Breathers and rogue waves excited by all-magnonic spin-transfer torque

In terms of Darboux transformation we investigate the dynamic process of spin wave passing through a magnetic soliton. It causes nonlinear excitations, such as Akhmediev breathers solution and Kuznetsov-Ma soliton. The former case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces. The spatial separation of adjacent magnetic petals increases rapidly, while one valley splits into two and the amplitude of valley increases gradually with the increasing amplitude of spin wave. The other case shows a localized process of the spin-wave background. In the limit case, we get rogue waves and clarify its formation mechanism.

Figure 1

Evolution of Akhmediev breathers for magnetization $\mathbf{m}=\left(m_1,m_2,m_3\right)$ in Eqs. (2) and (3). The component $m_3$ takes the spatial periodic distribution, which is characterized by two peaks and two valleys in each unit distribution. Parameters are given as follows: $A_s=0.8,k_s=1,\nu =k_s\sqrt{1-A_s^2}$, and $\mu =0.64$.

Figure 2

The formation of magnetic petal in the component $m_3$, shown in (a)–(d). As the spin wave amplitude $A_s$ increases, the connection line of two peaks in the component $m_3$ rotates clockwise and the two peaks move with the opposite direction. The one valley splits into two and the distance of two valleys increases with the increasing $A_s$. Parameters are given as follows: $k_s=1,\nu =k_s\sqrt{1-A_s^2}$, and $\mu =0.8A_s{k}_s$. The parameters $A_s$ is given by (a) $A_s=0.7$, (b) $A_s=0.9$, (c) $A_s=0.98$, and (d) $A_s=1$, respectively.

Figure 3

Evolution of Kuznetsov-Ma soliton for magnetization $\mathbf{m}=\left(m_1,m_2,m_3\right)$ in Eqs. (2) and (5). This soliton is spatially aperiodic and temporally periodic, while the component $m_3$ shows two peaks and two valleys in each periodic distribution. Parameters are given as follows: $A_s=1,k_s=1$, $\nu =k_s\sqrt{1-A_s^2}$, and $\mu =1.3$.

Figure 4

The asymptotic processes of the magnetic component $m_3$ in the limit processes $\mu \to A_s{k}_s$ and $\nu =k_s\sqrt{1-A_s^2}$ in Eq. (2). As $\mu \to A_s{k}_s$ the spatiotemporal separation between adjacent magnetic petal increases gradually, as shown in (a)–(f), where the parameters are $A_s=0.8$, $k_s=1$, (a) $\mu =0.9$, (b) $\mu =0.98$, (c) $\mu =0.9999$, (d) $\mu =1.2$, (e) $\mu =1.1$, and (f) $\mu =1.0001$, respectively.

Figure 5

The graphical evolution of rogue waves for the magnetization $\mathbf{m}=\left(m_1,m_2,m_3\right)$ in Eqs. (2) and (6), i.e., bright rogue waves (a)–(c) and dark rogue waves (d)–(f). The parameters are $A_s=\sqrt{3}/2,k_s=1$, $\nu =k_s\sqrt{1-A_s^2}$, and $\mu =\pm \sqrt{3}/2$ with the sign $\pm$ corresponding to the bright and dark rogue waves, respectively.