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Magnetization waves in the Landau-Lifshitz model
Volume 134, number 2
PHYSICS LETfERS A
19 December 1988
MAGNETIZATION WAVES IN THE LANDAU-LIFSHITZ MODEL R.F. BIKBAEV and R.A. SHARIPOV Mathematical Institute ofBashkir ScienttfIc Center, Academy of Sciences of the USSR, 450057 Ufa, Tukaeva 50, USSR Received 1 August 1988; accepted for publication 13 October 1988 Communicated by A.P. Fordy
The solutions of the Landau—Lifshitz equation with finite-gap behaviour at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.
1. The Landau—Lifshitz equation [1] describing the dynamics of the magnetization vector S for the one-dimensional ferromagnet of the “light-plane” type can be written in the following form St =SxS~~ +SxJS,
SI =
1, J=diag(O, 0,
2). —
(1)
16w
Lifshitz equation, i.e. the solutions S with the following behaviour as x—~’± ~,
S(x, t)—+S(x, t~ r, Dl(2)
öl(
2)),
x—~~t~co
,
(3)
where S(x, t I F, D, ö) denotes a real smooth g-gap solution of (1) with a phase ô constructed on a base of the hyperelliptic Riemann surface F with a fixed divisor D = P 2~~< Wthe =2 2g+1 of r, 1 + + Pg on it. Given branching points can 2~= define w <21<22<... < one the meromorphic function ~, ~ )...(2~2~g) on F and the pair ...
In refs. 12,3] (1)represented was shownas toabe completely integrable and eq. it was compatibility condition for the pair of linear equations
a~ w= u~i’, a, P= 1”?,
(2)
with 2 x 2 matrices U and V of the form 3
U=
—
i ~
SaWa o~,
—
~
ofinfinity points P,~with Y-~±2~÷ as P—i~P~. The Riemann surface F consists of two sheets: F~(upper sheet) and F_ (lower sheet). It admits of the hyperelliptic involution a, which does interchange sheets, and the antiholomorphic involution r,
a~I 3
2(tP)=~P~,Y(iP)=—,
3
V= 2i ~ w
1 w2w3 W~’sao~.—
a=I
~
[SXSx]’~waaa,
which does not. The boundary
a r’+ is a collection of
a=1
g cycles y~, ‘y~and the cycle ‘y~passing through two infinities Ps,. Let us choose the canonical basis of cycles a, b1, i=l, g on F as it is shown on fig. 1. The finitegap solution S(x, t I f, D, c5) then is given up to a phase shift by explicit formulae in terms of Riemann 0-functions, 2 C s’ C1 C2 C3 C4 s 1 C2 + C3 C4 C3 C2 C1 C4’ C3 C2 C1 C4’ ••~,
where
are the Pauli matrices and w1 = W2 = (02, w3 =2. Soliton-like solutions of (1) are well-known (see refs. [4,5]). The class of periodic and almost periodic wave-like solutions of (1) contains an important subclass of algebro-geometric (or finite-gap) solutions. They were constructed in refs. [5,6]. The study of algebro-geometric solutions for integrable equations was initiated by Novicov in ref. [7], it led to the well-developed theory of finite-gap integration (see review [8]). °a
J~-
In this paper we study the large-time asymptotics for “nearly finite-gap solutions” of the Landau—
...,
— —‘
—
—
—
—
—
—
3— C S
3C2+C4CL —
~3
~2
—
~1
105
Volume 134, number 2
PHYSICS LETFERS A
19 December 1988
iô/2
0(A(2)+Q+4)
et(P)=f~e
)
xexp[i(Qx+Qt2~t)J, e~(P)=f 2 0(A(2)+4) t1~x+Qt2~t)J. (6) xexp[i(Q Multiples~,~ are defined by fixing det e(P) and the condition eI(2 22g+1 ) =exp(iö). 0)/e2( Remark. The torus T 0 is an exceptional real torus in the following sense: the Baker—Akhiezer function e(P, x, t) is a regular bounded function in x, I, when PeôF’~.
I’
//
-
Fig. 1.
