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Solitons in a ferrimagnet
0038-1098/82/390411-02503.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 44, No. 3, pp. 411-412, 1982. Printed in Great Britain. SOLITONS IN A FERRIMAGNET
A.B. Borisov, V.V. Kiseliev and G.G. Talutz Institute of Physics of Metals, Ural Science Center GSP-170, Sverdlovsk, U.S.S.R.
(Received 12May 1982 by V.M.Agranovich) Exact N-soliton solutions of nonlinear Landau-Lifschitz equations are obtained by means of the modified form of the inverse scattering method ("dressing method") for the case of an isotropic ferrimagnet with two nonequivalent sublattices.
2 + where* 71,72 are gyromagnetic ratios, c = [~1 (Mdl ~. 2 2 M22) ~'yl'Y2(O~l + ot2 -- 20t3)] 1/2, ~ - [2(M01 --Mo~) (')"1 "-[-"~2)(M021 31-M~2) -1 .-[- "Y2- "/1] {~ ["~1"~2(0~1 "-~ 0~220~3)]-1} 1/2 2. In order to solve equation (3) we use the modified form of the inverse scattering method, "dressing" method [8, 9]. We found that the equation (3) is equivalent to the compatibility condition
1. Macroscopically the state of a ferrimagnet is described by sublattice magnetization vectors Ml(X, t), M2(x, t) as functions of coordinates and time. At sufficiently low temperatures lengths of M~, M2 do not change, i.e. M~ = Mo i (i = 1,2). We restrict ourselves with excitations propagating along the axis n = (1,0, 0) and consider a simple physical model (see, e.g. [7]) in which the magnetic energy density of a ferrimagnet is given by
W = f dx{f(Ml'M2)
+
c-lOtu(X)--
OxX(X) = U(X)X(X),
c-lotx(X)
M1 + M2 (2Mg 1 q._ 2Mg2)I/2,
(1)
Mt - |
3c-1~t I
= 0,
2
2
=
[1, Oxl]k 1--X 2
X2)l /
V = --87 3
(2)
(3)
m = [2(Mo21-bM22)-1]1/216(')'1 + "/2)]-111, ~)t1] 1
(6')
(1 -X2) 2 )r~r3,
These equations may be simplified supposing m 2 ~ 12 1. Then we obtain [1, c-20~1 -- 32xl1 +
(6)
= v(X)x(X).
X[I, c-l atl]k v
M2
(2M21 + 2M22)1/2.
(5)
Matrices U(X), V(X) are rational functions of the spectral parameter X and have the following form
Here 6 (6 > 0) is the uniform exchange constant; cq, a2, a3 are non-uniform exchange constants. It is convenient to write Landau-Lifschitz equations for vectors M1, M2 in terms of vectors of ferromagnetism m and antiferromagnetism I:
m
0xV(X) + [U(X), V(X)] = 0
for the system of linear differential equations for 2 x 2 matrix ×(X):
1[o~1(~xM1)2+ a2(OxM2) 2]
-t- o~3(0xM 1" OxM2)}.
(4)
-- "/'2(0~2 -- Ot3)][~('y 1 4- "/2)]-1[1, [|, ~211],
NONLINEAR MAGNETIZATION waves in magnetics with several sublattices possess a number of specific features [ 1-3] and in comparison with a ferromagnet [4-6] are investigated incompletely. In the present paper we study nonlinear localized waves (solitons) in the isotropic ferrimagnet with two non-equivalent sublattices.
(7)
[1, C-10tl]k X[I, a~l]~ 1 --)t 2 + 1 --X 2
2/3X21h t
where % = (i/2) of and ok are Pauli matrices. From system (7) it follows that X(X) satisfies the restriction: X(X) = - - 4 X ( X = 0)T1X*(~-~)T 1.
