Breather Bound States in a Parametrically Driven Magnetic Wire

Abstract

We report the results of a systematic investigation of localized dynamical states in the model of a one-dimensional magnetic wire, which is based on the Landau–Lifshitz–Gilbert (LLG) equation. The dissipative term in the LLG equation is compensated by the parametric drive imposed by the external AC magnetic field, which is uniformly applied perpendicular to the rectilinear wire. The existence and stability of the localized states is studied in the plane of the relevant control parameters, namely, the amplitude of the driving term and the detuning of its frequency from the parametric resonance. With the help of systematically performed simulations of the LLG equation, the existence and stability areas are identified in the parameter plane for several species of the localized states: stationary single- and two-soliton modes, single and double breathers, drifting double breathers with spontaneously broken inner symmetry, and multisoliton complexes. Multistability occurs in this system. The breathers emit radiation waves (which explains their drift caused by the spontaneous symmetry breaking, as it breaks the balance between the recoil from the waves emitted to left and right), while the multisoliton complexes exhibit cycles of periodic transitions between three-, five-, and seven-soliton configurations. Dynamical characteristics of the localized states are systematically calculated too. These include, in particular, the average velocity of the asymmetric drifting modes, and the largest Lyapunov exponent, whose negative and positive values imply that the intrinsic dynamics of the respective modes is regular or chaotic, respectively.

1. Introduction

Pattern formation in nonlinear dissipative media relies on the coexistence of two balance conditions: between nonlinear self-focusing and linear self-stretching, under the action of diffraction and/or dispersion, and between dissipative losses and a compensating mechanism. In physical systems governed by complex Ginzburg–Landau equations, the losses are balanced by the intrinsic gain [1,2], while in passively driven optical cavities the compensation is provided by an external pump, as modeled by the Lugiato–Lefever equation [3,4].

The parametrically driven damped nonlinear Schrödinger (PDNLS) equation [5,6] is known as a universal model of periodically forced dissipative systems. Solutions of this equation are known to produce various dynamical regimes, including stationary, periodic, and chaotic ones, such as Faraday waves [7,8,9,10], single solitons [6,11,12,13,14], two-soliton states [15,16], and spatiotemporal chaos [6,17]. The parametric instability of a vertically vibrated Newtonian fluid, modeled by the PDNLS equation, leads to notable hydrodynamic phenomena manifested by standing (Faraday) waves on its surface [7]. These standing waves are most sensitive to the parametric forcing at half its frequency (the 2:1 parametric resonance) [18]. In weakly dissipative systems, the PDNLS equation captures the development of the parametric instability occurring near the 2:1 resonance [19]. In this context, it is relevant to mention that stabilization of dissipative solitons by the parametric drive was studied in various contexts, such as the periodically time-modulated damped nonlinear Schrödinger equation for wave amplitudes [6,20] and chains of coupled pendula [21], just to mention a few. Chaotic dynamical states generated by the PDNLS equation have been studied in detail too [22,23]. Also investigated were bound states of solitons in the same model, chimeras and localized dynamical chaos in parametrically driven nonlinear lattices in the discrete version of the Equation [24], as well as periodic waves and multistable dissipative solitons produced by the parametrically driven version of the complex Ginzburg–Landau equation (in other words, the PDNLS equation with complex coefficients in front of the dispersion and nonlinearity terms) [25,26]. Patterns and localized structures produced by (generalized) PDNLS and related equations can be found in Refs. [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].

Among various physical settings, nonlinear phenomena are common in magnetic materials. In the classical regime, the appropriate model for the magnetization dynamics is provided by the Landau–Lifshitz–Gilbert (LLG) Equation [49] and its generalizations [50,51]. In the case of magnetic nanoparticles, where magnetization is represented by a single magnetic domain, both theoretical and experimental studies have explored different routes to periodicity, chaos, and multistability [29,52,53,54,55,56,57,58,59,60,61,62,63]. A relevant possibility is to include additional terms in the equation that account for magnetic inertia [64,65]. In the case of objects where spatial dimensions play a fundamental role, such as wires (both straight and curved ones), tubes, membranes, cubes, tori, etc., a wide variety of magnetic textures and a broad range of pattern-formation regimes have been discovered. Among them, well-known examples are vortices, skyrmions, and droplets [66,67,68,69,70,71,72,73,74,75,76,77,78,79,80].

