, S., Pogoryelov, Ye, Muduli, P.K., Iacocca, E., Eklund, A., Åkerman, J.: Magnetic droplet solitons in orthogonal nano-contact spin torque oscillators. Phys. B 435, 84–87 (2014)CrossRefGoogle Scholar
  9. 9.
    Chung, S., Mohseni, S.M., Sani, S.R., Iacocca, E., Dumas, R.K., AnhNguyen, T.N., Pogoryelov, Ye, Muduli, P.K., Eklund, A., Hoefer, M., Åkerman, J.: Spin transfer torque generated magnetic droplet solitons. J. App. Phys. 115, 172612 (2014)CrossRefGoogle Scholar
  10. 10.
    Maiden, M.D., Bookman, L.D., Hoefer, M.A.: Attraction, merger, reflection, and annihilation in magnetic droplet soliton scattering. Phys. Rev. B 89(18), 180409 (2014)CrossRefGoogle Scholar
  11. 11.
    Chung, S., Mohseni, S.M., Eklund, A., Dürrenfeld, P., Ranjbar, M., Sani, S.R., Anh Nguyen, T.N., Dumas, R.K., åkerman, J.: Magnetic droplet solitons in orthogonal spin valves. Low Temp. Phys. 41, 833 (2015)CrossRefGoogle Scholar
  12. 12.
    Bookman, L.D., Hoefer, M.A.: Perturbation theory for propagating magnetic droplet solitons. Proc. R. Soc. A 471(2179), 20150042 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chung, S., Eklund, A., Iacocca, E., Mohseni, S.M., Sani, S.R., Bookman, L., Hoefer, M.A., Dumas, R.K., Åkerman, J.: Magnetic droplet nucleation boundary in orthogonal spin-torque nano-oscillators. Nat. Commun. 7, 11209 (2016)CrossRefGoogle Scholar
  14. 14.
    Wang, C., Xiao, D., Liu, Y.: Merging magnetic droplets by a magnetic field pulse. AIP Adv. 8, 056021 (2018)CrossRefGoogle Scholar
  15. 15.
    Ivanov, B., Kosevich, A.: Bound-states of a large number of magnons in a ferromagnet with one-ion anisotropy. Zh. Eksp. Teor. Fiz. 72(5), 2000–2015 (1977)Google Scholar
  16. 16.
    Kosevich, A., Ivanov, B., Kovalev, A.: Magnetic solitons. Phys. Rep. 194(3–4), 117–238 (1990)CrossRefGoogle Scholar
  17. 17.
    Hoefer, M.A., Silva, T., Keller, M.: Theory for a dissipative droplet soliton excited by a spin torque nanocontact. Phys. Rev. B 82(5), 054432 (2010)CrossRefGoogle Scholar
  18. 18.
    Bonetti, S., Tiberkevich, V., Consolo, G., Finocchio, G., Muduli, P., Mancoff, F., Slavin, A., Åkerman, J.: Experimental evidence of self-localized and propagating spin wave modes in obliquely magnetized current-driven nanocontacts. Phys. Rev. Lett. 105, 217204 (2010)CrossRefGoogle Scholar
  19. 19.
    Ivanov, B.A., Stephanovich, V.A.: Two-dimensional soliton dynamics in ferromagnets. Phys. Lett. A 141(1), 89–94 (1989)CrossRefGoogle Scholar
  20. 20.
    Piette, B., Zakrzewski, W.J.: Localized solutions in a two-dimensional Landau–Lifshitz model. Physica D 119(3), 314–326 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ivanov, B.A., Zaspel, C.E., Yastremsky, I.A.: Small-amplitude mobile solitons in the two-dimensional ferromagnet. Phys. Rev. B 63(13), 134413 (2001)CrossRefGoogle Scholar
  22. 22.
    Hoefer, M.A., Sommacal, M.: Propagating two-dimensional magnetic droplets. Physica D 241, 890–901 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Hoefer, M.A., Sommacal, M., Silva, T.: Propagation and control of nano-scale, magnetic droplet solitons. Phys. Rev. B 85(21), 214433 (2012)CrossRefGoogle Scholar
  24. 24.
    Iacocca, E., Dumas, R.K., Bookman, L., Mohseni, M., Chung, S., Hoefer, M.A., Åkerman, J.: Confined dissipative droplet solitons in spin-valve nanowires with perpendicular magnetic anisotropy. Phys. Rev. Lett. 112, 047201 (2014)CrossRefGoogle Scholar
  25. 25.
    Demontis, F., Lombardo, S., Sommacal, M., van der Mee, C., Vargiu, F.: Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation (submitted)Google Scholar
  26. 26.
    Chen, A.-H., Wang, F.-F.: Darboux transformation and exact solutions of the continuous Heisenberg spin chain equation. Z. Nat. Teil A 69, 9–16 (2014)Google Scholar
  27. 27.
    Yersultanova, Z.S., Zhassybayeva, M., Yesmakhanova, K., Nugmanova, G., Myrzakulov, R.: Darboux transformation and exact solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials. Int. J. Geom. Methods Mod. Phys. 13, 1550134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)CrossRefzbMATHGoogle Scholar
  29. 29.
    Calogero, F., Degasperis, A.: Spectral Transforms and Solitons. North-Holland, Amsterdam (1982)zbMATHGoogle Scholar
  30. 30.
    Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  31. 31.
    Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Prob. 23, 2171–2195 (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Demontis, F.: Exact solutions to the modified Korteweg–de Vries equation. Theor. Math. Phys. 168, 886–897 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the Sine–Gordon equation. J. Math. Phys. 51, 123521 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Demontis, F., van der Mee, C.: Closed form solutions to the integrable discrete nonlinear Schrödinger equation. J. Nonlinear Math. Phys. 19(2), 1250010 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Demontis, F., Ortenzi, G., van der Mee, C.: Exact solutions of the Hirota equation and vortex filaments motion. Physica D 313, 61–80 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Demontis, F.: Matrix Zakharov–Shabat system and inverse scattering transform (2012), also: Direct and inverse scattering of the matrix Zakharov–Shabat system. Ph.D. Thesis, University of Cagliari, Italy (2007)Google Scholar
  37. 37.
    Dym, H.: Linear Algebra in Action, Vol. 78, Graduate Studies in Mathematics. American Mathematical Society, Providence (2007)Google Scholar
  38. 38.
    van der Mee, C.: Nonlinear evolution models of integrable type, vol. 11, e-Lectures Notes, SIMAI (2013)Google Scholar
  39. 39.
    Egorov, R.F., Bostrem, I.G., Ovchinnikov, A.S.: The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet. Phys. Lett. A 292, 325–334 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhao, Z., Han, B.: Lie symmetry analysis of the Heisenberg equation. Commun. Nonlinear Sci. Numer. Simul. 45, 220–234 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Olmedilla, E.: Multiple pole solutions of the non-linear Schrödinger equation. Physica D 25, 330–346 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Schiebold, C.: Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Mid Sweden University, Reports of the Department of Mathematics 1, 51 pages (2014)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  • F. Demontis
    • 1
  • G. Ortenzi
    • 2
  • M. Sommacal
    • 3
  • C. van der Mee
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CagliariCagliariItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano BicoccaMilanItaly
  3. 3.Department of Mathematics, Physics and Electrical EngineeringUniversity of Northumbria at NewcastleNewcastle upon TyneUK

Personalised recommendations