(8)
2
+ ~(Mo~ - Mo~)(Mo, + M , ~ ) - ~l - - ½[-,,~(o,~ - - o~3)
* If one sets al = a2, Mol = M02, 7~ = 72 then equations (3) and (4) describes the nonlinear waves in an antiferromagnet [3]. 411
The "dressing" method reduces the solution of the spectral problem (6), ( 6 ' ) and (7) to that of a HilbertRiemann problem and allows one to get analytic Nsoliton solutions of equation (3) via a purely algebraic
412
approach. According to [8, 9] we assume X = ~Xo, where Xo(X) is the particular solution of equations (6) and ( 6 ' ) and ~(X) is found from the matrix HilbertRiemann problem. Using the method [8, 9] to construct matrices qz(x) and relations (8) we obtain N-soliton solutions of equation (3). The final result has the form lkr ~ = rs'Iz(X = 0)Xo(X = 0), (nj)a(n*)~ ~ in detD
2N
* a 0 ( x ) = fiat~ + Y~
X--X~
i,./= 1
d~j
'
(9)
where D is the 2N x 2N matrix with elements dij =
(n*nj)(Xi -- X~')- 1, nk = ×o(X = X~)ck,nk+N = 2×o(X = 0)rln~, X~+N = X~(k = 1, 2 . . . . . N),
(10)
where Xh are complex numbers and ek are arbitrary constant complex vectors. 3. Let us choose the particular solution of equations (6) and (6') in the form Xo(X) = exp
{
2/3Xra } (1 --X2) 2 [ 2 X t + ( l + X 2 ) x l "
order to understand the character of this solution we consider the particular case when v equals zero (e = 7r/2). In this case the vector I precesses uniformly with frequency ~ =/3c/cosh2p around the axis x3. The nonuniformity is concentrated in the finite region. In the center of the soliton vector I approaches the equilibrium value (0, 0, -- 1). When the parameter/3 decreases (or tO[ increases) the soliton width cosh 2p/(/3 sinh p) increases and the frequency co decreases. Sublattice magnetizations M1, M2 represent localized waves which qualitatively resemble the soliton for an isotropic ferromagnet [5] if we neglect the third term in equation (4). When the group velocity v 4= 0 the soliton is dynamic and l 3 varies within limits 1 -- 2 sin 2 e ~< 13 ~< 1. At x ~-+ ~ vector I approaches the value (0, 0, 1). N-soliton solutions (9) describe the process of elastic scattering of the mentioned excitations. We proved that the asymptotic superposition principle holds. The authors are grateful to A.M. Kosevich, B.A. Ivanov, V.P. Silin for helpful and stimulating discussions.
Acknowledgements
(11)
REFERENCES
Then the one-soliton (N = 1) solution of equation (3) describes the localized magnetisation wave
1.
2 la = 1 - - - r '
2.
l 1 + il 2 =
exp {-- iS} r sin e sinh p
x [e y sin (e + ip) + e -y sin (e --iP)], y = In cA --13Im8,
x cosh (/9 + ie)
sinh 2 (p + ie) '
sinh 2 (p + ie) cosh 2y [-
sin 2 e
3. 4.
C2
ct
--
(12)
S = - - a r g cA+/3Re6,
C2
r
Vol. 44, No. 3
SOLITONS IN A FERRIMAGNET
sinh 2 y sinh 2 p _
5.
6.
_
7.
moving with the constant group velocity 8. cosh (p + i e) ]-1 V = c Im~~e)] Im[sinh(p+ie)]-2. 9.
Here P, e 4= 0 are arbitrary real constants (X~ = e°+te). In
A. Hubert, Theorie der Domdnenwiinde in geordneten Medien. Springer-Verlag. BerlinHeidelberg-New York (1974). A.K. Zvezdin, Zh. Eksp. Teor. Fiz. Pisma 29,605 (1979); V.G. Bar'yakhtar, B.A. Ivanov & A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 78, 1509 (1980). I.V. Bar'yakhtar & B.A. Ivanov, Fizika nizkikh temperatur 5, 352 (1979). A.M. Kosevich, B.A. Ivanov & A.S. Kovalev, Physica 3D, 1,363 (198 I); A.M. Kosevich, Fizika metallov i metallovedenie 53,420 (1982). M. Lakshmanan, Phys. Lett. 61A, 53 (1977); L.A. Takhtajan, Phys. Lett. 64A, 235 (1977). E.K. Sklyanin, Preprint LOMI E-3-1979, Leningrad ( 1979); A.E. Borovik & V.N. Robuk, Teor. i. Mat. Fiz. 46,371 (1981). E.A. Turov, Fizicheskie svoistva magnitouporyadochennikh kristallov. Izd-vo AN USSR, Moskva (1963). V.E. Zakharov & A.B. Shabat, Funct. Anal, Appl. 13, 13 (1979);V.E. Zakharov & A.V. Mikhailov, Soy. Phys. J E T P 47, 1017 (1979). Teorya solitonov (pod redaktsiey S.P. Noviokova). Nauka, Moskva (1980).
