Spintronic timing switches via magnetic breather oscillations

Abstract

Magnetization switching plays a pivotal role in many spintronic devices. Distinct from conventional field-driven approaches, we propose a self-induced magnetization reversal mechanism that exploits the intrinsic periodic energy focusing and dispersing behavior of magnetic breathers. In this paper, we explore magnetic breather excitations and potential applications of Kuznetsov−Ma (KM) breather in an anisotropic ferromagnet. Under the long-wavelength approximation, the magnetization dynamics of a ferromagnetic nanowire are governed by a generalized derivative nonlinear Schrödinger equation. By employing a two-fold Darboux transformation, we derive the general breather solution and classify it into five distinct types through modulation instability analysis. Numerical simulations reveal a reversible transition between magnetic solitons and KM breathers, achieved by tuning the amplitude of the background plane wave. The KM breather exhibits periodic magnetization oscillations capable of inducing time-dependent magnetoresistance. When coupled with an appropriately defined critical current, this feature facilitates the realization of autonomous timing switches. These results provide theoretical foundations for the advancement of breather-enabled spintronic technologies.

Graphical abstract

Keywords

magnetic breather / Darboux transformation / self-induced magnetization reversal / anisotropic ferromagnet

1 Introduction

Since the inception of spintronics, the discovery of the giant magnetoresistance (GMR) effect [1] in 1988 revolutionized data reading technologies in hard drives, propelling magnetic storage into rapid development over the past two decades [2–5]. The key to GMR’s success in addressing the challenge of high-density storage lies in the efficient utilization of the electron’s intrinsic spin and the magnetoresistive effect under weak magnetic fields: switching between parallel and antiparallel alignments of adjacent magnetic moments induces transitions between low and high resistance states [6–8]. Enhancing the efficiency and minimizing the energy consumption of free-layer magnetization switching remains a critical direction in magnetic storage and various spintronic device applications [9, 10].

Beyond the earliest approach of applying external magnetic fields directly, numerous theoretical and experimental strategies for magnetization switching have been explored. For instance, microwave-assisted magnetic reversal can reduce the switching field by more than 50% [11]. Spin-polarized currents leveraging spin-transfer torque and spin Hall effects have enabled magnetization reversal in increasingly miniaturized devices [12–17]. Further advancements using spin–orbit torque have achieved higher switching efficiency, ultralow current densities, and sub-picosecond switching speeds [18, 19]. Additionally, ultrafast voltage-pulse-induced switching allows for complete reversal of perpendicular magnetization driven purely by electric field pulses [20–23]. Other mechanisms — such as surface acoustic waves (SAWs), all-optical magnetization, and mechanical strain — have also been proposed [24–26]. Although these methods often exhibit higher switching error rates that limit their suitability for conventional logic operations, they hold considerable promise for fault-tolerant computing and emerging non-Boolean architectures, including high-performance microwave oscillators [24].

On the other hand, breathers — localized and temporally oscillating solutions — constitute a fundamental class of nonlinear phenomena that have been both theoretically predicted and experimentally observed across diverse physical systems, including optics, fluids, and Bose-Einstein condensates [27–34]. In recent years, they have attracted growing interest due to their remarkable dynamical properties [35–37]. From the perspective of excitation mechanisms, the periodic oscillations of breathers arise from internal energy exchange within the system. In magnetic materials, such oscillatory magnetization dynamics can induce periodic variations in magnetoresistance [38], offering a promising route for spintronic applications that function without external driving fields. Compared with conventional approaches relying on continuous external stimuli, self-induced magnetization switching enabled by magnetic breathers provides a compelling strategy for achieving ultra-low-power device designs.

To investigate the excitation, control, and potential applications of magnetic breathers, the remainder of this paper is organized as follows. Section 2 introduces a ferromagnetic nanowire model, where the magnetization dynamics under the long-wavelength approximation are governed by a generalized derivative nonlinear Schrödinger (GDNLS) equation. Section 3 presents a detailed analysis of magnetic breather solutions and their dynamical properties. Subsection 3.1 derives the breather solution via the Darboux transformation and examines the modulation instability underlying its excitation. In Subsection 3.2, we numerically demonstrate a controllable transition between magnetic solitons and Kuznetsov–Ma (KM) breathers by tuning the amplitude of the background wave. Subsection 3.3 investigates the periodic magnetization oscillations intrinsic to KM breathers and proposes a KM-breather-driven self-clocked switching mechanism, highlighting a promising route toward timing devices in spintronic systems. Finally, Section 4 concludes with a discussion of the implications of breather-based magnetic device technologies.