Two classes of physically relevant states in dissipative magnetic media, where losses can be counterbalanced by the parametric drive in the form of a periodically time-modulated (AC) external magnetic field, are magnetic solitons and Faraday waves. In this scenario, the parametrically driven LLG equation near the 2:1 parametric resonance can be approximated by the PDNLS Equation [6]. Therefore, all the phenomena predicted by the PDNLS equation can be also observed in the framework of its LLG counterpart. In particular, standing waves, pulses, solitons, breathers, and the interaction of in-phase and out-of-phase stationary solitons in magnetic wires, as well as solitons and magnetic textures in the magnetic plane, have been studied in this parametrically driven context [10,19,81,82,83,84]. However, the previous studies of PDNLS-based models did not address the complexity of nonstationary bound states of solitons, which is an essential direction for the extension of the work on this topic.

This work aims to tackle this possibility, addressing novel dynamical scenarios involving the interaction of two one-dimensional solitons in a parametrically driven magnetic chain, treated in the continuum approximation. Starting from the parametrically driven LLG model, we demonstrate that the interacting solitons either merge into a single mode or form a clearly identified bound state. Previously unexplored complex dynamical states emerge as an extension of the basic soliton dynamics. Specifically, by varying the model parameters, we observe that standing waves exhibit an oscillatory instability. Further changes in the parameters lead to a secondary oscillatory instability, which gives rise to spontaneous symmetry breaking of double breathers, which is accompanied by their drift, and, eventually, spatiotemporal chaos. The dynamics are quantified by several indicators, including bifurcation diagrams and Lyapunov exponents.

The manuscript is structured as follows. Section 2 and Section 3 present, respectively, the model and scheme of the adopted numerical analysis. Dynamical indicators, used for the characterization of various novel states produced by the model, are introduced in Section 4. Systematic numerical results and the corresponding discussion are reported in Section 5, and the paper is concluded in Section 6.

2. The Theoretical Model

We consider a magnetic wire, with the normalized magnetization field $\mathbf{m}=\mathbf{m}(\mathbf{r},t)$, where $\mathbf{r}$ and t represent the spatial coordinates and time, respectively. We concentrate on a ferromagnetic anisotropic long wire, assuming that the propagation of magnetization waves along the wire axis, i.e., in the direction represented by $\widehat{\mathbf{z}}=(0,0,1)$ [85], is governed by the LLG equation. In particular, this equation was used to model a one-dimensional magnetic wire driven by a time-modulated magnetic field applied in the perpendicular direction, to study the transition from regular solitons to regular bound states [16]. Here, we employ the same setup, as shown in Figure 1a. For this setting, the LLG equation can be cast in the form of:

$${\displaystyle \frac{\partial \mathbf{m}}{\partial t}}=-\mathbf{m}\times \Gamma +\alpha \mathbf{m}\times {\displaystyle \frac{\partial \mathbf{m}}{\partial t}}.$$

(1)

where $\Gamma$ is the effective torque and $\alpha$ is the damping coefficient. This equation is derived from the general LLG model by taking into regard that the scaled magnetization vector, $\mathbf{m}$, which represents the local orientation of constituent particles of the medium, keeps a constant absolute value, ${\left|\mathbf{m}\right|}^2=1$ and assuming the one-dimensional evolution of the $\mathbf{m}$ field along coordinate z. In this case, the effective torque in Equation (1), acting upon the magnetization in a long wire, may be approximated as:

$$\Gamma ={\displaystyle \frac{{\partial }^2\mathbf{m}}{\partial z^2}}-\beta (\mathbf{m}·\widehat{\mathbf{z}})\widehat{\mathbf{z}}+\mathbf{h},$$

(2)

which includes scaled parameter $\beta$ accounting for the anisotropy along the z-axis, which depends on properties of the magnetic medium, and the external modulated magnetic field composed of constant (DC) and time-modulated (AC) terms, namely:

$$\mathbf{h}\left(t\right)=(h_c+h_{0}cos(\Omega t))\widehat{\mathbf{x}},$$

(3)

where $h_c$, $h_0$, $\Omega$, and $\widehat{\mathbf{x}}$ are the modulation offset, AC amplitude, frequency, and the unit vector perpendicular to the wire, respectively [16].

In the application to a single magnetic particle, which does not admit spatial dependence, the LLG equation exhibits a variety of stable equilibrium solutions, including the fixed attractor with $\mathbf{m}=\widehat{\mathbf{x}}$, when the modulation is null (i.e., $h_0=0$). This means that the magnetization is aligned to the DC magnetic field. When the AC field is present, one can find periodic, quasi-periodic, and chaotic responses [86,87]. In particular, small perturbations around the stationary attractor produce damped oscillations with frequency ${\Omega }_0=\sqrt{h_c(h_c+\beta )}$. The attractor becomes unstable via an oscillatory instability when the magnetic field is modulated with a frequency close to the parametric-resonance value:

$$\Omega =2({\Omega }_0+\nu ),$$

(4)

where $\nu$ is the detuning from the resonance. Specifically, this bifurcation occurs when the modulation amplitude surpasses a critical value related to $\nu$ by the following equation:

$${\left(h_0^{\mathrm{crit}}\right)}^2={\left({\displaystyle \frac{4{\Omega }_0}{\beta }}\right)}^2\left({\nu }^2+{\left({\displaystyle \frac{\alpha (\beta +2h_c)}{2}}\right)}^2\right).$$

(5)

Under this condition, the magnetic particle responds by subharmonic oscillations along the x and y-axis, directly exhibiting the parametric resonance [83]. Equation (5) corresponds to a hyperbola in the $(\nu ,h_0)$ plane, which represents the first Arnold tongue in the system.

