A SMALL AMPLITUDE SOLITON AS A POINT ATTRACTOR OF THE LANDAU-LIFSHITZ EQUATION

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A SMALL AMPLITUDE SOLITON AS A POINT

ATTRACTOR OF THE LANDAU-LIFSHITZ

EQUATION

M. Grodecka, A. Sukiennicki

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, d h m b r e 1988

A

SMALL AMPLITUDE SOLITON AS

A

POINT ATTRACTOR OF THE

LANDAU-LIFSHITZ EQUATION

M. Grodecka and A. Sukiennicki

Institute of Physics, Warsaw Technical University, 00-662 Warsaw, Poland

Abstract. - It is shown that, if the driving force and damping are sufficiently small, a small amplitude phase-locked soliton is a fixed point attractor of the Landau-Lifshitz equation of motion which describes a uniaxial one-dimensional ferromagnet.

Recent studies have shown that a soliton can be a low dimensional attractor of the driven damped non- linear Schrijdinger equation [I]. In particular, if an oscillatory driving force is small enough, the first or- der perturbation theory gives a phase-locked solitan corresponding to a fixed point attractor [2]. If how- ever the driving force becomes larger, when the per- turbation theory breaks down and numerical analysis is necessary

-

a sequence of period doubling bifurca- tions appears and leads to a chaotic soliton as a strange attractor [I].

Recent studies have shown also that some other driven damped soliton systems, e.g. the driven damped sine-Gordon systems, have a solution in the form of the chaotic soliton, too [3]. One can expect that the phenomenon is more general and concerns a wide variety of soliton systems if they are damped and driven.

In this paper we investigate an attracting, small arn- plitude soliton in a uniaxial one-dimensional ferromag- netic system described by the Landau-Lifshitz (LL) equation of motion. The LL equation is known to de- scribe phenomenologically a dynamic behaviour of the magnetization vector in the ferromagnetically coupled systems of spins, e.g. in ferromagnetic resonance or in domain wall dynamics. Some years ago Sklyanin showed that the LL equation with no damping and no drive field is completely integrable [4]. Soliton prop- erties of solutions of the LL equation in such a case have been also investigated [ 5 ] . Here we show that with both damping and drive force included, but small enough for the application of perturbation theory, the small amplitude phase-locked soliton of the LL equa- tion becomes the fixed point attractor of this equation. Let us consider a uniaxial one-dimensional ferromag- net in the classical continuum approximation. The LL1 equation of motion has the form

cal damping constant. Here,

He

=. - S f ISM repre- sents the effective field acting on the magnetization vector and f - the free energy density. For the one- dimensional ferromagnet with an anisotropy of the easy axis type the energy density takes the form

where a! - the exchange parameter, ,B

-

the uniaxial

anisotropy constant (@

>

0) and

He,*

-

the external magnetic field taken as a sum of the constant bias field applied along the easy axis and the small amplitude ac circular driving field with the frequency w

We introduce the complex variable (see e.g. [6, 71)

thus MI = JM:

-

I$12.

In a small amplitude and long-wave approximation (i.e. for

[$I2

<<

M:) equa-

tion ( 1 ) takes the form (see [6, 71):

where p~ - Bohr magneton, T = Mo/XH, h - Planck constant. If we introduce the transformation of vari- ables [7]

$ = 2Mo exp [i ( 1

+

h) tlao] u (yl, t') ( 6 )

where

where M is the magnetization vector treated as a con- _{equation ( 5 ) can be written as} tinuous function of coordinates, Mo = /MI = const

- the length of the magnetization vector, y - the

_{- -}

_{au - 2 2 = -}

_-

_,

₍₈₎ giromagnetic ratio (y

>

0)

,

X - the phenomenologi-

at'

ayI2

C8 - 1592 JOURNAL DE PHYSIQUE

where

When

r

= 0 and E = 0, equation (8) is the nonlinear Schrodinger equation and has the standard l-soliton solution with a soliton amplitude and a soliton velocity as arbitrary parameters.

If I?

