Nonlinear generation modes in easy-axis anisotropy ferromagnetic spin chains with nearest-neighbor coupling

Abstract

In this paper, we consider the Heisenberg ferromagnetic lattices with single-ion easy-axis anisotropy to show the effects of the nearest-neighbor coupling on discrete modulation instability and localized energy in the forbidden gap. We use the multi-scale scheme to establish the nonlinear Schrödinnger equation from where it was carried out the static breather equation together with the threshold amplitude. Thus, one end of the spin chains was submitted to an external periodic boundary. It results that the nearest-neighbor coupling parameter induced instability in the forbidden frequency gap by increasing the amplitude of the plane wave. The most important feature of this investigation is the fact that the driven amplitude is considered below the threshold amplitude and the nonlinear supratransmission phenomenon arises. These outcome shed light on the fact that the Heisenberg ferromagnetic spin chains with a single-ion easy-axis anisotropy could be used to generate both long-lived temporal localized solitons and nonlinear supratransmission phenomenon.

1 Introduction

The nonlinear generation modes are essential in the fields of the solitary waves. Several studies have been carried out on localized modes in nonlinear lattice models during the last decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. To deal with this matter, Lai et al. [1] performed the effects of the nearest and next nearest neighbor on intrinsic localized spin resonance in ferromagnetic spin chains. They have shown that for the localized spin modes it can behave like soliton for small amplitudes and then spread out for strong enough amplitudes. In [2], the discrete breathers (DBs) as localized modes were highlighted in the weak ferromagnetic spin array. MI has been investigated in the frame of NS. It emerges that NNC induces unstable zones and leads to generate localized modes. Besides, in [3], they authors have associated the Glauber coherent state depiction with the Dyson–Maleev (DM) processing for local spin operators to derive a equation of motion in Heisenberg ferromagnetic chains with single-ion easy-axis anisotropy. They have investigated analytical localized bright solitons together with the MI growth rates. It results from this study that the single-ion anisotropy parameter can induce unstables zones as well-localized spin waves. Later, Xie et al. [4] point out the effects of the next NNC on localized bright soliton and MI growth rates. It was shown that the effects of the later reverse the propagation direction of the Brillouin zone boundary mode. Diverse other discrete structures were involved to show the behavior of the nonlinear localized modes.

The effects of the nonlinear self-modulated in optomechanical array have been shown on the localized bright solitons and it results equally the chaotic motion [4]. In [6, 7], they have used the discrete cubic–quintic nonlinear Schrödinger equation to exhibit the localized DB in optical fibers (OFs). Likewise, discrete soliton interaction in gyrotropic molecular chain have been fulfilled as well modulation growth rates of the plane wave with the variation of gyrotropy coefficient [15].

It should also be mentioned that if the study of the localized modes was a success in nonlinear systems, the generation of localized energies remains a major concern. At this purpose, several investigation have been carried out. The works of Geniet and Leon [9] were carried on the nonlinear supratransmission phenomenon, which leads to driving one end of the nonlinear chain by using the famous discrete Klein–Gordon equation. They have shown that energy can flows in the FG when the DA is considered above the threshold amplitude. More later, Tse Ve Koon et al. [14] have depicted the propagation of energy in nonlinear electrical transmission lines. They have shown the existence of the static breather and TA. In [19], it has been depicted numerically that the cubic–quintic parameter can engender the localized energy in OFs, despite the fact that the DA is below TA. Very recently, it has been noticed that the nonlinear supratransmission event can occur in discrete cold bosonic atoms zigzag optical lattice, where energy is flowing at different sites for specific time of propagation. Several applications of the nonlinear supratransmission were obtained throughout these studies, for example the generation of three-wave sharp interaction, the propagation of the signal in nonlinear electrical transmission line and the transmission of data in the case of quantum memory.