When the spatial coupling is incorporated into the model, the system exhibits a rich variety of responses, including the uniform state, $\mathbf{m}(z,t)=\widehat{\mathbf{x}}$, inside the Arnold tongue, subharmonic patterns predominantly located over the tongue, as well as domain walls, single solitons, and double solitons. A comprehensive summary of the single-soliton regime, which arises right under the Arnold tongue, is displayed in Ref. [84]. For small values of the modulation drive $h_0$, the soliton shows the usual behavior with subharmonic oscillations. However, for higher values of the drive, starting from $h_0\approx 0.44$, the solitons exhibit features typical for breathers [88], such as the emission of low-amplitude dispersive waves and the oscillation amplitude that varies on a long timescale. The distinction between the standard and breather solitons in the space of parameters is revealed by the numerical analysis based on the fast-Fourier-transform algorithm. It identifies a well-defined boundary between these dynamical regimes, without any coexistence region [84].

Following the pattern of Ref. [16], we here intend to perform the analysis of the double-soliton response for different values of the modulation parameters in Equations (3) and (4), scanning the $(\nu ,h_0)$ plane. The other parameters are fixed as in Ref. [16], namely, $\alpha =0.015$, $\beta =20$, and $h_c=3$, and the wire’s length is chosen as $2L=250$ (these are values relevant to the respective experimental setup). As shown in Figure 1, the double solitons may exhibit either the usual (panel (b)) or breathing (panels (c,d)) behavior, depending of the values of $\nu$ and $h_0$. Similar to single-breather states, double breathing solitons emit dispersive waves. In some cases (panels (b,c)), amplitudes of the bound solitons remain identical in the course of the long evolution, i.e., the response exhibits symmetry with respect to the midpoint between the solitons. However, in other cases (panel (d)), the symmetry is spontaneously broken, and the amplitudes of the bound solitons may differ significantly. The asymmetric double breathers also exhibit a slow steady drift, not found in the other single- and double-soliton regimes, on top of random fluctuations of their positions. Naturally, the one-sided drift of the breather is made possible by its spontaneously emerging asymmetry.

3. The Numerical Framework

We simulated Equation (1) by means of FORTRAN 90, using the fifth-order, double-precision, variable-step Runge–Kutta method, with monitoring of the local truncation error [89]. Equation (1) is supplemented by the Neumann’s boundary conditions imposed at edges of the integration domain, namely, $\partial \mathbf{m}/\partial z=0$ at $z=\pm L$. The accuracy or tolerance level was adopted at the level of ${10}^{-5}\%$. The spatial discretization was implemented with $dz=1/6\approx 0.17$, in the domain of size $-125<z<+125$. The accuracy of the numerical scheme was earlier established in Refs. [84,90]. The simulations were run for different values of parameters $(\nu ,h_0)$ in Equations (3) and (4), with $\nu$ and $h_0$ ranging from $-0.7$ to 0 and from $0.3$ to $0.9$, respectively. For both these parameters, the variation step was $0.002$, except inside the region of the standard single soliton, where the step was $0.01$ (five times coarser), as dynamical indicators of this regime are well known [84,90]. The simulations were run in parallel using MPI FORTRAN. To assess the reproducibility of the model and the reliability of the algorithm, simulations were also run using Python 3.11, accelerated with Numba 0.56.

The initial conditions were set symmetrically with respect to $z=0$, approximately corresponding to two solitons separated by distance $\Delta z=7$, while keeping the unitary field magnitude ($\left|\mathbf{m}\right|\equiv 1$) everywhere along the wire:

$$\begin{array}{cc}\hfill m_y(z,t=0)& =\frac{1}{2}\left(sech(z+3.5)+sech(z-3.5)\right),\hfill \\ \hfill m_z(z,t=0)& =0.95m_y,\hfill \\ \hfill m_x(z,t=0)& =\sqrt{1-m_y^2-m_z^2}.\hfill \end{array}$$

(6)