2 Modeling

A quasi one-dimensional anisotropic ferromagnetic slab lying in the z-axis is considered, as shown in Fig. 1. The ferromagnetic slab is magnetized to saturation by an in-plane external field {\mathit{H}}_0 directed along the propagation z-axis. The evolution of the magnetic field \mathit{H} and magnetization \mathit{M} are governed by the Maxwell equation (in the absence of currents and charges) [39–43] coupled with the Landau–Lifshitz–Gilbert equation [44]:

(1)\begin{array}{rl}& -\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathit{H}\right)+\mathrm{\Delta }\mathit{H}=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\left(\mathit{H}+\mathit{M}\right),\\ & \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathit{M}=-\gamma \mathit{M}\times {\mathit{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}+\frac{\alpha }{M_s}\mathit{M}\times \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathit{M}.\end{array}

Here c=1/\sqrt{{\mu }_0\tilde{\epsilon }} is the speed of light, with {\mu }_0 being the magnetic permeability of the vacuum and \tilde{\epsilon } being the scalar permittivity of the medium. \gamma is the gyromagnetic ratio, \alpha >0 denotes the Gilbert-damping constant, M_s is the saturation magnetization. The expression for the effective magnetic field {\mathit{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}} is given by {\mathit{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{H}}_0+\frac{2A}{M_s}{\mathrm{\nabla }}^2\mathit{M}-\frac{2K}{M_s}\mathit{n}\left(\mathit{n}\cdot \mathit{M}\right), which includes the external field, exchange field, and the anisotropy field. Here, A and K are the exchange and anisotropy constants, respectively. The unit vector \mathit{n}=(0,0,1) is directed along the anisotropy axis.

While investigating the propagation of electromagnetic waves in the ferromagnetic nanowire, Sathishkumar and Jin et al. [42, 45] have constructed the following nonlinear evolution system:

(2)\mathrm{i}\eta {\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\tau }}-{\displaystyle \frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\zeta }^2}}+\mathrm{i}\delta {\displaystyle \frac{\mathrm{\partial }(|\psi {|}^2\psi )}{\mathrm{\partial }\zeta }}=0

by using a non-uniformly expanding scheme M^p= {\epsilon }^{1/2}\left(M_1^p+\epsilon M_2^p+{\epsilon }^2{M}_3^p+...\right),M^z=M_0+\epsilon M_1^z+{\epsilon }^2{M}_2^z+{\epsilon }^3{M}_3^z+..., H^p={\epsilon }^{1/2}\left(H_1^p+\epsilon H_2^p+{\epsilon }^2{H}_3^p+...\right),H^z=H_0+\epsilon H_1^z+{\epsilon }^2{H}_2^z+{\epsilon }^3{H}_3^z+..., where p=x,y, \eta =2\gamma c^2{M}_0/(c^2-v^2{)}^2, \delta =\gamma c^2/[2v{M}_0(c^2-v^2)], when \zeta =\epsilon (z-vt) and \tau ={\epsilon }^{2}t represent the space and time variables. The constants M_0 and H_0 characterize the unperturbed (initial) state of the system. Here the complex field \psi is defined as \psi =M_1^x-\mathrm{i}{M}_1^y. Considering the relation for conservation of length of the magnetization vector, M_0^2=(M^x{)}^2+(M^y{)}^2+(M^z{)}^2, it is easy to obtain M_1^z=-|\psi {|}^2/(2M_0).

This derived model equation allows for systematic investigation of nonlinear excitations in anisotropic ferromagnetic nanowires, while providing direct insights into their potential applications in spintronic devices such as magnetic storage. In some literature, magnetic solitons and magnetic rogue wave solutions of this equation have been obtained, and their potential applications in magnetic storage and electromagnetic interference were explored [42, 45].

3 The solution and potential application of magnetic breathers

3.1 Magnetic breather solution and modulation instability

In our previous work, the Lax pair of the GDNLS (2) has been constructed as:

(3)\begin{array}{rl}& {\mathrm{\Phi }}_{\xi }=\mathit{U}\mathrm{\Phi },\\ & {\mathrm{\Phi }}_{\tau }=\mathit{V}\mathrm{\Phi },\end{array}

where \mathrm{\Phi }=(\varphi ,\phi {)}^{\mathrm{T}} is the vector function (the superscript {}^{\mathrm{T}} denotes matrix transpose) and

(4)\begin{array}{rl}\mathit{U}& =\frac{{\eta }^2}{\delta }\mathit{J}{\lambda }^2+\eta \mathit{Q}\lambda ,\\ \mathit{V}& =\frac{2{\eta }^3}{{\delta }^2}\mathit{J}{\lambda }^4+\frac{2{\eta }^2}{\delta }\mathit{Q}{\lambda }^3+\eta \mathit{J}{\mathit{Q}}^2{\lambda }^2+(\delta {\mathit{Q}}^3-\mathrm{i}{\mathit{Q}}_x)\lambda ,\end{array}

with

(5)\mathit{J}=\left(\begin{array}{cc}\mathrm{i}& 0\\ 0& -\mathrm{i}\end{array}\right),\mathit{Q}=\left(\begin{array}{cc}0& \psi \\ -{\psi }^{\ast }& 0\end{array}\right).