In this present work, we carry out the effects of the NNC on localized modes and we drive one end of the ferromagnetic spin chains at the cutoff frequency. As a result, we use the single-ion easy-axis anisotropy ferromagnetic spin lattice. We depicted through analytical and numerical setup the propagation of the nonlinear localized modes and the propagation of the periodic boundary in the FG as energy. We show that for specific conditions on the NNC parameter, the energy can flow in the FG despite the fact that the DA is consider under the TA. These results shown new features of the single-ion easy-axis anisotropy ferromagnetic spin chains as compared to [1,2,3,4].

The rest of the paper is constructed as follows: In Sect. 2, the mathematical model is described. The linear analysis is carried out to depict the dispersion relation and MI growth in Sect. 3. In Sect. 4, we predicted that the NNC can induce unstable zones and generate instability by growing the amplitude of the plane wave. To confirm the conjecture made through the analytical investigation, we use the NS, to show the effects of the NNC. The periodic boundary condition has been used to drive one end of the chains within the FG. In Sect. 5, we carry out the conclusions.

2 Model and description

The Hamiltonian operator of the ferromagnetism system can be set as [1,2,3,4]:

$$\begin{aligned} \mathbf {H}=-2J\sum _j\overrightarrow{S}_j\bullet \overrightarrow{S}_{j+1}-D\sum _j(S^z_j)^2, \end{aligned}$$

(1)

with $\overrightarrow{S_j}=(S^x_j,\, S^y_j,\, S^z_j)$ the sectional spin operator on trellis site j, $J>0$ the nearest-neighbor interaction and $D>0$ the positive unique-ion uniaxial anisotropy parameter which is equally call the Dzyaloshinsky–Moriya interaction. So, to maintain the ferromagnetic spin array system in the basic state, it is considered the latter without loss of generality and one can undertake that all local spins align along the simple Z-axis direction. Inserting the DM processing for local spin operators in all trellis positions in the Hamiltonian operator as in Refs. [1,2,3,4], we get the bosonic Hamiltonian for the Planck constant $\hbar =1$:

$$\begin{aligned} \mathbf {H}&=-2JS\sum _j\left( b^+_{j+1}b_j+b^+_jb_{j+1}-2b^+_jb_j\right) +j\sum _j\left( b^+_{j+1}b^+_jb_jb_j+b^+_jb_{j+1}b_{j+1}-2b^+_{j+1}b_{j+1}b^+_jb_j\right) \\&\quad +D\left( 2S-1\right) \sum _jb^+_jb_j-D\sum _jb^+_jb^+_jb_jb_j, \end{aligned}$$

(2)

where $b^+_j$ and $b_j$ are, respectively, the creation and local boson annihilation operators on j-position. However, the spin amplitude is designated by S and $S^\pm _j=S^x_j\pm iS^y_j$. It is worth highlighting here that the ground state energy of the spin trellis has been disregarded to oversimplify the mathematic manipulation. Considering the representation of the lattice system by $|\beta _j\rangle$ as it is depicted by the procedure of the Glauber coherent state [2, 3], the lattice quantum system state $|\Phi (t)\rangle$ should be a product of:

$$\begin{aligned} |\Phi (t)\rangle =\prod _j|\beta _j\rangle , \end{aligned}$$

(3)

where $\langle \Phi (t)|\Phi (t)\rangle =1.$ After some mathematical consideration, the equation of movement for the consistent state magnitude is derived as follows [2,3,4]:

$$\begin{aligned} i\frac{\mathrm{d}\beta _j}{\mathrm{d}t}&=\omega _0\beta _j-2JS\left( \beta _{j-1}+\beta _{j+1}\right) +J[|\beta _{j-1}|^2\beta _{j-1}+|\beta _{j+1}|^2\beta _{j+1}\\&\quad + \left( \beta ^\star _{j+1}+\beta ^\star _{j-1}\right) \beta ^2_j-2(|\beta _{j-1}|^2+|\beta _{j+1}|^2)\beta _j] -2D|\beta _{j}|^2\beta _j, \end{aligned}$$

(4)

where $\omega _0=4JS+D\left( 2S-1\right)$. Recently, Eq. (4) has been used to highlight the effects of the single-ion anisotropy parameter on MI [2, 3]. It results that the MI growth rates with the variation of single-ion. They established the existence of the localized bright soliton and the intrinsic localized spin-wave mode solutions in the system by using the semi-discrete multi-scale method. So, it is well known that the solid founded quantum is deeply enroll in quantum memory and several applications are concerned such as big-data storage and absolute data transfer [2,3,4, 20].

