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Rogue wave solution in ferromagnetic nanowires

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## Rogue wave solution in ferromagnetic nanowires

Li Zai-Dong, Guo Qi-Qi
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• ### Abstract

In this paper, we introduce some new excited states of magnetization in ferromagnetic nanowires, including Akhmediev breathers, Kuznetsov-Ma soliton and rogue wave in isotropic ferromagnetic nanowires, and rogue wave in anisotropic ferromagnetic nanowires driven by spin-polarized current. The isotropic case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces and a localized process of the spin-wave background. In a limit case, we get rogue waves and clarify its formation mechanism. In the case of anisotropy, it is found that the generation of rogue waves mainly comes from the accumulation of energy and rapid dispersion in the center. In addition, rogue waves are unstable, the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background. These results can be useful for the exploration of nonlinear excitation in Bosonic and fermionic ferromagnet.

### Authors and contacts

###### Corresponding author: Li Zai-Dong, lizd@hebut.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61774001) and the Natural Science Foundation of Hebei Province, China (Grant No. F2019202141)

### Cited By

• 图 1  方程(3)在$\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$的极限条件下, 磁化分量${m_3}$的渐近过程, 参数如下: As = 0.9, ks = 1　(a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001

Figure 1.  The asymptotic processes of the magnetic component ${m_3}$ in the limit processes $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}$ and $\nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$ in Eq. (3), where the parameters are as follows: As = 0.9, ks = 1: (a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001, respectively.

图 2  方程(3)和(7)中磁化强度${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$的怪波演化图, 即(a)−(c)为亮怪波, (d)−(f)为暗怪波. 参数: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}, \mu = \pm 0.75 \sqrt 3$, ±分别表示亮怪波和暗怪波[54]

Figure 2.  The graphical evolution of rogue waves for the magnetization ${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$ in Eq. (3) and (7), i.e., bright rogue waves (a)−(c) and dark rogue waves (d)−(f). The parameters are as follows: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$, and $\mu = \pm 0.75 \sqrt 3$ with the sign $\pm$ corresponding to the bright and dark rogue waves, respectively[54].

图 3  方程(3)在As = 1, $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$的条件下, 磁化分量${{m_3}}$四片花瓣结构的演化图. 参数如下: As = 1, ${k_{\rm{s}}} = 0.9, \mu = 0.8999$

Figure 3.  The formation of magnetic petal in the component ${{m_3}}$ of Eq.(3) under the special condition of As = 1, $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$. The parameters are as follows: ${A_{\rm{s}}} = 1, {k_{\rm{s}}} = 0.9, \mu = 0.8999$.

图 4  不同的参数${\mu _1}$下的磁振子密度分布图[53], 范围从0.09到0.29间隔0.05. 插图为怪波形成时的磁振子密度. 其余参数为${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_c} = 0.1$

Figure 4.  The magnon density distribution against the background for the different parameter ${\mu _1}$, which ranges from 0.09 to 0.29 in 0.05 steps[53]. The inset figure is the magnon density distribution against the background for the excited formation of magnetic rogue wave. Other parameters are ${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_{\rm{c}}} = 0.1$.

图 5  (a)−(e)不同电流值激发怪波在区域$\left( {x, t} \right)$的分布图[53], 电流从0到0.8, 间隔为0.2; (f)不同电流怪波形成时的图形, 插图为磁振子积聚的最大时情况. 临界电流${A_{\rm{J}}} = 2{k_{\rm{c}}}$, 其它参数${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$

Figure 5.  (a)−(e) The formation region in space $\left( {x, t} \right)$ for magnetic rogue wave with different current[53]. The parameter ${A_J}$ ranges form 0 to 0.8 in 0.2 steps; (f) The nonuniform exchange of magnons between rogue wave and background for the different spin current. The inset figure in (f) denotes the maximal accumulation (or dissipation) process for the critical current value${A_{\rm{J}}} = 2{k_{\rm{c}}}$. Other parameters are ${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$.

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• Abstract views:  3574
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##### Publishing process
• Received Date:  06 September 2019
• Accepted Date:  02 December 2019
• Available Online:  17 December 2019
• Published Online:  05 January 2020

## Rogue wave solution in ferromagnetic nanowires

###### Corresponding author: Li Zai-Dong, lizd@hebut.edu.cn;
• 1. Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China
• 2. School of Science, Tianjin University of Technology, Tianjin 300384, China
Fund Project: Project supported by the National Natural Science Foundation of China (Grant No. 61774001) and the Natural Science Foundation of Hebei Province, China (Grant No. F2019202141)

Abstract: In this paper, we introduce some new excited states of magnetization in ferromagnetic nanowires, including Akhmediev breathers, Kuznetsov-Ma soliton and rogue wave in isotropic ferromagnetic nanowires, and rogue wave in anisotropic ferromagnetic nanowires driven by spin-polarized current. The isotropic case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces and a localized process of the spin-wave background. In a limit case, we get rogue waves and clarify its formation mechanism. In the case of anisotropy, it is found that the generation of rogue waves mainly comes from the accumulation of energy and rapid dispersion in the center. In addition, rogue waves are unstable, the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background. These results can be useful for the exploration of nonlinear excitation in Bosonic and fermionic ferromagnet.

Reference (65)

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