Here \ast denotes complex conjugate, \lambda is the complex spectral parameter.

By applying the two-fold Darboux transformation as shown in the appendix, we successfully obtain the general solution that contain various solutions of the GDNLS equation (2):

(6){\psi }^{[2]}={\displaystyle \frac{{\mathrm{\Theta }}_1\cdot {\mathrm{\Theta }}_2}{{\mathrm{\Theta }}_3^2}}{\psi }^{[0]},

with

(7)\begin{array}{rl}{\mathrm{\Theta }}_1=& Q_1\cos (g_1)+Q_2\cosh (g_2)+Q_3\sin (g_1)+\mathrm{i}{Q}_4\sinh (g_2),\\ {\mathrm{\Theta }}_2=& [Q_1-q\beta Q_2]\cos (g_1)+[Q_2-q\beta Q_1]\cosh (g_2)\\ & +[Q_3-\mathrm{i}q\alpha Q_4]\sin (g_1)+[\mathrm{i}{Q}_4-q\alpha Q_3]\sinh (g_2),\\ {\mathrm{\Theta }}_3=& Q_1\cos (g_1)+Q_2\cosh (g_2)-Q_3\sin (g_1)-\mathrm{i}{Q}_4\sinh (g_2),\\ {\psi }^{[0]}=& a\exp \left[\mathrm{i}k\left(\zeta +(k-a^2\delta )\tau /\eta \right)\right],\end{array}

and

(8)\begin{array}{rl}& Q_1=\alpha (A_2{B}_1+A_1{B}_2),Q_2=\alpha (A_1{A}_2+B_1{B}_2),\\ & Q_3=\beta (A_2{B}_1-A_1{B}_2),Q_4=\beta (A_1{A}_2-B_1{B}_2),\\ & A_1=\frac{1}{2}\mathrm{i}\left(k-\frac{2{\eta }^2{\lambda }_1^2}{\delta }\right)-{\mathrm{\Lambda }}_1,B_1=a\eta {\lambda }_1,\\ & A_2=\frac{1}{2}\mathrm{i}\left(k-\frac{2{\eta }^2{\lambda }_2^2}{\delta }\right)-{\mathrm{\Lambda }}_2,B_2=a\eta {\lambda }_2,\\ & {\mathrm{\Lambda }}_1=-\frac{1}{2\delta }\sqrt{4{\eta }^{2}k{\lambda }_1^2\delta -4a^2{\eta }^2{\lambda }_1^2{\delta }^2-4{\eta }^4{\lambda }_1^4-k^2{\delta }^2},\\ & {\mathrm{\Lambda }}_2=\frac{1}{2\delta }\sqrt{4{\eta }^{2}k{\lambda }_2^2\delta -4a^2{\eta }^2{\lambda }_2^2{\delta }^2-4{\eta }^4{\lambda }_2^4-k^2{\delta }^2},\\ & K=\sqrt{(4{\alpha }^2{\eta }^2+a^2{\delta }^2)(4{\beta }^2{\eta }^2-a^2{\delta }^2)},{\lambda }_2=-{\lambda }_1^{\ast },\\ & g_1=\frac{4\alpha \beta \eta K\tau }{{\delta }^2},g_2=\frac{K\zeta }{\delta }+\frac{4\eta K({\alpha }^2-{\beta }^2)\tau }{{\delta }^2},q=\frac{4\eta }{a\delta }.\end{array}

This exact analytical solution provides an effective framework for analyzing the fundamental characteristics of broad classes of nonlinear excitations in the ferromagnetic nanowire. Through analytical simplification, we reveal that the solution constitutes a unified expression capable of describing multiple types of nonlinear wave phenomena, specifically magnetic solitons, breathers, and rogue waves, superimposed on a plane-wave background in anisotropic ferromagnetic nanowires. A key structural feature of the solution lies in its composition: it integrates localized hyperbolic functions with periodic trigonometric functions. This hybrid structure implies that the nature of the solution — whether it is localized or periodic in time and space — can be finely tuned by varying the system parameters.