#

0 and E

#

0, equation (8) represents the per- turbed nonlinear Schrodinger equation with the damp- ing I? and the drive force E as perturbations. In the case of small I' and E , the effect of the damping is that the soliton amplitude has a tendency to decrease while the effect of the ac driven field can work either with or against the damping, depending on the instan- taneous phase difference between the soliton and the driving term. In such a situation it is possible for soli- ton to "phase-lock7' onto the drive field by adjusting the frequency [2]. Such a synchronized soliton has a form

u (y', t') =

f i

sech

(fig1)

exp (iwotl

-

ixo

-

i- 2

"

or, in view of transformation (6),

$ (9, t) = 2Mo (1

+

h

-

I

(11) 2

r

sin

xo

=

- -

_"

6.

E (12) where

Here xo has the meaning of the phase-locked difference of soliton and drive phases. This solution may exist only if 2 r E

>

-

(1

+

h

-

aow)l/' 7r (13) and if w

<

(1

+

h) lao. ₍₁₄₎ If we analyse this solution in the phase space (q, X) of the soliton parameters ( r ) - the soliton amplitude,

x

- the difference of the soliton and drive phases), we find that our soliton corresponds to the fixed point attrac-

1

tor of the equation (8) with the coordinates q =

-

6

2

x

=

xo

+

272". Our result shows that the small ampli- tude soliton can be a fixed point attractor of the LL equation in the system considered here. The meaning of this result is that in our model only soliton can gain energy in a resonant fashion from positive frequency forcing ac field and survive against dissipation (though its velocity vanishes due to dissipation). For that the frequency w of the drive field should be only slightly smaller than the upper limit in the inequality (14), so that both inequalities (13) and (14) are fulfilled. (14) tells that the allowed frequencies of the drive field are smaller than the homogeneous resonance frequency

(1

+

h) lao. This reflects the fact that the own frequen- cies of the soliton in our small amplitude model are smaller than this resonant value (see [6]). The model presented here gives a good description of the dynam- ics of the considered system in a low-temperaure re- gion, where solitons seem to be dominant excitations as they are energetically more favorable than magnons (see [91).

Another question is what happens to this fixed point attractor when E becomes larger, where the pertur- bation theory breaks down. The numerical analysis performed for equation (8) in reference [I] shows that if I? and w are fixed while E increases and becomes greater than a critical value the fixed point bifurcates in the phase space to a periodic cycle corresponding t o an amplitude oscillating soliton and the further in- crease of E causes a sequence of further period dou- bling bifurcations. For E larger than the final criti- cal point the strange attractor appears corresponding to a chaotic soliton. Unfortunately, in our case, one can expect that with E increasing the soliton ampli- tude is not longer small, thus our derivation of driven damped nonlinear Schrodinger equation breaks down and a more sophisticated nonlinear equation appears [8]. -413 analysis in this case is under preparation. Acknowledgment

This work was supported by the Institute for Low Temperatures and Sructure Researches of the Polish -4cademy of Sciences, Wroclaw, project CPBP 01.12.

[I] Bekki, N., Nozaki, K., Dynamical Problems in Soliton Systems, Ed. S. Takeno, Springer Ser. Synergetics, 30 (1985) 268.

[2] Kaup, D. J., Newell, A. C., Phys. Rev. B 18

(1978) 5162.

[3] Bishop, A. R., Fesser, K., Lomdahl, P. S., Kerr, W. C., Wiliams, M. B., Trullinger, S. E., Phys. Rev. Lett. 50 (1983) 1095.

[4] Sklyanin, E. K., On complete integrability of the Landau-Lifshitz equation, preprint of LOMI, E3, Leningrad (1979) p. 32.

[5] Biegala, L., Borovik, A. E., J. Magn. Magn. Mater. 21 (1980) 269.

[6] Kosevich, A. M., Ivanov, B. A., Kovalev, A. S., Nonlinear Waves of Magnetization. Dynamical and Topological Solitons, Naukova Dumka, Kiev (1983) in Russian.

[7] Blaszak, M., Proc. V. Conf. Phys. of Magn., Poznan (1987) p. 454.

[8] Gill, T . L., Zachary, W. W., J. Appl. Phys. 61 (1987) 4130.