In this current study, our main objective is to investigate the nonlinear generation modes in the system by driven one end of the site in the FG. We will derive the TA from the static breather equation. We check whether the NNC can generate the localized energy in the FG. Furthermore, we overview the effects of the single-ion, NNC and the amplitude spin parameters on the MI growth rates. To our knowledge, no study has yet been carried out on the effects of the NNC in ferromagnetic chains with single-ion easy-axis anisotropy. We compare the effectiveness of the NNC on MI growth rates with the others parameters.

3 Modulation instability

This section is devoted to the MI analysis in Heisenberg ferromagnetic chains. As we have mentioned above, we overview the effects of the different parameters of the system on the MI growth rates and then compare its effectiveness. As a result of this fact, we start our assumption by considering the linear analysis. We set the plane wave solution of Eq. (4) as [2,3,4,5, 8, 12, 15, 17]:

$$\begin{aligned} \beta _j=\Gamma _0e^{i(k\delta {j}-\omega t)}, \end{aligned}$$

(5)

where $\Gamma _0$ is the initial amplitude of $\beta _j$, and k and $\omega$ are, respectively, the wave number and angular frequency of the plane wave. Substituting Eq. (5) into Eq. (4) gives the dispersion law as follows:

$$\begin{aligned} \omega = 4J\left( \Gamma ^2_0-S\right) \cos \left( k\delta \right) -2\Gamma ^2_0\left( 2J+D \right) +4SJ+D\left( 2S-1 \right) . \end{aligned}$$

(6)

Fig. 1

Dispersion curve (showing lower FG). The parameters used are $\beta _0=0.1,\,D=2.5,\,J=1\,S=20$ and $\delta =1$

Figure 1 shows the dispersion relation where it is shown the lower FG. At the center of the Brillouin zone $k=0$, the minimum lower frequency is expressed by $\omega _{\min }=D(2S-2\Gamma ^2_0-1)$, and for $k=\pi$, the corresponding maximum upper frequency reads $\omega _{\max }=D(2S-2\Gamma ^2_0-1)+8J(S-\Gamma ^2_0).$ At the corresponding cutoff frequency, the GV vanishes, which is a good argument to investigate the nonlinear supratransmission in the system. This matter will be entirely concerned in the next section. Now, we use a small perturbations in the plane wave solution to analyze the MI growth rates. We consider the solution of Eq. (4) as:

$$\begin{aligned} \beta _j=\left( \Gamma _0+B_j\right) e^{i(k\delta {j}-\omega t)}, \end{aligned}$$

(7)

where $B_j$ is the perturbed complex amplitude of $\beta _j$. Inserting Eq. (7) into Eq. (4), we set the linearized expression as follows:

$$\begin{aligned}&i\frac{\partial B_j}{\partial t}+2JS\left[ B_{{j+1}} e^{ik\delta }+B_{{j-1}} e^{-ik\delta }-2B_{j}\cos (k\delta ) \right] -2J\Gamma ^2_0 \big [B_{{j+1}}e^{ik\delta }+B_{{j-1}}e^{-ik\delta }\\&\quad +(\cos (k\delta )-1)\big (B^*_{{j-1}}+B^*_{{j+1}}\big )-B_{{j+1}}-B_{{j-1}}\big ] +2D\Gamma ^2_0\big (B_{{j}} +B^*_{{j}}\big )=0. \end{aligned}$$

(8)

We consider the solution of Eq. (8) in the form:

$$\begin{aligned} B_j=e_1\cos (Kj-\Omega {t})+ie_2\sin (Kj-\Omega {t}), \end{aligned}$$