By carefully analyzing the relationship between localization and periodicity inherent in the solution (6), we classify the resulting nonlinear excitations into five distinct types, depending on the parameters’ phase. When the background amplitude satisfies the condition and the imaginary and real parts of the spectral parameters are equal, i.e., \alpha =\beta, the solution manifests as a KM breather. In this regime, the excitation is localized in the transverse spatial direction but remains periodic in the propagation (temporal) direction. The temporal breathing period of this solution is given by T=\pi {\delta }^2/(2\alpha \beta \eta K), reflecting the periodic energy exchange between the localized excitation and the plane-wave background. When but \alpha \ne \beta, the solution corresponds to a velocity-modulated KM breather, which we refer to as a VKM breather. In this case, the breather structure moves with a finite velocity v=4({\alpha }^2-{\beta }^2)\eta /\delta, resulting from the asymmetry in the spectral parameters. When the background amplitude exceeds the threshold value, a>2\beta \eta /\delta, the nature of the solution changes qualitatively. The excitation now becomes localized in the temporal direction while exhibiting spatial periodicity. This configuration corresponds to an Akhmediev breather (AB), which arises from the modulation instability of a plane wave. In this regime, the envelope of the breather solution effectively propagates with an infinite group velocity, while each periodically spaced wave packet (or fundamental unit) within the structure moves at a finite speed v=4({\alpha }^2-{\beta }^2)\eta /\delta. The temporal recurrence period of the AB is also given by T=\pi {\delta }^2/(2\alpha \beta \eta K), analogous to that of the KM breather. At the critical value a=2\beta \eta /\delta, both modulation parameters g_1 and g_2 vanish, leading to a degeneracy in the breather solution. Taking the limiting case a\to 2\beta \eta /\delta in solution (6) yields the Peregrine rogue wave, a highly localized and singular excitation that appears both in time and space before vanishing. The rogue wave solution marks the boundary between periodic and isolated behavior, and is the common limit form of AB and KM breathers, representing the most localized form of a modulation instability-induced structure. Finally, when the background amplitude vanishes, i.e., a=0, the solution describes a bright soliton. In summary, this classification framework not only elucidates the rich variety of nonlinear magnetic excitations supported by the system but also provides critical insights into the parameter-dependent transitions among different types of nonlinear waves. These findings serve as a theoretical foundation for future experimental investigations and the development of functional magnetic devices exploiting controllable nonlinear excitations.

We performed a detailed analysis of the system’s modulation instability (MI) characteristics to investigate the dynamical behavior of nonlinear wave excitations on a plane-wave background. According to the theory of linear stability analysis, we consider small-amplitude Fourier-mode perturbations superimposed on a plane-wave solution of the GDNLS equation (2). Specifically, we consider a plane-wave background of the form {\psi }^{[0]}=a{\mathrm{e}}^{\mathrm{i}k[\xi +(k-a^2)\tau ]} and introduce a weak perturbation as {\psi }_p=(a+\epsilon p){\mathrm{e}}^{\mathrm{i}k[\xi +(k-a^2)\tau ]}, where p(\xi ,\tau ) is a small complex-valued disturbance and \epsilon \ll 1. By substituting this perturbed ansatz into the governing equation and retaining only linear terms in p, we obtain the linearized evolution equation governing the dynamics of the perturbation. The resulting equation is: \mathrm{i}{p}_{\tau }-p_{\zeta \zeta }+2\mathrm{i}(a^2-k)p_{\zeta }+\mathrm{i}{a}^2{p}_{\zeta }^{\ast }-a^{2}k(p+p^{\ast }). To analyze the modulation instability, we consider a lowest-order Fourier-mode perturbation of the form: p(\xi ,\tau )=f_{+}{\mathrm{e}}^{\mathrm{i}(K\xi +\mathrm{\Omega }\tau )}+f_{-}{\mathrm{e}}^{-\mathrm{i}(K\xi +\mathrm{\Omega }\tau )}, where f_{+} and f_{-} are small complex amplitudes satisfying |f_{\pm }|\ll a, while K and \mathrm{\Omega } denote the perturbation wavenumber and frequency, respectively. Substituting this ansatz into the linearized equation yields a coupled system of linear algebraic equations for f_{+} and f_{-}. The existence of nontrivial solutions requires that the determinant of the coefficient matrix vanish, leading to a dispersion relation between the perturbation frequency \mathrm{\Omega } and the wavenumber K. The resulting dispersion relation is given by \mathrm{\Omega }=(2k-2a^2\delta )K+|K|\sqrt{a^4{\delta }^2-2a^2\delta k+K^2}/\eta. From this expression, it becomes evident that when the inequality is satisfied, the argument of the square root becomes negative, and \mathrm{\Omega } acquires an imaginary part. This indicates that the perturbation grows exponentially in time — a signature of modulation instability. The imaginary component of \mathrm{\Omega } quantifies the rate of this exponential growth and defines the MI gain G=\mathrm{I}\mathrm{m}(\mathrm{\Omega })=|K|\sqrt{a^4{\delta }^2-2a^2\delta k+K^2}/\eta. Figure 2(b) presents the modulation instability gain spectrum for the GDNLS equation (2) mapped on the (a,K) parameter plane. The deep blue region corresponds to the modulation-stable (MS) regime with G=0, while the other colored regions signify modulation-instable (MI) regime.