The timeline of each simulation was divided into three different stages. The first one is the transient stage, when the system starts its evolution from input (6). The sampling timestep adopted for this stage was $dt=5^3/2^6\approx 1.95$, and its duration was $\Delta t=5\times {10}^3\approx 2560\times dt$, after which we can safely assume that the system has reached a steady state. During the next (steady-state) stage, a number of dynamical indicators were computed to gather relevant information about the system, including its response type. This stage had duration $\Delta t={10}^3$, while the sampling timestep was adopted as $dt=5^3/2^{14}\approx 0.0076\approx \Delta t/\mathrm{131,600}$, being much shorter than the modulation period of the AC magnetic field, see Equation (3). Data for the computation of the largest Lyapunov exponent (LLE) for the dynamical regimes were collected in the course of the longer third stage, with duration $\Delta t=14\times {10}^3$ and sampling time step $dt=5^3/2^7\approx 0.98\approx \Delta /\mathrm{14,300}$.

4. Dynamical Indicators

Addressing the last part of the transient stage of the simulations, namely, $t\in [4000,5000]$, spatially uniform solutions and fast-moving localized modes were excluded from the analysis. Those states are irrelevant for the present work, as they are not related to the single- and double-soliton regimes, which are the target of the consideration. To distinguish the irrelevant states from the relevant ones, the coordinate of the center of mass (CM) of the soliton (or solitons) potentially appearing in the magnetization field was calculated as:

$$z_{\mathrm{CM}}\left(t\right)={\displaystyle \frac{{\int }_{-L}^{+L}z·m_y^2(z,t)dz}{{\int }_{-L}^{+L}{m}_y^2(z,t)dz}}.$$

(7)

We focused on the $m_y$ component of the magnetization field, as $m_y=0$ in the uniform state, while solitons give rise to a local excitation of $m_y$. The CM mean velocity of the drifting solitons was found by fitting the time series $z_{\mathrm{cm}}\left(t\right)$ to a linear dependence, and taking its slope. Modes with the drift velocity exceeding $0.005$ are considered as fast-moving ones and dismissed from the subsequent analysis due to artifacts expected from their collisions with the boundaries of the simulation box. This problem will be eliminated if a ring-shaped wire is considered, with periodic boundary conditions, which will be the subject of a separate work.

Additionally, the following integral characteristic was computed:

$$I_D={\int }_{4000}^{5000}{\left[{\displaystyle \frac{1}{2L}}{\int }_{-L}^{+L}{\left(m_y(z,t)\right)}^{2}dz\right]}^{2}dt,$$

(8)

which provides a measure of the departure of the given state from the uniform one, which has $I_D\equiv 0$. Subsequently, $I_D$ can be used to roughly estimate what share of the entire wire is occupied by the solitons or patterns. States with $I_D<{10}^{-10}$ are assumed to be uniform ones, while those with $I_D>{10}^{-2}$ are considered as patterns.

While considering the steady-state stage, in the course of interval $t\in [5000,6000]$, attention was again focused the $m_y$ component. The short-timescale dynamics of $\mathbf{m}$ in response to the external field modulation was eliminated by computing the envelope of $m_y$. The evolution of the envelope is represented by values $m_y(z,t_i)$, where $t_i$ are time points maximizing the space average of $m_y$:

$$\langle m_y\rangle \left(t\right)={\displaystyle \frac{1}{2L}}{\int }_{-L}^{+L}{m}_y(z,t)dz.$$

(9)

Thus, only variations of the response amplitude on the long timescale are taken into account. Figure 2 shows examples of the envelope evolution for different responses observed at the steady-state stage for different values of the control parameters, $(\nu ,h_0)$. Panels (a,b) show the standard single-soliton and breather regimes, which are similar to those previously reported in Ref. [16]. Panel (c) shows an example of the standard two-soliton pair. In that case, individual solitons in the pair keep the amplitude and width similar to those of the single-soliton states.

On the other hand, panels (d,e) in Figure 2 display novel double-breather soliton solutions, whose amplitudes oscillate on the long timescale, unlike the nonbreathing solitons, which remain completely stationary. Both single- and double-breather solitons emit radiation. Furthermore, the double breather in panel (e) in Figure 2 is drifting away from its initial position, which is closely related to the noticeable breaking of its inner symmetry. The drift is exhibited by all double-breather solitons (it is not visible in panel (d), as the timescale is not long enough for that). Indeed, the asymmetric emission of radiation resulting from the spontaneously emerging asymmetry in the structure of the soliton complexes gives rise to unequal left and right recoil forces, which drives the drift. Finally, panel (f) in Figure 2 shows an example of the dynamical regime in which the entire wire is occupied by an apparently randomized subharmonic pattern.