(9)

where the perturbed wave number is K and the angular frequency of the MI is $\Omega .$ The parameters $e_s\,(s=1,2)$ are constants. Substituting Eq. (9) into Eq. (8), we get the following matrix:

$$\begin{aligned} \left( \begin{array}{cccc}i\left( \Omega -H_0\right) &{}\quad iH_1\\ H_1-H_2&{}\quad \Omega -H_0 \end{array} \right) \left( \begin{array}{c} e_{{1}}\\ e_{{2}}\end{array}\right) = \left( \begin{array}{c} 0\\ 0\end{array} \right) , \end{aligned}$$

(10)

with the nonlinear dispersion equation

$$\begin{aligned} i(\Omega -H_0)^2-iH_1(H_1-H_2)=0, \end{aligned}$$

(11)

where

$$\begin{aligned} H_0&=4\,J \left( -\Gamma ^2_0+S \right) \sin \left( K\delta \right) \sin \left( k\delta \right) ,\\ H_1&=4\,JS\cos \left( k\delta \right) \left( \cos \left( K\delta \right) -1\right) ,\\ H_2&=8J\Gamma ^2_0 \left( \cos \left( k\delta \right) -1 \right) \cos \left( K\delta \right) -4D\Gamma ^2_0. \end{aligned}$$

(12)

In Fig. 2a–d, the evolution of the MI growth rates was depicted with the variation of the single-ion anisotropy parameter. In Fig. 2a, b, we set $D=0$ and, we have exhibited three large side lobes and three additional small side lobes which have reduced the MI bands and generate unstable plane wave. We increase the single-ion value $D=0.5$, and Fig. 2c, d shows that the small side lobes at the middle decrease in amplitude. This behavior could be a good argument to increase once again the value of the single-ion to well emphasize its effects on MI growth rates.

Fig. 2

Illustration of the MI growth rate with the variation of the strength of the different single-ion anisotropy (D). a–d Strength values. a, b $D=0$ and c, d $D=0.5.$ The parameters used are $\Gamma _0=10.25,\,J=0.1,\,S=5$ and $a=1$

Thus, we have increased in Fig. 3a, b the single-ion value to $D=1.5$. It is shown how the amplitude of the additional small side lobes has been reduced, while the three large side lobes are keeping fixed amplitude. In Fig. 3c, d, the value of the single-ion was increased strong enough, and we have pointed out vanishing small side lobes which leads to stable plane wave solution under MI. It emerges that for strong values of the single-ion the system could stable, and it is very interesting for the dynamics of solitary waves. These results confirmed the investigation done in [14]. As we have highlighted at the beginning of this section, it is subservient to seek the effects of the NNC and spin magnitude on MI growth rates and perform a comparison with the previous analysis. For this, Fig. 4a–d shows the growth of the MI with unstable zones for two different values of the NNC. For $J=1.5$, we have noticed that the amplitude of the MI bands has increased than the case of null single-ion (see Fig. 4a, b). We have increased the value of the coupling strong enough to $J=5.5$, and Fig. 4c, d shows how the amplitude of the MI bands increases and the small side lobes bands also have increased to reduce the stable zones. We have to turn to the effects of the amplitude spin next. Figure 5c, d shows the effects of the spin amplitude on the MI growth rates. It is shown that the spin amplitude increases both the side lobes and the MI bands, while the stable zones are reducing. One observes that the both the NNC and the amplitude of the spin generate instability in the system and could be suitable during the investigation of the nonlinear localized energy compared to the single-ion parameter.

Fig. 3

Illustration of the MI growth rate with the variation of the strength of the different single-ion anisotropy (D). a–d Strength values. a, b $D=1.5$ and c, d $D=2.5.$ The parameters used are $\Gamma _0=10.25,\,J=0.1,\,S=5$ and $a=1$

Fig. 4

Influence of the exchange coupling of antiferromagnetic (J) on MI growth rate. a–d Exchange coupling values. a, b $J=1.5$ and c, d $J=5.5.$ The parameters used are $\Gamma _0=10.25,\,D=0.01,\,S=5$ and $a=1$