Solid State Communications, Vol. 44, No. 3, pp. 411-412, 1982. Printed in Great Britain. SOLITONS IN A FERRIMAGNET
A.B. Borisov, V.V. Kiseliev and G.G. Talutz Institute of Physics of Metals, Ural Science Center GSP-170, Sverdlovsk, U.S.S.R.
(Received 12May 1982 by V.M.Agranovich) Exact N-soliton solutions of nonlinear Landau-Lifschitz equations are obtained by means of the modified form of the inverse scattering method ("dressing method") for the case of an isotropic ferrimagnet with two nonequivalent sublattices.
2 + where* 71,72 are gyromagnetic ratios, c = [~1 (Mdl ~. 2 2 M22) ~'yl'Y2(O~l + ot2 -- 20t3)] 1/2, ~ - [2(M01 --Mo~) (')"1 "-[-"~2)(M021 31-M~2) -1 .-[- "Y2- "/1] {~ ["~1"~2(0~1 "-~ 0~220~3)]-1} 1/2 2. In order to solve equation (3) we use the modified form of the inverse scattering method, "dressing" method [8, 9]. We found that the equation (3) is equivalent to the compatibility condition
1. Macroscopically the state of a ferrimagnet is described by sublattice magnetization vectors Ml(X, t), M2(x, t) as functions of coordinates and time. At sufficiently low temperatures lengths of M~, M2 do not change, i.e. M~ = Mo i (i = 1,2). We restrict ourselves with excitations propagating along the axis n = (1,0, 0) and consider a simple physical model (see, e.g. [7]) in which the magnetic energy density of a ferrimagnet is given by
W = f dx{f(Ml'M2)
+
c-lOtu(X)--
OxX(X) = U(X)X(X),
c-lotx(X)
M1 + M2 (2Mg 1 q._ 2Mg2)I/2,
(1)
Mt - |
3c-1~t I
= 0,
2
2
=
[1, Oxl]k 1--X 2
X2)l /
V = --87 3
(2)
(3)
m = [2(Mo21-bM22)-1]1/216(')'1 + "/2)]-111, ~)t1] 1
(6')
(1 -X2) 2 )r~r3,
These equations may be simplified supposing m 2 ~ 12 1. Then we obtain [1, c-20~1 -- 32xl1 +
(6)
= v(X)x(X).
X[I, c-l atl]k v
M2
(2M21 + 2M22)1/2.
(5)
Matrices U(X), V(X) are rational functions of the spectral parameter X and have the following form
Here 6 (6 > 0) is the uniform exchange constant; cq, a2, a3 are non-uniform exchange constants. It is convenient to write Landau-Lifschitz equations for vectors M1, M2 in terms of vectors of ferromagnetism m and antiferromagnetism I:
m
0xV(X) + [U(X), V(X)] = 0
for the system of linear differential equations for 2 x 2 matrix ×(X):
1[o~1(~xM1)2+ a2(OxM2) 2]
-t- o~3(0xM 1" OxM2)}.
(4)
-- "/'2(0~2 -- Ot3)][~('y 1 4- "/2)]-1[1, [|, ~211],
NONLINEAR MAGNETIZATION waves in magnetics with several sublattices possess a number of specific features [ 1-3] and in comparison with a ferromagnet [4-6] are investigated incompletely. In the present paper we study nonlinear localized waves (solitons) in the isotropic ferrimagnet with two non-equivalent sublattices.
(7)
[1, C-10tl]k X[I, a~l]~ 1 --)t 2 + 1 --X 2
2/3X21h t
where % = (i/2) of and ok are Pauli matrices. From system (7) it follows that X(X) satisfies the restriction: X(X) = - - 4 X ( X = 0)T1X*(~-~)T 1.
