Conservation laws, modulation instability and rogue waves for the localized magnetization with spin torque

Under investigation in this paper is an integrable equation which can describe the localized magnetization with spin torque under the long-wavelength approximation. In order to obtain the rogue wave solutions, a new Lax pair is derived. Infinitely-many conservation laws are also constructed. Based on the generalized Darboux transformation, the first-, second- and third-order rogue wave solutions are derived. Property of the rogue waves are analyzed and influence of parameters α, β and separating function f(ϵ) on the rogue waves and spatial-temporal structures are also discussed (the meaning of α and β can be found in the paper). For the case of f(ϵ)=0, the modulus of the kth-order rogue wave (k=1,2,3) is irrelevant to parameter α. Parameter β influences the spatial-temporal range where the rogue wave appears. Spatial-temporal range enlarges with the increase of β. In addition, β also produces a skew angle and the skew angle rotates in the counter clockwise direction with the increase of β. f(ϵ) influences the spatial-temporal structures of the second- and third-order rouge waves. If f(ϵ) ≠ 0, the second-order rogue wave will split into three single first-order rouge waves and the triangular pattern can be formed, while the third-order rogue wave will split into six ones and the triangular pattern and pentagon pattern can be formed. The linear stability analysis is carried out, which shows that the modulation instability process is influenced by the amplitude of the harmonic wave and the wave number.