Fig. 5

Influence of the spin magnitude (S) on MI growth rate. a–d Spin magnitude values. a, b $S=50$ and c, d $S=500.$ The parameters used are $\Gamma _0=10.25,\, J=1,\, D=0.01,\,S=5$ and $a=1$

4 Numerical simulation

This section carries out full NSs of Eq. (4) by considering two cases. We first need to consolidate the above analytical investigations depend on perturbation consideration to highlight the effects of the NNC as well as the other parameters of the system. As it is known that when the perturbation extends substantially, the analytical assumption might not be well founded. Then, we focus on generation of the localized energy in the lower forbidden frequency gap, where the DA is taken above the threshold.

4.1 Investigation of unstable/stable modes

We use the following perturbed plane waves as initial condition

$$\begin{aligned} \beta _j=\Gamma _0\left( 1+0.01\cos (Kj)\right) \cos \left( kj\right) , \end{aligned}$$

(13)

We set the cell index $j=150$ and the excitation wave number $K=1$ with the periodic boundary condition. As we have proceeded in analytical investigation, we set the initial amplitude $\Gamma _0=0.4$, the coupling parameter $J=0.1$ and the amplitude of the spin $S=5$. Figure 6a shows the the density of the wave $|\beta _j|^2$ for the single-ion anisotropy $D=0.$ At site ($j=10$), the propagation of the train of pulse with symmetric amplitude is shown in the right panel of in Fig. 6b. We have carried the value of the single-ion anisotropy strong enough as we did in analytical section, to confirm whether the periodic boundary will obey the stable MI. So, we set $D=2.5$, it results soft oscillations and breathing characteristics with the amplitude around the case of null single-ion parameter in Fig. 6c, d. One can confirm here that when the single-ion value increases the system is modulationally stable, which sound to be new behavior compare to the analytical prediction made for $k=0$ in [3, 4].

Fig. 6

Numerical illustration of the spatiotemporal evolution of the $|\beta _j|^2$ with the variation of the single-ion anisotropy parameter. a–d Values of the single-ion anisotropy parameter. a, b $D=0$ and c, d $D=2.5.$ The parameters used are $\Gamma _0=0.25,\,K=1.5,\, k=0.2,\,J=1$ and $S=2$

Let get to effects of the NNC parameter and spin amplitude on modulational unstable/stable modes. For this fact, we keep fix the value of single-ion anisotropy and excitation wave number is $K=1.5$. Figure 7a–d shows mild oscillations, breathing features and a train of pulse with asymmetric amplitude for specific cell index. For $J=1$ in Fig. 7a, we have not get the modulational unstable modes, due to the small value of the coupling coefficients. Maybe we have to increase its value strong enough to corroborate our analytical prediction. We increase the NNC value to $J=1.5$, and one observes higher oscillations in left panel (see Fig. 7c), and the right panel (see Fig. 7d) shows how the amplitude of the plane wave grows asymmetrically. Beside, to clearly depict unstable modes brought the NNC, we increase strong enough the NNC values to $J=10$ and $J=15$ in Fig. 8a–d. For $J=10,$ we have shown unstable modes in Fig. 8a, while in Fig. 8b the plane wave amplitude grows to reach $5\times 10^{-1}$. In addition, we have exhibited this behavior in Fig. 8c, d for $J=15$ where it is shown that the amplitude of the plane wave grows to 0.01. We can now confirm that the NNC of ferromagnetic chains with anisotropy easy axis can generate modulational unstable modes.

Fig. 7

Numerical illustration of the spatiotemporal evolution of the $|\beta _j|^2$ with the variation of NNC. a–d Values of the coupling parameter. a, b $J=1$ and c, d $J=5.$ The parameters used are $\Gamma _0=0.25,\,K=1.5,\; k=0.2,\,D=1$ and $S=2$

Fig. 8

Numerical illustration of the spatiotemporal evolution of the modulational unstable modes brought by the NNG. a–d NNC values. a, b $J=10$ and c, d $J=15.$ The parameters used are $\Gamma _0=0.25,\,K=1.5,\; k=0.2,\, D=1$ and $S=1$

Proceeding as in the previous case, in Fig. 9a–d, we have exhibited features of the plane waves with the variation of spin amplitude, which leads to appear clearly unstable modes when its value increases. These features were shown through the stable and growing amplitude of the plane waves as well as the breathing behavior which is depicted by the Fermi–Pasta–Ulam mechanism [19]. To our knowledge, it is the first time that these features are obtained in the ferromagnetic chains with NNC. We now turn to modulated waves patterns and nonlinear localized energy phenomenon where one end of the ferromagnetic chains is driven in the lower forbidden gap.