To gain deeper insight into the excitation characteristics of the KM breather, we introduce the concept of effective perturbation energy \epsilon, defined as {\epsilon }_0={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}(|\psi {|}^2-a^2)\mathrm{d}t. This quantity measures the excess energy introduced by the perturbation relative to the unperturbed plane wave background a\exp (\mathrm{i}\theta ). A positive value \epsilon >0 indicates that additional energy is injected into the system due to the perturbation, while {\epsilon }_0=0 implies that the evolution of the perturbation draws entirely from the energy of the plane wave background without external input. Based on this definition, we compute the perturbation energies of various nonlinear excitations on the plane wave background. The KM breather exhibits an effective perturbation energy of 4{\delta }^{-1}\arctan (\eta K/{\delta }^3{a}^2) whereas both the AB breather and the Peregrine rogue wave yield zero perturbation energy. The soliton corresponds to a finite-energy excitation emerging from a zero background. Figures 2(a)−(f) illustrate the temporal evolution of magnetization distributions for various types of nonlinear magnetic excitations.

3.2 Controllable transition between KM breather and soliton

The above theoretical analysis reveals that the Kuznetsov−Ma (KM) breather emerges as a perturbative nonlinear excitation sustained on a finite-amplitude plane-wave background. Intriguingly, in the limiting case where the background amplitude vanishes a\to 0, the KM breather solution degenerates into a bright soliton propagating on a zero background, highlighting its dual nature as both a localized and extended nonlinear mode. To provide a connection between the two solutions, this section employs numerical simulations to illustrate this transformation process, thereby providing a controllable protocol for the experimental generation of KM breathers in physical systems.

In our simulations, we initialize the system with the magnetic soliton solution and adiabatically introduce the plane-wave background during temporal evolution. The key stages of this transition, as captured in Fig. 3, demonstrate the following:

Stage I — Soliton regime (): In the absence of a background field, the system retains a stable soliton characterized by its canonical sech-shaped amplitude profile. The magnetization M_1^z remains localized, exhibiting no appreciable oscillatory behavior. Stage II — Background induction (): A continuous-wave background with amplitude a=0.4 is gradually imposed. As the background field is introduced, the solitary structure begins to evolve into a breathing excitation. The energy density redistributes, and the M_1^z component starts exhibiting amplitude oscillations, signifying the onset of the KM breather. The frequency and modulation depth of these oscillations are governed by the nonlinear interplay among dispersion, background amplitude, and the system’s intrinsic parameters. Stage III — Breather formation (): The system dynamically stabilizes into a well-defined KM breather. The magnetization oscillations attain a periodic pattern, with a temporal period T=\pi {\delta }^2/(2\alpha \beta \eta K), which matches the analytical prediction derived from the exact breather solution. This confirms the theoretical framework for describing breather dynamics on a continuous-wave background. Stage IV — Steady breather propagation (): The KM breather persists with sustained periodic energy exchange between the central peak and the surrounding field. The wave maintains its spatial localization while exhibiting temporal modulation, highlighting the robustness of the breather state. Stage V — Reversion to soliton (\tau >140): Upon removal or attenuation of the background field, the breather gradually reverts to a solitonic structure. The amplitude oscillations diminish, and the system returns to a stable, localized state analogous to its initial configuration.

This numerical experiment verifies the controllable transformation between bright solitons and KM breathers in ferromagnetic systems. Moreover, it provides a feasible strategy for their experimental realization in spintronic and photonic platforms. In particular, the adiabatic background modulation employed here mimics practical techniques such as parametric pumping and staggered magnetic field application, thereby offering valuable insights for the design and implementation of breather-based functional devices.

