3 Solution of Eq. (5) with initial condition η = 0 is where ω$_{c}$ = αγh 0 1+ α 2 is the critical frequency that separates two modes: η approaches exponentially a fixed value sin − $^{1}$( − ω/ω$_{c}$ ) in a time scale of 1 / √ ω 2 c − ω 2 for ω < ω$_{c}$ . This is the fully synchronized motion. Since T$_{p}$ = ( α + 1 α ) ω is always larger than T$_{d}$ = αω sin θ , the HH DW propagates to the left as mentioned early. η keeps rotating around z-axis with a period of 2 π/ √ ω 2 − ω 2 c and a variable velocity for ω > ω$_{c}$ , corresponding to an in- complete synchronization. For ω < ω$_{c}$ , the steady DW propagation velocity is given by Eq. (6) when ˙ η = 0 is reached, GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s32> GLYPH<s99> GLYPH<s47> GLYPH<s104> GLYPH<s48> GLYPH<s86> GLYPH<s101> GLYPH<s108> GLYPH<s111> GLYPH<s99> GLYPH<s105> GLYPH<s116>GLYPH<s121> GLYPH<s32>GLYPH<s32>GLYPH<s40> GLYPH<s65> GLYPH<s47> GLYPH<s48> GLYPH<s41> GLYPH<s49>GLYPH<s47>GLYPH<s50> GLYPH<s41> GLYPH<s70>GLYPH<s114>GLYPH<s101>GLYPH<s113>GLYPH<s117>GLYPH<s101>GLYPH<s110>GLYPH<s99>GLYPH<s121>GLYPH<s32>GLYPH<s40> GLYPH<s77> GLYPH<s115> GLYPH<s41> cot η 2 =   $^{−}$$^{ω}$$^{c}$ ω − √ ( ω c ω ) 2 − 1 coth √ ω 2 $_{$_{c}$−}$ω 2 2 t, ω < ω$_{c}$ − ω c ω − 1 − ( ω c ω ) $^{2}$cot √ ω $^{2}$$_{−}$ω2 c 2 t, ω > ω$_{c}$ (7) υ = ˙ Q ( t → ∞ ) = − ω Δ α (1 − sin θ 0 ) . (8) ¯ υ = − ω − √ ω 2 − ω 2 c α (1 − sin θ 0 ) Δ , (9) Similar to the low current velocity, DW propagation ve- locity is linear in ω as shown by the solid lines in Fig. 2 for low frequency. For ω > ω$_{c}$, the precession velocity of DW plane change with time. According to Eq. (6), DW velocity will also change with time. Since the aver- age angular velocity of η is Ω = √ ω 2 − ω 2 $_{c}$, the averaged DW velocity is find that the DW position line is a straight line at low frequency, and it becomes an oscillation curve at high frequency. FIG. 2: DW velocity versus CPMF frequency with α = 0 . 1 and K = 8. Symbols are numerical data for field amplitude h 0 = 3 (squares), 1 (circles), and 0 . 5 (stars). Solid curves are the theoretical results of Eqs. (8) and (9). Inset: Ratio ω c /γh 0 versus α . Solid curve is the theoretical formulation α 1+ α 2 as we get. The symbols in Fig. 2 are numerical data for K = 8, α = 0 . 1 and various h 0 = 3 , 1 and 0 . 5 . It is clear that h 0 affects DW velocity. The larger h 0 is, the higher the velocity will be, as given by Eqs. (8) and (9) because the DW width is increased by a factor 1 / (1 − sin θ 0 ). The agreement between Eq. (8) (straight lines in Fig. 2 with- out any fitting parameters ) and numerical results for low- frequency are good. Eq. (9) (curves in Fig. 2) captures qualitatively the decay feature for large frequency, but systematically smaller than the numerical values. This should not be a surprise since Eqs. (5) and (6) are de- rived from rigid DW assumption that, strictly speaking, does not hold for ω > ω$_{c}$ . To further test our assump- tions for Eqs. (5) and (6), we compare numerical values of ω$_{c}$/ ( γh 0 ) with theoretical prediction of α/ (1+ α $^{2}$) (curve in the inset of Fig. 2). The symbols (squares, circles, and stars for h 0 = 3 , 1 , and 0 . 5, respectively) in the inset of Fig. 2 are numerical data for various α that compares well with the theoretical prediction. monotonically decreasing with ω shown by the solid curves in Fig. 2 for large ω . The results can also be un- derstood from the energy consideration[4]. For ω < ω$_{c}$ , the non-conservative field does not do any work to the system for a rigid DW propagation, and the energy dissi- pation must be compensated by the energy released from the DW propagation[4]. According to Ref. [4], the DW velocity is proportional to DW width and the axial field which is the frequency in the present case. For ω > ω$_{c}$ , DW plane precesses around the wire while it propagates along the wire. The non-conservative field pumps energy to the system so that the system needs to release less en- ergy, corresponding to a low DW velocity shown in Eq. (9). To test validity of our analytical results, we solve Eq. (1) numerically for a uniaxial 1D wire of magnetic anisotropy U = − 1 $_{2}$K cos 2 θ . We first scale time, length, energy density and field amplitude in units of ( γM$_{s}$ ) − 1 , √ A/µ 0 M 2 $_{s}$, µ 0 M 2 s and M$_{s}$, respectively, so that velocity is in the unit of γ √ A/µ 0 . We then adopt a standard arithmetic Method-of-Lines to discretize space with an adaptive time-step control. The length of the wire is 100 and the mesh size is 0.2. The density plot of m$_{z}$ in z − t plane determines the DW position. The DW velocity is extracted from the slope of the DW position line. We It is predicted and confirmed[21, 22] that a DW motion with its plane precessing around wire axis can generate an electromotive potential of ℏ e dφ dt between the two sides of the DW, where dφ dt is the precession velocity of the DW. However, a rigid body propagation along the wire will not generate an electromotive force. A naive applica- tion of this theory to our DW motion seems lead to zero electromotive force in the rotation frame and non-zero electromotive force in the laboratory frame. This obvi-