Fig. 9

Numerical illustration of the spatiotemporal evolution of the $|\beta _j|^2$ with the variation spin amplitude. a–d Values of the spin amplitude. a, b $S=1$ and c, d $S=6.$ The parameters used are $\Gamma _0=0.25,\,K=1.5,\; k=0.2,\,D=1$ and $J=1$

4.2 Modulated waves patterns

By means of the multi-scale method, we establish the standard NLS equation from where we can latter derive the static breather equation. We assume the solution of Eq. (4) as follows:

$$\begin{aligned} \beta _j=\epsilon \Psi _j(\tau ,T,\xi )e^{i(kj\delta -\omega t)}, \end{aligned}$$

(14)

where $\tau$, T and $\xi$ are multi-scale variable with $\tau =\epsilon t,$ $T=\epsilon ^2 t$ and $\xi =\epsilon x\,$. Inserting Eq. (14) into Eq. (4) and using the continuum approximation [14], we get the standard NLSE after considering that $\xi =\zeta -V_g\tau$ and $\Psi =A/\epsilon ,\, \zeta =\epsilon X$

$$\begin{aligned} i\frac{\partial A}{\partial t}+P \frac{\partial ^2 A}{\partial X^2}+Q|A|^2A=0, \end{aligned}$$

(15)

with $P=2JS\delta ^2\cos \left( k\delta \right) ,\,\, Q=\left( J+6J\sin ^2\left( \frac{k\delta }{2}\right) +2D\right)$. Depending on the sign of the product PQ, Eq. (15) admits bright and dark soliton solutions. Figure 10 shows focusing (defocusing) zones which exhibits clearly the existence of bright and dark solitons, respectively. Thus, we can write the solution of the predicted bright and dark soliton in the form of:

$$\begin{aligned} \beta _j&=2\epsilon \mu {{\,\mathrm{sech}\,}}\left[ \epsilon \sqrt{|\frac{Q}{2P}|}\left( j\delta -V_gt\right) \right] \exp \left[ i(kj\delta -\omega t)\right] , \end{aligned}$$

(16)

$$\begin{aligned} \beta _j&=2\epsilon \mu \tanh \left[ \epsilon \sqrt{\mid \frac{Q}{2P}\mid }\left( j\delta -V_gt\right) \right] \exp \left[ i(kj\delta -\omega t)\right] , \end{aligned}$$

(17)

and $\mu$ the soliton’s width. Here, assume Eqs. (16) and (17) as initial conditions ($t=0$) and the cell index at $j=400$ to depict the evolution of the bright and dark solitons, respectively, with the effects of the NCC.

Fig. 10

Product of the dispersion and nonlinear coefficient PQ. The parameters used are $D=2.5,\,J=1$ and $S=1$

Figures 11a–d and 12a–d show the density of stable bright and dark soliton, respectively. To examine the effects of the NNC, we have pointed out in Fig. 11a, b for small value of the NNC $J=1$ and for large value $J=5$ in Fig. 11c, d, as we did previously. One can observe that bright soliton move very fast from left to right and preserves stable amplitude, despite the fact that its amplitude was slightly decreased in Fig. 11d.

We now move to verify the outcome of the NNC on the dark soliton which is usually propagated in defocusing zone. Figure 12a–d shows through top and bottom panels the evolution of dark solitons with the variation of NNC and wave number. Our results indicate that dark soliton can be generate and moves from left to right with stable amplitude. Finally, we can say that our numerical results attested openly consistent with the analytical findings for small values of NNC.