In the steady stage, the CM position of the envelope was defined in the same way as in Equation (7), with the difference that here we consider only time moments $t_i$ at which $\langle m_y\rangle \left(t\right)$ has a maximum. The average CM velocity is then estimated, as said above, by means of the linear fit to $z_{\mathrm{CM}}\left(t_i\right)$. To provide a simple criterion distinguishing single and double solitons in the response, the time average of the centered envelope was computed:

$${\overline{m}}_{\mathrm{env}}\left(z\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{m}_y(z-z_{\mathrm{CM}}\left(t_i\right),t_i).$$

(10)

In the framework of the time-averaging procedure, the radiation waves emitted by the breathers were flattened, and the number of solitons was identified by directly counting the number of local-maximum points in the $\langle m_y\rangle \left(z\right)$ profile.

Regardless of the number of solitons, the distinction between fundamental ones and breathers was determined by computing the normalized power spectral density (PSD) of $\langle m_y\rangle \left(t\right)$, i.e., the squared absolute value of the complex fast Fourier transform, normalized to its maximum. Figure 3 presents plots of both $\langle m_y\rangle \left(t\right)$ and its PSD corresponding to the same responses which are displayed in Figure 2. In the usual regimes (panels (a,c)), the soliton’s oscillation amplitude remains constant, and the PSD exhibits a global peak at $\omega =\Omega /2$ (i.e., half the driving frequency in the parametrically forced Equation (3)), as the response is subharmonic, and it also exhibits lower peaks at odd multiples of $\Omega /2$. If the breathing is present (panels (b,d)), the soliton’s size varies on the long timescale, and PSD exhibits a multitude of low peaks agglomerated around the odd multiples of $\Omega /2$, representing a large set of oscillation modes. The simple peak count makes it possible to conclude if the breathing is present or not in the dynamical regime. To this end, only the peaks at frequencies $\omega <3\Omega$ are considered, as peaks beyond this range are extremely weak.

The microscopically defined magnetic free energy Q of the system is the sum of the exchange energy, anisotropy energy, and the energy of the interaction with the external magnetic field [49], that is:

$$Q={\displaystyle \frac{1}{2L}}{\int }_{-L}^{+L}\left({\displaystyle \frac{1}{2}}{\left({\partial }_z\mathbf{m}\right)}^2-{\displaystyle \frac{\beta }{2}}(1-m_z^2)-\mathbf{h}·\mathbf{m}\right)dz.$$

(11)

The last dynamical indicator employed in our analysis is the largest Lyapunov exponent (LLE), ${\lambda }_{max}$, which provides a deeper understanding of the dynamical system. In particular, ${\lambda }_{max}$ determines the average exponential rate of divergence or convergence of neighboring orbits in the system’s phase space. Thus, condition ${\lambda }_{max}>0$ delineates a region of chaotic dynamics, suggesting rapid loss of the system’s predictability, which is the hallmark of dynamical chaos. The magnitude of positive ${\lambda }_{max}$ defines the timescale on which the dynamics become unpredictable. Conversely, exponential convergence, corresponding to ${\lambda }_{max}<0$, indicates stability of periodic orbits. In the neutral case, ${\lambda }_{max}\approx 0$, the system is defined as a marginally stable or quasi-periodic one.

The LLE can be calculated as [91]:

$${\lambda }_{max}=\underset{t\to \infty }{lim}{\displaystyle \frac{1}{t}}ln{\displaystyle \frac{\left|\right|\delta \mathbf{m}(z,t)\left|\right|}{\left|\right|\delta \mathbf{m}(z,t_0)\left|\right|}},$$

(12)

where $\left|\right|•\left|\right|\equiv ({\int }_{-L}^{+L}{|•|}^2{dz)}^{1/2}$. The vector function $\delta \mathbf{m}$ in Equation (12), which represents the distance in the phase space between close orbits, obeys the linear equation:

$${\displaystyle \frac{\partial \left(\delta \mathbf{m}\right)}{\partial t}}=\overline{\mathbf{J}}·\delta \mathbf{m},$$

(13)

where $\overline{\mathbf{J}}$ is the Jacobian matrix of Equation (1). In our numerical analysis, ${\lambda }_{max}$ was calculated from $t_0=6\times {10}^3$ up to $t_{max}=2\times {10}^4$. At each time step, $\left|\right|\delta \mathbf{m}(z,t)\left|\right|$ was rescaled to restore its initial norm, $\left|\right|\delta \mathbf{m}(z,t_0)\left|\right|$. Finally, ${\lambda }_{max}$ is evaluated as the average value between times $t=1.8\times {10}^4$ and $t=2.0\times {10}^4$. Let us comment that this method is the most common one used in the literature to characterize chaotic states [92,93,94,95,96,97,98,99,100].

