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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.

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 [

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 self- organizing state for PBS exists even at three-dimensional motion.

To analyze magnetic solitons in a ferromagnet at the first-order transition in the presence of periodic magnetic field, as in [

and corresponding expression for the density of thermodynamic potential (as in [

Here H_{z} > 0, _{z},

We consider the PBS in a flat plate perpendicular to the Z-axis, use the following dimensionless values:

Equation (2), in comparison with article [

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.

In this paper, the PBS is considered in a spherical coordinate system, when its beginning coincides with the center of the soliton:

In this case

The phase of precession of magnetic moments for localized excitation depends on a radius, i.e.

The (7) and (8) equations correspond to the expressions describing the time dependencies of the density parameter

(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

For spherical solitons (PBS) at

here

(In (11): for, and for.)

However if

tatively. Kinetic energy of PBS

where

of movement. We expand the expression (9) and (10) in powers of the quantities

leave only the first degree of decomposition, and believing that at the initial time, at

ence of

There expressions can be split to two parts, proportional to

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

formation of the soliton and redistribution of

Note that during the deformation of PBS due moving, and magnetic moment

and energy

without taking into account the kinetic energy (see [

For

expression

to the initial state. In accordance with the expression

For

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 high- frequency field. Note that if

Integrating the expression (21) on the values of the angle

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

It is believed that similar to momentum (21), the value

To evaluate the effect of motion on the PBS, we consider value

using (9) where there is the most direct connection between

This expression can be split into two parts:

• quantity related with

• quantity related with

We see that the changes of

Note that

Moreover, integral value

(here

Therefore, self-organizing state of PBS does not disappear, but continues to live, in spite of the thermal vibrations in the crystal.

Like to consideration for

at

For

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 self- organizing PBS, when dissipative losses of energy are compensated by the influx of energy from the high- frequency field, continues to exist at mechanical motion of PBS inside the crystal.

V. V.Nietz, (2015) Movement of Self-Organizing Solitons in Ferromagnet. Applied Mathematics,06,1747-1754. doi: 10.4236/am.2015.610155