3.3 Timing switches via magnetic KM breather

Figure 4(a) presents the cross-sectional profiles of the KM breather at \tau =0. It is evident that, under a fixed background amplitude, an increase in the spectral parameter \alpha leads to a higher peak amplitude and a narrower width of the KM breather, indicating a larger maximum deflection angle of the central magnetization. On the other hand, Fig. 4(b) illustrates the temporal evolution of the peak amplitude of the M_z^1 component of the KM breather. It shows that, for a fixed \alpha, larger background amplitudes result in greater breathing amplitudes and longer periods. As the background amplitude approaches zero, the oscillations of M_z^1 diminish and stabilize, signaling a transition from a KM breather to a magnetic soliton. We further computed the energy density \mathcal{E}(\zeta ,\tau )=\mathcal{A}({\mathit{m}}_x{)}^2 of the KM breather in the ferromagnetic nanowire, whose evolution is shown in Fig. 4(c). The results reveal a clear temporal periodicity in the energy density, characterized by alternating energy concentration and dispersion. Figure 4(d) displays snapshots of the breather’s energy density at various moments within a single period. For this set of parameters, spatial integration of the energy density at different times consistently yields a constant value of 10.7, indicating that the total energy of the breather is conserved. The localized peaks in energy density at the breathing center arise from the nonzero plane-wave background. This temporal oscillation and spatial localization are sustained by a dynamic balance between nonlinear and dispersive effects.

The unique properties of magnetic KM breathers motivate our investigation of their potential spintronic applications. Next,as schematically demonstrated in Fig. 5, we propose a clocked switching device comprising a ferromagnetic layer initially saturated along the horizontal axis, where controlled excitation of KM breathers induces periodic deviations and recoveries of the central magnetic moment from its initial orientation. This oscillation between distinct magnetic states modulates the magnetoresistance, thereby generating periodic current variations through the magnetic thin film. Critically, as established in Section 3.1, the oscillation period can be electrically tuned through several parameters: the background wave amplitude a, the propagation velocity v of the wave in the medium, saturation magnetization M_0, and spectral parameters \alpha and \beta. The breather period is determined by the background amplitude a. By applying an external magnetic field of varying strength at the initial state, the background amplitude can be modified, thereby enabling the excitation of KM breathers with different periods. On this basis, it is sufficient to simply define a critical current threshold (I_c), where currents exceeding I_c trigger an “ON” state while those below it maintain an “OFF” state, we achieve field-free, self-sustained timing control driven intrinsically by KM breather dynamics, enabling auto-switching without external field assistance.

Given that white noise is unavoidable in experiments, we examine the stability of the breather under such perturbations. The results show that breathers propagate stably in the absence of noise, while their robustness under noise depends strongly on system parameters. As illustrated in Figs. 6(c) and (d), random perturbations induce modulation instability during breather evolution, with lower-amplitude breathers exhibiting greater resistance to noise. Besides noise, damping is another unavoidable factor in experiments. The presence of damping may lead to additional excitations through modulation instability, and the higher the damping, the faster the breather decays during propagation.

4 Discussion and conclusion

This work investigates magnetic breather excitations and their potential applications in anisotropic ferromagnets. Under the long-wavelength approximation, the magnetization dynamics of the anisotropic ferromagnetic layer can be described by a generalized derivative nonlinear Schrödinger equation. Starting from the Lax pair of this equation, we employ the Darboux transformation with a plane wave seed solutiuon to derive the system’s general breather solution. Based on modulation instability analysis, we classify the solution into five distinct types: KM breathers, ABs, general breathers, rogue waves, and soliton solutions, and identify their excitation regions by introducing physical quantities such as perturbation energy and growth rate.

In particular, we numerically examine the reversible transformation between magnetic solitons and KM breathers. The results demonstrate that this transition can be achieved by tuning the amplitude of the background plane wave. We focus on the characteristics and potential applications of KM breathers. The periodic magnetization oscillations arising from the energy exchange between the KM breather and its background can induce periodic variations in magnetoresistance. By appropriately setting the critical current, this effect may enable the realization of an automatic timing switch. For typical material parameters of Co/Pt ferromagnets \gamma =1.76\times {10}^{11}\mathrm{r}\mathrm{a}\mathrm{d}/(\mathrm{s}\cdot \mathrm{T}),M_s=1.4\times {10}^6\mathrm{A}\cdot \mathrm{m},c=3\times {10}^8\mathrm{m}/\mathrm{s}, v=0.1c, the switching period is estimated to be approximately t\approx 0.1614\mathrm{s}.

These findings reveal the excitation and controllability of magnetic breathers in anisotropic ferromagnets and provide a theoretical foundation for designing timing switches based on magnetic KM breathers.