5. Results

5.1. Existence Regions

Figure 4a summarizes all localized-state responses in the $(\nu ,h_0)$ map for the initial condition given by Equation (6). The existence regions lie right under the boundary of the first Arnold tongue, above which the system exhibits only delocalized patterns covering the entire spatial domain. Among the existence regions, the lowest and largest ones are those where the system responds with the creation of a standard single soliton (1—pink) or a single breather soliton (2—purple). Both regions are significantly larger than their counterparts found in Ref. [90], almost doubling their area. This happens because a different initial condition, in the form of a simple soliton, was used in that work. Surprisingly enough, such initial condition falls outside the attraction basin of the single-soliton response for lower values of $h_0$, yet the double-soliton initial condition, defined above by Equation (6), belongs to the respective basin.

On the other hand, the double-soliton initial condition does produce a stable double-soliton response above the aforementioned regions, still under the boundary of the Arnold tongue, between $\nu =-0.5$ and $\nu =0.2$. Three regions are distinguished here. In the middle one (3—orange), the double soliton is of the usual type, with no breathing, while near the lower-left border and in the upper part (4—red) the double soliton exhibits breathing (thus, this response is found in a disconnected set). Some of these modes exhibit the drift at a relatively small velocity. The region in the upper-left corner (5—blue) corresponds to the responses that are identified as fast-drifting localized structures, according to the measurement of their CM velocity, as outlined at the beginning of Section 4. Here, the system exhibits breathing double solitons. Drifting at larger velocities, they hit edges of the spatial domain, during either the transient stage of the steady one. The simulations reveal that, as a result of hitting the edge, the double soliton stays stuck near the edge for an uncertain time, and then drifts back. In the double-soliton regions, the single-soliton initial condition still leads to the stable single-soliton responses [90], which implies the system’s multistability, as the single- and double-soliton regimes coexist with each other and with the uniform state, $\mathbf{m}=\widehat{x}$.

An unexpected response was observed in a very small region around $(\nu ,h_0)=(-0.20,0.45)$, just under the Arnold tongue (region 6—green in Figure 4a). Here, the system exhibits a localized intermittent state where a small region in the wire is filled by solitons whose number varies in time, as they may spontaneously appear, disappear, or collide. Figure 5a illustrates an example of the intermittent state. The two initially launched solitons collide and merge into a single soliton, which suddenly splits in a set of seven solitons, which occupy the region between $z=-50$ and $z=+50$ with a relatively smooth distribution. Shortly afterwards, the solitons drift towards the center, leading to disappearance of the central soliton, and thus, decreasing the soliton number to six. Next, two adjacent solitons collide and merge, which further reduces the number to five. As the solitons keep drifting to the center, the above-mentioned soliton, produced by the merger, also disappears, while two others, adjacent to it, collide and merge once again. The remaining three solitons become closer and suddenly split in seven solitons, then repeating the cycle of the soliton metamorphoses. This dynamical regime is very robust and regular in its evolution. It is spatially symmetric and does not exhibit a global drift. The regularity of this regime is also confirmed in Figure 5b by the respective picture of the evolution of the spatial average $\langle m_y\rangle \left(t\right)$, as well as by its PSD. As the number of solitons decreases by steps, so does the oscillation amplitude of $\langle m_y\rangle \left(t\right)$, while the regime remains periodic on the long timescale, covering the entire cycle of the metamorphoses. The corresponding PSD exhibits pronounced peaks at the subharmonic response frequency $\omega =\Omega /2$, as well as its odd multiples, similar to the regimes based on single and double standard solitons, which are outlined above.

The boundaries between the existence regions of localized states plotted in Figure 4 are well defined in most cases, except for the upper-left corner, where the boundaries are fuzzy. The reason for the fuzziness is that, as mentioned above, stability regions of the respective solutions overlap, and the multistability takes place. As the attraction basins are different for different areas in the $(\nu ,h_0)$ plane, the initial condition given by Equation (6) evolves towards different attractors. The fuzzy boundaries involve breathing solutions only, suggesting that the true attraction basins may have a fractal structure, and the attractors are chaotic [101,102,103].

In the rest of the region under the Arnold tongue, the system responds with either a spatially uniform state or delocalized patterns.

5.2. The Power Spectral Density

Panel (b) in Figure 4 shows the number of peaks in the power spectral density from $\omega =0$ to $\omega =3\Omega$ in the existence regions of localized states in the $(\nu ,h_0)$ plane. There is a large region with a reduced number of peaks $(\le 5)$. This region coincides with the union of those where the system exhibits either a single soliton or double standard ones in Figure 4a. The chart is in agreement with the examples displayed in Figure 3a,c. Single- and double-soliton regimes with no breathing feature only three peaks at $\omega =\frac{1}{2}\Omega$, $\omega =\frac{3}{2}\Omega$ and $\omega =\frac{5}{2}\Omega$ (with some additional tiny peaks arising as the level of the numerical errors). On the other hand, the number of peaks is suddenly increased in the breather-soliton regions. Most of the single-breather-soliton region features the number of peaks ∼50–100. In most of the lower-left double-breather region, the results are similar. In the upper-right double-breather region, four well-delimited zones can be recognized, where the numbers of peaks are on the order of 1, 10, and 100. The last of these zones surround the region of fast-drifting localized modes, where the number of peaks is counted in hundreds and thousands, with a maximum of ≈3200 peaks. This is also consistent with the examples displayed in Figure 3b,d.

