Movement of Self-Organizing Solitons in Ferromagnet

Precessing ball solitons (PBS) in a ferromagnet during the first order phase transition induced by a magnetic field directed along the axis of anisotropy, while the additional action of high-frequency field perpendicular to the main magnetic field, are analyzed. It is shown that the spatial motion of solitons, associated with thermal fluctuations in the crystal, does not destroy the equilibrium of self-organized PBS.


Introduction
Precessing ball solitons (PBS) that may occur at the first order phase transition in a uni-axis ferromagnet under the action of a magnetic field along the easy axis has been considered in several papers [1]- [11].The main characteristics of PBS have been investigated in articles [10] [11].In [12] it was shown that with the additional effect of high-frequency magnetic field perpendicular to the main field, there may be of self-organizing state of PBS, when the decrease of the energy associated with dissipation, fully offset by the influx of energy from the source of the high-frequency field.But in this case, only immobile PBS were considered, i.e. without three-dimensional motion.The question of the existence of equilibrium self-organizing states of moving PBS remained open.
In this paper, an analysis was conducted and it was shown that the existence of self-organizing state for PBS had a dynamic character, in which strict observance of the equilibrium was disrupted, but immediately the movement was damped, and thereby the equilibrium state of self-organized PBS restored.i.e. actually selforganizing state for PBS exists even at three-dimensional motion.

Analysis of PBS Movement
To analyze magnetic solitons in a ferromagnet at the first-order transition in the presence of periodic magnetic field, as in [12], we use the Landau-Lifshitz equation with a Gilbert form of dissipative term [13]: and corresponding expression for the density of thermodynamic potential (as in [12]) Here H z > 0, ⊥ H is the periodic field directed perpendicular to the Z-axis and to the main field H z , 1 0 K > , 0 κ > ; m is a non-dimensional vector of ferromagnetism equal in the absolute value to 1; x y m m im , in initial state : 1 z m = − .We consider the PBS in a flat plate perpendicular to the Z-axis, use the following dimensionless values: ; and note: The change of sign of the self-demagnetizing field for PBS, ( ) 2), in comparison with article [12], does not lead to a qualitative change in results, but is logically more correct.
If added periodic field is the Equation ( 1) can be present in form In the case if the movement of PBS is directed along Z-axis, i.e.
( ) , the expression for magnetic component is In this paper, the PBS is considered in a spherical coordinate system, when its beginning coincides with the center of the soliton: (In these equation, and ρ θ i i are the unit vectors of spherical coordinate system.)The ( 7) and ( 8) equations correspond to the expressions describing the time dependencies of the density parameter z m and energy e: ( ) (It does not show here the gradient of these expressions related to "domestic" parts of these parameters, which are vanish when integrated over the volume of the soliton.) We see that for unmoving PBS, i.e. if k z = 0 and d 0 d (see [12]).In this case For spherical solitons (PBS) at 0 z k ≡ , the (7) equation for equilibrium state takes the form: here (In ( 11):   , at 0 0.01 ω = , and corresponding dependency of parameter β (from [12]).

z k ≅
. Consider how parameters of z m and e of equilibrium PBS are changed at occurrence and decreasing of movement.We expand the expression ( 9) and (10) in powers of the quantities cos There expressions can be split to two parts, proportional to Here is important dependence on the θ .Excitation of PBS movement due to thermal fluctuations in a crystal are random and chaotic.Evolution of PBS can be presented in two stages.In the first stage, movement due to fluctuations of the kinetic energy arises.
In this case 0 k τ ∂ > ∂ and 0 k > .For example at 0 z k > , in correspondingly with ( 14) and (15), there is a de- formation of the soliton and redistribution of z m and e densities between the front and the back parts of PBS: at cos 0 θ > there are decreasing of z m but increasing of e-on the front part of PBS, and contrary, increasing of z m but decreasing of e-on the back part of PBS.Of course, strictly speaking, PBS ceases be in equilibrium in this stage.However further, in the second stage of process, restoration of the initial configuration of PBS is carried out, also in accordance with ( 14) and ( 15 Note that during the deformation of PBS due moving, and magnetic moment and energy without taking into account the kinetic energy (see [12]), do not change.

Angle Parameter as Invariant of Landau-Lifshitz Equation for Equilibrium PBS
Let us consider one more invariant of (4) equation, namely density of a momentum: ( ) ( ) Using ( 6) expression, we obtain for the density of momentum in spherical coordinate system: ( This expression refers to the case of moving precessing ball soliton at the presence of high-frequency field when the energy dissipation is almost completely offset by the influx of energy from the source of the highfrequency field.Note that if 0 Integrating the expression (21) on the values of the angle θ , we obtain the expression for total momentum of PBS: Differentiating (21) and using (9), we have where ( ) Note that (24) equation also satisfies all the conditions for equilibrium of PBS in self-organizing state, if d 0 d It is believed that similar to momentum (21), the value β ρ ∂ ∂ is invariant for the PBS, moving along the axis z: , , , cos sin 1 To evaluate the effect of motion on the PBS, we consider value β τ ∂ ∂ obtained from ( 9) and (12).We are using (9) where there is the most direct connection between z m and z k .From here, the change in the angular parameter near equilibrium is equal to This expression can be split into two parts: ( ) ( ) • quantity related with d d • quantity related with z k .Here values 0 β and 0 z m relate to initial equilibrium state, at 0 τ = .
We see that the changes of d d , i.e. on front part (fore-part) of soliton, the values β are increasing, but decreasing with the iden- tical value on back part of soliton, i.e. for condition π 2 π θ < < .There is contrary change of z m (see the sign in ( 14)).Thus, we have: Moreover, integral value ( ) (here 0m p is amplitude of PBS) in equilibrium state of PBS does not change.Therefore, self-organizing state of PBS does not disappear, but continues to live, in spite of the thermal vibra-

Conclusions
The analysis of moving precessing ball solitons, arising at the first-order phase transition in ferromagnet at the combined action of main field, causes the transition, and transverse high-frequency magnetic field.
It is shown that the movement caused by thermal fluctuations in a crystal causes a small change in the parameters of solitons and thereby to a very small distortion of equilibrium self-organizing state of PBS.However, this distortion is eliminated very quickly and verily equilibrium state is restored.In practice, equilibrium selforganizing PBS, when dissipative losses of energy are compensated by the influx of energy from the highfrequency field, continues to exist at mechanical motion of PBS inside the crystal.
equilibrium state of PBS disappears.Let us consider this situation quantitatively.Kinetic energy of PBS

Figure 1 andFigure 1 .Figure 2 .
Figure 1.Frequency dependencies of energy and amplitude sm p of PBS for 0.998 h = , first degree of decomposition, and believing that at the initial time, at 0 τ = , all the conditions for equilibrium are fulfilled, we obtain the expression for the change of parameters directly related to the pres- e. accurate to linear values by κ and h ⊥ , equilibrium state of PBS is restored.

(
initial value.Can be estimated the time t ∆ during which this change is carried out at return to the initial state.In accordance with the expression 2

.
Note that in this case initial value is 0 2.5 β ≅  .The time required to return from ( )