Fig. 11

a, b $J=1$ and c, d $J=5$ give the evolution of the stable bright soliton. The other parameters are $D=1,\, S=1,$ and $k=0.25$

Fig. 12

a, b $J=1,$ and c, d $J=3$ give the evolution of dark soliton. The corresponding wave number are $k=0.2$ and $k=0.3$, respectively. The other parameters are $D=1$ and $S=1$

4.3 Driven ferromagnetic chains

In this section, we have to submit one end of the ferromagnetic chains to an external harmonical boundary. This phenomenon is well known in the nonlinear system and holds the name of nonlinear supratransmission. It was carried out theoretically and experimentally by the pioneers Geniet [9, 19]. As it is the matter, we submit one end of the chains on-site ($j=0$) to an external periodic driving

$$\begin{aligned} \beta _0=\mu \cos (\omega t), \end{aligned}$$

(18)

with $\mu$ the DA to be considered above the TA and the driven frequency $\omega$ slightly below the lower cutoff frequency $\omega _{\min }=97.514$. So, the static breather solution of Eq. (15) that synchronizes and adjusts to the driven is given by [9, 19, 20]

$$\begin{aligned} \phi (X,t)=\sqrt{\frac{2\left( \omega _{\min }-\omega \right) }{2D+7J}}e^{-i(\omega -\omega _{\min })t}{{\,\mathrm{sech}\,}}\left[ \sqrt{\frac{\omega -\omega _{\min }}{2JS}}(X-X_0)\right] . \end{aligned}$$

(19)

The corresponding nonlinear and dispersion coefficients read $P=2JS\delta ^2,\,\, Q=2D+J$. Substituting Eq. (19) into Eq. (15) gives the TA from where we can deduce the DA

$$\begin{aligned} \mu _{\mathrm{th}}=2\sqrt{\frac{2\left( \omega _{\min }-\omega \right) }{2D+7J}}. \end{aligned}$$

(20)

Now, we are going to seek whether driving the NLS equation with Eq. (18) can generate instability and formation of solitons in the chains. Before going to this point, we have to indicate that the threshold depends mostly of the NNC and single-ion parameter. As we predict above, the single-ion does not get more control on the periodic plane wave than the CNN. So, to avoid carry any variables during the driven mechanism, we keep fix the single-ion as well as the spin amplitude. One can see that the threshold will decrease or increase with the value of the NNC. As it was mentioned above the NNC $J>0$ in this work, the TA will increase in amplitude for $J<1$ and decreases for $J>1.$

• Case 1: The nearest-neighbor coupling $J<1$

For more clarification, we consider $J<1$ and drive the chains to see whether the localized energy can occur in the system. Figure 13a, b shows the driven ferromagnetic chains for two different values of the NNC. We have set the DF $\omega =96.95$ below the lower cutoff frequency and the DA. For $J=0.5$, we found the corresponding threshold $\mu _{\mathrm{th}1}=1.20$, while the DA is taken above the latter at $\mu _{d1}=1.22$. One observes the production of the localized energy in the chains in Fig. 13a. We have carried up the value of the CNN to $J=0.9$ and the threshold amplitude decreases to $\mu _{\mathrm{th}2}=1.16<\mu _{\mathrm{th}1}.$ As we have chosen the previous DA above enough the threshold, we have maintained its value. In Fig. 13b, the energy transmission occurs by the way of generation of localized system which propagate through the ferromagnetic chains with single-ion easy axis. In Fig. 13b, the energy transmission occurs by the way of generation of localized system which propagate through the ferromagnetic chains with single-ion easy axis.