5.3. The Center-of-Mass (CM) Drift of Solitons

Figure 6 summarizes indicators related to the CM mobility of the soliton structure, at different values of $(\nu ,h_0)$. For each point of the map, the time series $z_{\mathrm{CM}}\left(t_i\right)$, accounting for the CM position, is extracted from the envelope data in the course of the steady-state stage (i.e., at $t_i\in [5000,6000]$), and the following dynamical indicators are computed. The absolute value of the average CM velocity, which is estimated as the slope of the linear fit for $z_{\mathrm{CM}}\left(t_i\right)$, is shown in panel (a). The departure of the velocity from its average is estimated as the standard deviation of the linear-fit slope, considering $z_{\mathrm{CM}}\left(t_i\right)$ terms as normally distributed random variables [89]. The deviation is shown in panel (b). The absolute value of the time-average CM position and its standard deviation are shown in Figure 6c and d, respectively.

In Figure 6, four well-delimited zones can be identified in terms of these dynamical indicators. Zone 1 (in purple) is the largest one, encompassing all usual-single-soliton regimes, almost all single-breathing-soliton ones, almost all regimes of usual double solitons, and most cases of double breathing solitons. Here, the average soliton speed is ≲${10}^{-12}$, the other indicators exhibiting extremely low values too, which may be regarded as zero for all practical purposes. Zone 2 is a disconnected set at the upper-left corner of Figure 6. It includes a narrow edge occupied by single breathing solitons under (but disconnected from) a narrow edge occupied by the usual double solitons, whose left end is connected to the lower end of the edge occupied by single breathing solitons. The solitons in this zone exhibit average speeds ∼${10}^{-6}$ with deviations ∼${10}^{-9}$. Such speeds imply a displacement ∼${10}^{-3}$ during the steady stage, which is much smaller than the mesh size $dz=1/6$ of the numerical scheme. Therefore, these solitons are also regarded as nondrifting ones. However, the average displacement of the solitons from $z=0$ ranges between $0.1$ and 10. This occurs because, although the solitons do not drift during the steady-state stage, they do so during the preceding transient stage, thus producing an initial position at the beginning of the steady-state stage far from $z=0$.

Zone 3 is the right diagonal end of the breathing double-soliton region. It has the same features as zone 2 in terms of the average drift speed, but the average CM position is closer to zero (with a distance ∼${10}^{-4}$), which means that the CM remains close to $z=0$ in the course of both the transient and steady-state stages. On the other hand, both the CM position and CM velocity exhibit larger, yet still small deviations, ∼${10}^{-1}$ and ${10}^{-5}$, respectively. These tiny fluctuations are caused by the complexity of the breathing double-soliton structure. Zone 4 is located at the upper-left corner in Figure 6, around the region of the fast-moving localized modes, see Figure 4a. It is occupied exclusively by the breathing double solitons, being the only zone that exhibits nonnegligible drift. The respective drift speeds exhibit average values in the range of ${10}^{-4}$ to ${10}^{-1}$, with the maximum average speed found to be $0.032$, which allows the double breather to travel the distance of $32.0$ during the steady-state stage. Considering that the average CM positions have absolute values in the range of 1 to 10, with a maximum of $39.28$, which implies that the double breathers remain far enough from the wire’s edges at $z=\pm 125$. All in all, some double breathers exhibit a persistent drift with an almost constant velocity, while others move with a sign-changing (oscillating) velocity.