(8)
2
+ ~(Mo~ - Mo~)(Mo, + M , ~ ) - ~l - - ½[-,,~(o,~ - - o~3)
* If one sets al = a2, Mol = M02, 7~ = 72 then equations (3) and (4) describes the nonlinear waves in an antiferromagnet [3]. 411
The "dressing" method reduces the solution of the spectral problem (6), ( 6 ' ) and (7) to that of a HilbertRiemann problem and allows one to get analytic Nsoliton solutions of equation (3) via a purely algebraic
412
approach. According to [8, 9] we assume X = ~Xo, where Xo(X) is the particular solution of equations (6) and ( 6 ' ) and ~(X) is found from the matrix HilbertRiemann problem. Using the method [8, 9] to construct matrices qz(x) and relations (8) we obtain N-soliton solutions of equation (3). The final result has the form lkr ~ = rs'Iz(X = 0)Xo(X = 0), (nj)a(n*)~ ~ in detD
2N
* a 0 ( x ) = fiat~ + Y~
X--X~
i,./= 1
d~j
'
(9)
where D is the 2N x 2N matrix with elements dij =
(n*nj)(Xi -- X~')- 1, nk = ×o(X = X~)ck,nk+N = 2×o(X = 0)rln~, X~+N = X~(k = 1, 2 . . . . . N),
(10)
where Xh are complex numbers and ek are arbitrary constant complex vectors. 3. Let us choose the particular solution of equations (6) and (6') in the form Xo(X) = exp
{
2/3Xra } (1 --X2) 2 [ 2 X t + ( l + X 2 ) x l "
order to understand the character of this solution we consider the particular case when v equals zero (e = 7r/2). In this case the vector I precesses uniformly with frequency ~ =/3c/cosh2p around the axis x3. The nonuniformity is concentrated in the finite region. In the center of the soliton vector I approaches the equilibrium value (0, 0, -- 1). When the parameter/3 decreases (or tO[ increases) the soliton width cosh 2p/(/3 sinh p) increases and the frequency co decreases. Sublattice magnetizations M1, M2 represent localized waves which qualitatively resemble the soliton for an isotropic ferromagnet [5] if we neglect the third term in equation (4). When the group velocity v 4= 0 the soliton is dynamic and l 3 varies within limits 1 -- 2 sin 2 e ~< 13 ~< 1. At x ~-+ ~ vector I approaches the value (0, 0, 1). N-soliton solutions (9) describe the process of elastic scattering of the mentioned excitations. We proved that the asymptotic superposition principle holds. The authors are grateful to A.M. Kosevich, B.A. Ivanov, V.P. Silin for helpful and stimulating discussions.
Acknowledgements
(11)
REFERENCES
Then the one-soliton (N = 1) solution of equation (3) describes the localized magnetisation wave
1.
2 la = 1 - - - r '
2.
l 1 + il 2 =
exp {-- iS} r sin e sinh p
x [e y sin (e + ip) + e -y sin (e --iP)], y = In cA --13Im8,
x cosh (/9 + ie)
sinh 2 (p + ie) '
sinh 2 (p + ie) cosh 2y [-
sin 2 e
3. 4.
C2
ct
--
(12)
S = - - a r g cA+/3Re6,
C2
r
Vol. 44, No. 3
SOLITONS IN A FERRIMAGNET
sinh 2 y sinh 2 p _
5.
6.
_
7.
moving with the constant group velocity 8. cosh (p + i e) ]-1 V = c Im~~e)] Im[sinh(p+ie)]-2. 9.
Here P, e 4= 0 are arbitrary real constants (X~ = e°+te). In
A. Hubert, Theorie der Domdnenwiinde in geordneten Medien. Springer-Verlag. BerlinHeidelberg-New York (1974). A.K. Zvezdin, Zh. Eksp. Teor. Fiz. Pisma 29,605 (1979); V.G. Bar'yakhtar, B.A. Ivanov & A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 78, 1509 (1980). I.V. Bar'yakhtar & B.A. Ivanov, Fizika nizkikh temperatur 5, 352 (1979). A.M. Kosevich, B.A. Ivanov & A.S. Kovalev, Physica 3D, 1,363 (198 I); A.M. Kosevich, Fizika metallov i metallovedenie 53,420 (1982). M. Lakshmanan, Phys. Lett. 61A, 53 (1977); L.A. Takhtajan, Phys. Lett. 64A, 235 (1977). E.K. Sklyanin, Preprint LOMI E-3-1979, Leningrad ( 1979); A.E. Borovik & V.N. Robuk, Teor. i. Mat. Fiz. 46,371 (1981). E.A. Turov, Fizicheskie svoistva magnitouporyadochennikh kristallov. Izd-vo AN USSR, Moskva (1963). V.E. Zakharov & A.B. Shabat, Funct. Anal, Appl. 13, 13 (1979);V.E. Zakharov & A.V. Mikhailov, Soy. Phys. J E T P 47, 1017 (1979). Teorya solitonov (pod redaktsiey S.P. Noviokova). Nauka, Moskva (1980).