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12ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
Khusainova Galina Vladimirovna
Candidate of Science, associate professor of the Ural State
University of Architecture and Art, 620075 Russia, Ekaterinburg, Karl Liebknecht Street 23
E-mail: [email protected]
Khusainov Damir Zinnurovich
Candidate of Science, associate professor of the Ural State
University of Architecture and Art, 620075 Russia, Ekaterinburg, Karl Liebknecht Street 23
E-mail: [email protected]
DEGENERATE SOLITON IN FERROMAGNET.
ABSTRACT
The exact degenerate soliton solution of Landau-Lifshitz equation is constructed by procedure based on Hirota method. It has shown that the solution is described the bound state of two domain walls. The evolution of this pair result in dynamic growing domain formation.
Keywords: exact solution, soliton
Хусаинова Галина Владимировна
канд. физ.-мат. наук, доцент Уральского государственного архитектурно-художественного университета, 620075 Россия, Екатеринбург, ул. Карла Либкнехта, 23.
E-mail: [email protected] Хусаинов Дамир Зиннурович.
канд. физ.-мат. наук, доцент Уральского государственного архитектурно-художественного университета, 620075 Россия, Екатеринбург, ул. Карла Либкнехта, 23.
E-mail: [email protected]
ВЫРОЖДЕННЫЙ СОЛИТОН В ФЕРРОМАГНЕТИКЕ.
АННОТАЦИЯ
Получено точное вырожденное солитонное решение для уравнения Ландау-Лиф-шица с помощью процедуры, основанной на методе Хироты. Показано, что это решение описывает связанное состояние двух доменных границ. Эволюция этой пары приводит к образованию динамически растущего домена. Ключевые слова: точное решение, солитон
Let us consider the Landau-Lifshitz (L-L) equation:
S t = |S x S xx J+|s X JS J , (1)
where S _ (S1A,S3) , J=diag(J1 J ,J3 ) , Jl < J < J3.
Here S is a three-dimensional spin-vector with amplitude equal to unity. In the work [1] it is shown that eq.(1) are transformed into bilinear form:
(„* „ * \ f • f+g • g )=
(91* * *
iDt + Dx )f • g - a(1 + b)f • g - a(1 - b)f • g =
(iDt - Dx )(f * • f - g * • g )
0
, (4)
through the transformation
Si +iS2 =
-t-
2f • g
*
f • f + g • g
S* =
*
f • f - g • g
r- *
f • f + g • g
(5)
a = 1(J3 - J1 ) '
b =
J3 - J2 U - Ji
Here g=g(x,t), f=f(x,t) .
Expanding g,f in a power series in a parameter
(6)
3 5 g = £gi +s g3 + s g5 + ••••
f = 1 + S 2f2 +S4f4 + •••• ^
gives one the set of equations
(iDt + DX - a(1 + b))l •
(* * \
f2 • 1 +1 • f2 + g* • gi )
+ b) j1 • g 1 = a(1 - b)g1 , (8a)
, (86)
(iDt - DX )(f2* • 1 +1 • f2 - g *• g1 )= 0
0
, (8B)
S
It is known [2] that N-soliton solutions of eq.(1) are obtained by a choice of the starting
solution g1 in the form: N
g1 = S exp(^i) i=1 ,
where kiX ^^ + , N=1,2,..., and the soliton parameters ki and are restricted by the dispersion relation:
®2
2 =(k2 - 2ab)(2a - lq )
To obtained the simplest rational-exponential (RE solution) we must choose the starting solutiong1 (by analogy as it was been done in ref.[3]) as follows g1 = Aexp(X1) (9)
where for convenience we introduced the notations
A = œjx + 2k^2 - a(1 + b)]t - ikx + q X1 = k1x - ro1x + d + if
(k1 , Ю1 принимают действительные значения, а d, f и C1 - вещественные постоян-
gi,fi
recursively. As a result, we have
Hbie). The substitution of (9) into system (8) yields g(x,t) = Aexp(X1) (10) f(x,t) = 1 -a exp(2X1) , (11)
a = ^a(1 - b)
where 2 .
Thus we found the simplest RE solution of the L-L equation for the two-axis ferromagnet. Note, that RE solutions (or degenerate soliton solutions) correspond to multiple poles in the scattering data of the inverse scattering problem [4].
S
From (5) the analytical form 3 is given by
S = 1 --
2AA exp(x1 +x*)
[1 -aexp(2X1)]"[1 -aexp(2X*)] + AA exp(^1 +x*) (12) .
Also, we find value of deviation from equilibrium position for vector S:
S
л/sf + S
2
AA*eXl +Xl (1 - ae2xi )(1 - ae2Xl )
>2Xi
1/2
(1 - ae2xi )(1 - a e2xi ) + AA*eXl +Xl These solutions are shown in Fig.1and Fig.2.
(13).
2
Fig.l Solution S_(x,t) given in eq.(13) for t = 0, t = 10, t = 20. (ki=L26 ; raj =0,7 ;
a(l+b)=1.5 ; a(l-b)=0,5 ; d = f= ci = 0).
Thus, the obtained RE solution of eq.(1) describes magnetic soliton (degenerate soliton )
with the same ground state S;3 (x ^ ^1 out of localization range. We can interpret this soliton as bound state of two 1800-domain walls. The evolution of this pair shows that for large x, t we have the dynamic growing domain.
References
1. Косевич А.М., Иванов Б.А., Ковалев А.С. Нелинейные волны намагниченности. Динамические и топологические солитоны. - Киев: Наук. Думка, 1983 -192с.
2. Богдан М.М., Ковалев А.С. Точные многосолитонные решения одномерных уравнений Ландау - Лифшица для неизотропного ферромагнетика. // Письма в ЖЭТФ, -1980 -Т.31, Вып.8-С.453- 457.
3. Bezmaternih G.V. (Khusainova G.V.), Borisov A.B. Rational - Exponential Solutions of Nonlinear Equations. // Lett.Math.Physics, - 1989 -V.18- P.1- 8.
4. Poppe C. Construction of solutions of the sine - Gordon equation by means of Fredholm determinants//Physica D -1983 -Vol.9 -P.103 - 139.