5.4. The Micromagnetic Energy

Figure 7a summarizes the computation of the maximum value of the micromagnetic free energy along the line which was shown above in Figure 4, defined as $h_0\left({\nu }_{h_0}\right)=2.857{\nu }_{h_0}+1.729$ and ranging between points $(\nu ,h_0)=(-0.43,+0.5)$ and $(\nu ,h_0)=(-0.36,+0.7)$. For 2048 points uniformly distributed along the line, the peak values of the time series for the energy were found. Along the line, a number of segments can be identified in terms of the peak energies. First, in the segment $0.5<h_0<0.553$, the peak energies increase with a uniform slope of $\approx 0.98$, always lying in a band of width $\Delta Q=0.025$. This segment is the one where the line passes through the lower region of the breathing single solitons. At $h_0\approx 0.553$, there is a sudden increase in the peak energies, as the line is now passing through the region of the breathing double solitons. The energy increases here, as a longer section of the wire is now occupied by the excited state. As $h_0$ grows, so do the peak energies, but the width of the band decreases. Close to $h_0=0.579$, the peak-energy branch becomes a line, which extends up to $h_0=0.609$ with a slope of ≈0.97. This makes sense because the respective segment passes through the region of the standard (usual) double solitons. As the long-timescale dynamics remain steady, the peak energies are identical to one another. At $h_0=0.609$, the line enters the upper region populated by double breathing solitons, and the peak energies again appear as bands with a width that tends to increase with $h_0$. All in all, the peak energies keep increasing with $h_0$ at a fairly constant rate. At $h_0=0.662$, the peak energies increase suddenly from about $-12.3$ to $-11.3$, as the line enters the region where the system switches into the subharmonic pattern, the entire wire being now occupied by the excited state.

5.5. The Largest Lyapunov Exponent (LLE)

The LLE was computed for the same set of 2048 points along the same line plotted in Figure 4. The results are summarized in Figure 7b. Similar to the peak energies, changes in the LLE along the line can be associated with its segments traversing different existence regions, see Figure 4a. For $h_0\in [0.5,0.553]$, in the region of the breathing single solitons, the LLE shows values ∼${10}^{-4}$. These small positive values imply that the system is weakly chaotic in this case. Similar LLE values are observed for $h_0\in [0.561,0.579]$, in the lower region of the breathing double solitons. Between these two intervals, i.e., at $h_0\in [0.553,0.561]$, the LLE exhibits much larger values, up to $0.026$. This part of the line covers the lower edge of the lower region of the breathing double solitons, where the respective PSD exhibits significantly more peaks than in the rest of the region (about 10 times more, as shown in Figure 4b). This difference is related to the drastic increase in the LLE values in this segment. For $h_0\in [0.579,0.604]$, in the region of usual double solitons, the LLE takes negative values ∼$-{10}^{-5}$. This means that the usual double-soliton states are nonchaotic. For $h_0\in [0.604,0.662]$, in the upper region of the breathing double solitons, the LLE takes positive values ∼$5\times {10}^{-4}$. At $h_0>0.662$, in the region of the subharmonic patterns, the LLE drastically increases to about $0.045$, which is expected, given the chaotic shape of these patterns occupying the entire wire. All in all, the LLE analysis shows that the dynamics of the usual solitons is regular (nonchaotic), while it is chaotic for the breathing solitons, and subharmonic patterns are chaotic too. This is consistent with the results for single solitons reported in Ref. [84].

6. Conclusions

The subject of this work is the systematic study of dynamical regimes in the one-dimensional rectilinear magnetic wire modeled by the LLG (Landau–Lifshitz–Gilbert) equation with the Neumann’s boundary conditions, driven by the spatially uniform magnetic field, including DC and AC components, which is directed perpendicular to the wire. The AC parametric drive maintains magnetic excitations at the subharmonic frequency. By means of the systematic numerical analysis, which includes the computation of PSD (power spectral density) with the help of the Fourier transform, areas in the parameter plane of the drive’s amplitude and frequency detuning from the parametric resonance are identified, in which the driven LLG equation supports different robust localized dynamical states. These include usual single- and double-soliton modes, single and double breathers (which emit small-amplitude dispersive waves), fast-moving modes, and intermittent multisoliton complexes, which exhibit periodic metamorphoses with the change in the number of solitons. The results reveal the presence of multistability in the system, which includes the coexistence of stable localized states and delocalized ones that cover the entire wire. Basic dynamical characteristics of these states are found, such as the average magnetization, LLE (largest Lyapunov exponent), and velocity of the CM (center of mass) drift, in the case of the double breathers with spontaneously broken inner symmetry. The states with negative and positive values of LLE feature, respectively, regular and chaotic evolution.

The dynamical states addressed in the present work do not exhaust the variety of complex modes that the LLG equation may generate. More sophisticated states, such as multisoliton bound states, may be a subject for further investigation.

It may also be relevant to extend the work by considering the model based on the LLG equation with periodic boundary conditions, corresponding to the ring-shaped magnetic wire. Besides, it may be interesting to study collisions between the fast-moving modes. In these cases, a comparison with the simplified model based on the PDNLS (parametrically driven damped nonlinear Schrödinger) equation can help to establish the universality of the phenomena. A challenging possibility is to develop a two-dimensional version of the setting. Indeed, it is known that many models predict the existence of stable two-dimensional solitons maintained by the balance of gain and loss (see, e.g., Ref. [104]).