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Acta Physica Sinica
Vol. 69, Issue 1, 017501-1 (2020)

Rogue wave solution in ferromagnetic nanowires

Zai-Dong Li^1,2,* and Qi-Qi Guo¹

Author Affiliations

¹Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China

²School of Science, Tianjin University of Technology, Tianjin 300384, China

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DOI: 10.7498/aps.69.20191352 Cite this Article

Zai-Dong Li, Qi-Qi Guo. Rogue wave solution in ferromagnetic nanowires[J]. Acta Physica Sinica, 2020, 69(1): 017501-1 Copy Citation Text

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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.

Keywords

breathers magnetic rogue wave modulation instability spin-transfer torque

$ \frac{{\partial {{m}}}}{{\partial t}} = - {{m}} \times {{{h}}_{{\rm{eff}}}} + {\rm{\alpha }}{{m}} \times \frac{{\partial {{m}}}}{{\partial t}} + {A_{\rm{J}}}\frac{{\partial {{m}}}}{{\partial x}}, $

(1)

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$ \frac{{\partial {{m}}}}{{\partial t}} = - {{m}} \times \frac{{{\partial ^2}{{m}}}}{{\partial {x^2}}}. $

(2)

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$ {{m}} \cdot {{\sigma}} = {{K}}\left( {{{{m}}_0} \cdot {{\sigma}}} \right){{{K}}^{ - 1}}, $

(3)

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$ {{K}} = \frac{1}{{{{\left| \xi \right|}^2}\left( {P + Q} \right)}}\left(\!\!\!{

\begin{matrix} ξ^{*} P + ξ Q & - μ R^{*} e^{- i δ} \\ μ R e^{i δ} & ξ P + ξ^{*} Q \end{matrix}

}\!\!\!\right), $

()

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$

\begin{aligned} ξ = i μ / 2 + ν / 2, N = \sqrt{{(k_{s} - 2 ξ m_{03})}^{2} + 4 ξ^{2} A_{s}^{2}}, \\ β = - i 2 ξ - i m_{03} k_{s}, \\ P = h_{11} h_{11}^{*}, Q = h_{12} h_{12}^{*}, R = - i e^{- i δ} h_{11}^{*} h_{12}, \\ h_{11} = i (C_{1} e^{B} - C_{2} e^{- B}) e^{- i δ / 2}, \\ h_{12} = (C_{1} e^{- B} - C_{2} e^{B}) e^{i δ / 2}, \\ C_{1} = \sqrt{(μ m_{03} + i (A_{s}^{2} k_{s} - N)) / 2}, \\ C_{2} = \sqrt{(μ m_{03} + i (A_{s}^{2} k_{s} + N)) / 2}, \\ B = - i N (x + i β t) / 2. \end{aligned}

$

()

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$

\begin{aligned} P = A_{s} (k_{s} c o s h θ - N s i n h θ) - μ c o s ϕ - N m_{03} s i n ϕ, \\ Q = A_{s} (k_{s} c o s h θ + N s i n h θ) - μ c o s ϕ + N m_{03} s i n ϕ, \\ R = μ c o s h θ + i N m_{03} s i n h θ - A_{s} (k_{s} c o s ϕ + i N s i n ϕ), \end{aligned}

$

(4)

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$

\begin{aligned} m_{+} (0, t) & \approx (- 1 \pm i ϵ \frac{4 μ N}{k_{s}^{2}} s i n ϕ) e^{i k_{s} x}, \\ m_{3} & \approx \pm ϵ \frac{4 N^{2}}{k_{s}^{2}} s i n ϕ, \end{aligned}

$

(5)

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$

\begin{aligned} P = μ c o s h θ + ζ m_{03} s i n h θ - A_{s} (k_{s} c o s ϕ + ζ s i n ϕ), \\ Q = μ c o s h θ + ζ m_{03} s i n h θ - A_{s} (k_{s} c o s ϕ - ζ s i n ϕ), \\ R = A_{s} (k_{s} \cosh θ + i ζ \sinh θ) - μ \cos ϕ + i ζ m_{03} \sin ϕ, \end{aligned}

$

(6)

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$

\begin{aligned} P = & {(2 t A_{s} k_{s}^{2} + A_{s} k_{s} m_{03} x \pm 1)}^{2} \\ + A_{s}^{3} k_{s}^{2} (A_{s} x^{2} - 3 A_{s} k_{s}^{2} t^{2} \mp 6 t), \\ Q = & {(2 t A_{s} k_{s}^{2} + A_{s} k_{s} m_{03} x \mp 1)}^{2} \\ + A_{s}^{3} k_{s}^{2} (A_{s} x^{2} - 3 A_{s} k_{s}^{2} t^{2} \pm 6 t), \\ R = & i 2 A_{s}^{2} k_{s} (x + 3 t k_{s} m_{03}) + (P + Q) / 2 - 2, \end{aligned}

$

(7)

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$

\begin{aligned} m_{+} = - e^{i k_{s} x} [1 - (8 x^{2} k_{s}^{2} - i 4 x k_{s} (F_{1} - 2)) / F_{1}^{2}], \\ m_{3} = \pm 8 t x k_{s}^{3} / F_{1}^{2}, \end{aligned}

$

(8)

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$ {\rm{i}}\frac{{\partial q}}{{\partial t}} = \frac{{{\partial ^2}q}}{{{\partial ^2}x}} + \frac{1}{2}q|q{|^2} + {\rm{i}}{A_{\rm{J}}}\frac{{\partial q}}{{\partial x}} - {\omega _0}q, $

(9)

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${Q_1} = {A_{\rm{c}}}{{\rm{e}}^{{\rm{i}}\varphi }}\left[ {\frac{{4\left( {1 - {\rm{i}}tA_{\rm{c}}^2} \right)}}{{{t^2}A_{\rm{c}}^2\eta + 2txA_{\rm{c}}^2\zeta + \varepsilon }}} \right], $

(10)

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$ \rho = 8A_{\rm c}^2\frac{{{\varGamma _1} - {\varGamma _2}}}{{{{\left( {{\varGamma _1} + {\varGamma _2}} \right)}^2}}}, $

(11)

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${\delta _1}(x, t) \equiv \mathop {\lim }\nolimits_{lQ \to \pm \infty } {\left| {{Q_1}(x, t) - {Q_1}\left( {x = {l_Q}, t} \right)} \right|^2},$

()

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$ {\delta _1}(x,t) = 16A_{\rm{c}}^2\frac{{{\varGamma _1}}}{{{{\left( {{\varGamma _1} + {\varGamma _2}} \right)}^2}}}. $

(12)

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${\xi _1}(x, t) =\displaystyle\int_{ - \infty }^{ + \infty } {{\delta _1}} (x, t) = 8\pi {A_{\rm{c}}}/{\left( {1 + {t^2}A_{\rm{c}}^4} \right)^{1/2}}$

()

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Zai-Dong Li, Qi-Qi Guo. Rogue wave solution in ferromagnetic nanowires[J]. Acta Physica Sinica, 2020, 69(1): 017501-1

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