• Case 2: The nearest-neighbor coupling $J>1$

We consider now that the NNC is $J>1$. Through Fig. 14a–d, we have shown the propagation of the localized energy for $J=1.5$ and $J=1.9$, respectively. More explicitly, we have fixed the NNC $J=1.5$ and the TA is set at $\mu _{\mathrm{thr}3}=1.10<\mu _{\mathrm{thr}2}<\mu _{\mathrm{thr}1}$. So, to drive the chains, we should consider the DA above the threshold. Thus, we have considered the latter $\mu _{d2}=1.15$ slight below the $\mu _{d1}$. In Fig. 14a, b, we have depict the propagation of the bright soliton in the system of ferromagnetic chains for $J=1.5$. Increasing the NNC to $J=1.9$, the TA decreases to $\mu _{\mathrm{thr}4}=1.07<\mu _{\mathrm{thr}3}<\mu _{\mathrm{thr}2}<\mu _{\mathrm{thr}1}$. However, we have maintained the same value of the DA $\mu _{d2}=1.15$, as it is above the TA. In Fig. 14c, d, the propagation of the localized energy happens once again in the system. It emerges that, despite the fact that the DA is considered below the TA set in the case of ($J=0.5$ and $J=0.9$), it is shown the existence of the localized energy in the chains. For more accuracy, we next consider two strong enough values of the CCN and set the DA to $\mu _{d5}=1$, less than the previous ones, and the localized energy amplitude grows rapidly for $J=3.9$ and $J=4.9$ (see Fig. 15a–d). In addition, these features are in agreement with the results given in Refs. [1, 2], where intrinsic localization modes like bright solitons, DBs have been successful fulfilled. Otherwise, our results will open other perspectives in the nonlinear fields such as the ferromagnetic spin chain with biquadratic isotropic exchange and one-dimensional discrete faintly ferromagnetic with the crystal effects. It is equally worth highlighting that these results could be used for development of many devices in quantum information and during long-distance quantum information as well as for magneto-optic recording and high density storage devices [21, 22].

Fig. 13

Driven ferromagnetic chains with the NNC $J<1$ and the DA $\mu _{d1}=1.22$. a Driven chains for $J=0.5$ and the corresponding threshold $\mu _{\mathrm{thr}}=1.20$. b Propagation energy with $J=0.9$ and the corresponding threshold is $\mu _{\mathrm{thr}}=1.16$. The other parameters are $D=1$ and $S=1$

Fig. 14

Driven ferromagnetic chains with the NNC $J>1$ and the DA $\mu _{d3}=1.20$. a Driven chains for $J=1.5$ and the corresponding threshold $\mu _{\mathrm{thr}3}=1.10$. b Propagation of energy with $J=1.9$ and the corresponding threshold is $\mu _{\mathrm{thr}4}=1.07$. The other parameters are $D=1$ and $S=1$

Fig. 15

Driven ferromagnetic chains with the NNC $J>1$ and the DA $\mu _{d5}=1$. a Driven chains for $J=3.5$ and the corresponding threshold $\mu _{\mathrm{thr}3}=0.943$. b Propagation of energy with $J=4.9$ and the corresponding threshold is $\mu _{\mathrm{thr}4}=0.89.$ The other parameters are $D=1$ and $S=1$

5 Conclusion

In this study, we exhibited the effects of the nearest-neighbor coupling and single-ion anisotropy parameter on localized energy and discrete modulation instability in ferromagnetic spin chains. We use the linear analysis to point out the dispersion relation and modulation instability growth rates. By means of the dispersion law, we have pointed out lower forbidden frequency gap, which stimulate the investigation of the localized energy. We have shown how the nearest-neighbor coupling can increase the amplitude of the plane wave. To consolidate our analytical results, we use the numerical experiment. It is shown that the nearest-neighbor coupling generates unstable modes as well-localized energy in the forbidden gap despite the fact that the driven amplitude is considered sometime below the threshold amplitude. A common feature of all the values of nearest-neighbor coupling is that the plane wave has been successfully drive within the forbidden gap and these can open new investigation on the effects of nearest-neighbor coupling and next-nearest-neighbor coupling parameters in easy-axis weak ferromagnetic spin chains. As we have used the Glauber’s coherent state method to derive the generalized discrete nonlinear Schrodinger which describes the dynamics of ferromagnetic spin on different site, the future study will consider the momentum representation to investigate the energy propagation in the structure.