5 Appendix A: Solution

To derive the breather solution of the equation, it is necessary to provide its Lax pair, which has been presented previously in Eq. (3). The GDNLS equation (2) can be derived from the compatibility condition,

(A{\text{-)\begin{array}{}\text{(A{text{-}}1)}& {\mathit{U}}_t-{\mathit{V}}_x+[\mathit{U},\mathit{V}]=0.\end{array}

Since the matrices U and V contain variables, a diagonal matrix P is introduced to transform the matrices U and V into constant matrices \hat{U},\hat{V}. The resulting Lax pair after this transformation takes the following form:

(A{\text{-)\begin{array}{}\text{(A{text{-}}2)}& \begin{array}{rl}& (P\mathrm{\Phi }{)}_{\zeta }=\hat{U}(P\mathrm{\Phi }),\\ & (P\mathrm{\Phi }{)}_{\tau }=\hat{V}(P\mathrm{\Phi }).\end{array}\end{array}

The Lax pair can be rewritten as

(A{\text{-)\begin{array}{}\text{(A{text{-}}3)}& \begin{array}{rl}& \hat{U}=PU{P}^{-1}+P_{\zeta }{P}^{-1},\\ & \hat{V}=PV{P}^{-1}+P_{\tau }{P}^{-1}.\end{array}\end{array}

We choose P in the following form:

(A{\text{-)\begin{array}{}\text{(A{text{-}}4)}& P=\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}(k\zeta +\omega \tau )/2}& 0\\ 0& {\mathrm{e}}^{\mathrm{i}(k\zeta +\omega \tau )/2}\end{array}\right).\end{array}

Therefore, \hat{U},\hat{V} can be expressed as

(A{\text{-)\begin{array}{}\text{(A{text{-}}5)}& \hat{U}=\left(\begin{array}{cc}{\hat{U}}_{11}& {\hat{U}}_{12}\\ {\hat{U}}_{21}& {\hat{U}}_{22}\end{array}\right),\end{array}

(A{\text{-)\begin{array}{}\text{(A{text{-}}6)}& \hat{V}=\left(\begin{array}{cc}{\hat{V}}_{11}& {\hat{V}}_{12}\\ {\hat{V}}_{21}& {\hat{V}}_{22}\end{array}\right),\end{array}

where

(A{\text{-)\begin{array}{}\text{(A{text{-}}7)}& \begin{array}{rl}& {\hat{U}}_{11}=-\frac{\mathrm{i}k}{2}+\frac{\mathrm{i}{\eta }^2{\lambda }^2}{\delta },\\ & {\hat{U}}_{12}=a\eta \lambda ,\\ & {\hat{U}}_{21}=-a\eta \lambda ,\\ & {\hat{U}}_{22}=\frac{\mathrm{i}k}{2}-\frac{\mathrm{i}{\eta }^2{\lambda }^2}{\delta },\\ & {\hat{V}}_{11}=\frac{2\mathrm{i}{\eta }^3{\lambda }^4}{{\delta }^2}-\mathrm{i}{a}^2\eta {\lambda }^2+\frac{\mathrm{i}k{a}^2\delta -\mathrm{i}{k}^2}{2\eta },\\ & {\hat{V}}_{12}=\frac{2a{\eta }^2{\lambda }^3}{\delta }+(ka-a^3\delta )\lambda ,\\ & {\hat{V}}_{21}=-\frac{2a{\eta }^2{\lambda }^3}{\delta }-(ka-a^3\delta )\lambda ,\\ & {\hat{V}}_{22}=-\frac{2\mathrm{i}{\eta }^3{\lambda }^4}{{\delta }^2}+\mathrm{i}{a}^2\eta {\lambda }^2-\frac{\mathrm{i}k{a}^2\delta -\mathrm{i}{k}^2}{2\eta }.\end{array}\end{array}

The eigenvalue equation of the constant matrix \hat{U} can be obtained from \mathrm{d}\mathrm{e}\mathrm{t}[U-\lambda I]=0, which has two solutions. Thus, the eigenvalues of \hat{U} are given by

(A{\text{-)\begin{array}{}\text{(A{text{-}}8)}& {\chi }_{\pm }=\pm {\displaystyle \frac{\sqrt{-4{\eta }^4{\lambda }^4+4(k-a^2\delta ){\eta }^2\delta {\lambda }^2-k^2{\delta }^2}}{2\delta }}.\end{array}

For computational convenience, it is further necessary to transform the matrices \hat{U},\hat{V} into diagonal matrices U_d and V_d. To this end, a transformation matrix D is introduced, which satisfies the following condition:

(A{\text{-)\begin{array}{}\text{(A{text{-}}9)}& \begin{array}{rl}& {\mathrm{\Phi }}_{0\zeta }=U_d{\mathrm{\Phi }}_0,\\ & {\mathrm{\Phi }}_{0\tau }=V_d{\mathrm{\Phi }}_0,\end{array}\end{array}

(A{\text{-)\begin{array}{}\text{(A{text{-}}10)}& {\mathrm{\Phi }}_0=D^{-1}P\mathrm{\Phi }.\end{array}

After computation, D takes the following form:

(A{\text{-)\begin{array}{}\text{(A{text{-}}11)}& D=\left(\begin{array}{cc}{\displaystyle \frac{2a\delta \eta \lambda }{\mathrm{i}(k\delta -2{\eta }^2{\lambda }^2)+2\delta {\chi }_{+}}}& 1\\ 1& {\displaystyle \frac{2a\delta \eta \lambda }{\mathrm{i}(k\delta -2{\eta }^2{\lambda }^2)-2\delta {\chi }_{-}}}\end{array}\right).\end{array}

Next, we solve for the eigenfunctions of the matrix \hat{U}, that is, the elements of the diagonal matrix U_d. According to Eq. (A-9), the elements of {\mathrm{\Phi }}_0 denoted as {\varphi }_0,{\phi }_0

(A{\text{-)\begin{array}{}\text{(A{text{-}}12)}& \begin{array}{r}\left(\begin{array}{c}{\varphi }_0\\ {\phi }_0\end{array}\right)=\left(\begin{array}{c}{\mathrm{e}}^{\mathrm{\Gamma }}\\ {\mathrm{e}}^{-\mathrm{\Gamma }}\end{array}\right).\end{array}\end{array}

Therefore, after a single Darboux transformation, the eigenfunction {\mathrm{\Phi }}_1=({\varphi }_1,{\phi }_1{)}^{\mathrm{T}} corresponding to the original Lax pair can be expressed as

(A{\text{-)\begin{array}{}\text{(A{text{-}}13)}& \left(\begin{array}{c}{\varphi }_0\\ {\phi }_0\end{array}\right)=\left(\begin{array}{c}\left[({\displaystyle \frac{2a\eta \delta {\lambda }_j}{\mathrm{i}k\delta -2\mathrm{i}{\eta }^2{\lambda }_j^2+2\delta \chi }}){\mathrm{e}}^{\mathrm{\Gamma }}+{\mathrm{e}}^{-\mathrm{\Gamma }}\right]{\mathrm{e}}^{(\frac{\mathrm{i}}{2}\theta )}\\ \left[{\mathrm{e}}^{\mathrm{\Gamma }}+({\displaystyle \frac{2a\eta \delta {\lambda }_j}{\mathrm{i}k\delta -2\mathrm{i}{\eta }^2{\lambda }_j^2+2\delta \chi }}){\mathrm{e}}^{-\mathrm{\Gamma }}\right]{\mathrm{e}}^{(-\frac{\mathrm{i}}{2}\theta )}\end{array}\right),\end{array}

where

(A{\text{-)\begin{array}{}\text{(A{text{-}}14)}& \begin{array}{rl}& \mathrm{\Gamma }=\chi \left(\xi +\frac{2{\eta }^2{\lambda }_j^2+k\delta -a^2{\delta }^2}{{\eta }^2{\delta }^2}\tau \right),\\ & \theta =k\xi +\frac{k^2-a^{2}k\delta }{\eta }\tau .\end{array}\end{array}

Then the new solution {\psi }^{[1]} can be given by

(A{\text{-)\begin{array}{}\text{(A{text{-}}15)}& {\psi }^{[1]}={\displaystyle \frac{2\mathrm{i}\eta {\lambda }_1}{\delta }}{\displaystyle \frac{{\phi }_1}{{\varphi }_1}}+{\displaystyle \frac{{\phi }_1^2}{{\varphi }_1^2}}{\psi }^{[0]}.\end{array}

Base on Eq. (A-15), the breather solutions can be derived by employing the two-fold Darboux Transformation from a seed plane wave solution using the following formulate:

(A{\text{-)\begin{array}{}\text{(A{text{-}}16)}& {\psi }^{[2]}={\displaystyle \frac{{\mathrm{\Omega }}_1^2}{{\mathrm{\Omega }}_3^2}}{\psi }^{[0]}+{\displaystyle \frac{2\mathrm{i}\eta }{\delta }}{\displaystyle \frac{{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2}{{\mathrm{\Omega }}_3^2}},\end{array}

where

(A{\text{-)\begin{array}{}\text{(A{text{-}}17)}& {\mathrm{\Omega }}_1=\left|\begin{array}{cc}{\lambda }_1{\phi }_1& {\varphi }_1\\ {\lambda }_2{\phi }_2& {\varphi }_2\end{array}\right|,{\mathrm{\Omega }}_2=\left|\begin{array}{cc}{\lambda }_1^2{\varphi }_1& {\varphi }_1\\ {\lambda }_2^2{\varphi }_2& {\varphi }_2\end{array}\right|,{\mathrm{\Omega }}_3=\left|\begin{array}{cc}{\lambda }_1{\varphi }_1& {\phi }_1\\ {\lambda }_2{\varphi }_2& {\phi }_2\end{array}\right|.\end{array}

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