C~=0[n, 0](Q+A+z), C, = —0[n, 0](Q+4—z), C3O(Q+~1+z), C4=0(Q+A—z), n = ~(1, 0
0)
O(A(2)+4)
t)
,
2. In order to construct a scattering theory for S(x, of the form (3) let us define the vectorial Jost
The change of phase ö is equivalent to the rotation ofthe vector S around the third coordinate axis. The vector 4 is connected with the divisor by the Abel map A: div(F)—~Jac(F),A1(P)=f~’0w,,PelT, according to the formula 4= —A(D)—Kwhere Kis the vector of Riemann constants. Real solutions S(x, t) corresponding to real divisors are determined by the restrictions
functions ~(P) and 9’(P) solving (2) and having asymptotics ~(P)—~e~(P, D1 , ô1) as x—+ +cc,
A(D)_A(rD)=A(2o
~(P)= ~‘(P)a(P)+ Y’(aP)b(P), PeÔF+.
+22g+1
—P~—P;)=0.
(4)
Vector Q=i ( V ~~x+ V (2 ~t) is composed of two vectors v (1) and V (2) the vectors of b-periods of two normalized abelian differentials of the second kind withthe only poles at infinities P~having main parts of the form 2) ±42d2+.... ~ Q( Vector zeJac(F) is equal toA(P~)where the path of integration y is shown on fig. i.
~P(P)—*e~(P, D2, ô2)
The functions ck(P), W(P) are bounded with each other by scattering data a(P), b(P),
i
106
inIlrPraPI
w(F)
2xi a(22g+I)+b(22g÷I)\
~
on which the 0-function O(A (2~)+ Q+ 4) does not vanish [9]. The main instrument in constructing finite-gap solutions is the matrix Baker—Akhiezer function e(P) = (e~(F), e + (oF)) solving (2). 4, efl eqs. is given, Thetofirst column of it,f(x, e + (F) up scalar multiples t), =I =t(e 1,j 2, by
j
A(D 2D1)
(5)
(7)
In this paper we study the non-soliton case, i.e. a (F) ~ 0, if Fe F~. Starting from (7) we obtain a scattering theory for (1), (3) most similar to that of ref. [10] for the fast-decreasing case. The only difference consists in the existence of relations between the asymptotic divisors D1, D2, phases ô1, ô2 and scattering data a(P), b(P),
The reality condition (4) defines 2g disjoint real toriT~,v==0,...,2~_1inJacJ(F).Wechooseonly one of them: torus T0 with Re[4+A(20)]=0
as x—~—cx.
— ~2
=
—
in
(
a(2~)—b(2~) )‘
(8)
where r(P) = b(P) Ia(P) is the reflection coefficient. It should be pointed out that for or choice of divisors D1, D2 (i.e. torus T~)1 —r(P)r(aP) is a real and positive function on ar’~. For the integral asymptotical analysis of functions (1), (3) similar we use a singular equation for Jost to that of ref. [11]. Our method is a generalisation
Volume 134, number 2
PHYSICS LETTERS A
of the asymptotical construction of ref. [121. The final result of our investigation is the following: the main term of the asymptotics for S(x, t) as t—~+ cx is given by the finite-gap solution S(x,
t)
=S(x,
t~D(~),ö(~))+e(~~, t)
~ t)=o(l) , (9) with the phase ö(~)and divisor D(~)depending on the “slow variable” ~=x/t according to
19 December 1988
ref. [9]). The function F(P) is given by ?(P) = r(F), FE Q ( ~) 0, P~Q (~). The value of the rest term of asymptotics (9) depends on 2), whetherinP0e thear’+ second or not. f=o(t_N), In the first VN>0. case ~(~=O(t” The scattering problem studied here describes the in=
teraction of two magnetization waves with the same spectrum F. After finishing all the “transition pro-
A (D(~)) =A (D
—
2)
J
1 2xi
cesses two interacting waves” consolidate into one asymptotical wave (9) with slowly changing phases.
ln[l—r(F)r(aP)]w(F),
2 ( ~)
References
)~
“A~(22g+i)l+?(22g+I)\
A(20)
l—?(2~)
(10)
[1] L.D. Landau and E.M. Lifshitz, Phys. Z. Sowjetunion 8 (1935) 153.
Here the path of integration Q(~)is a part of the contour a r’÷ which is situated to the left of the unique stationary point F0 ( ~) (see fig. 1), (Qt I Q (F0) = 0. The function A(F) is given by
[2]A.E. Borovik, JETP Lett. 28 (1978) 629. [31E.K. Skluanin, preprint LOMI E-3-79, Leningrad (1979). [4] A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetization non-linear waves (Naukova Dumka, Kiev, 1983). [5] R.F. Bikbaev, A.I. Bobenko and A.R. Its, preprint Don FTI-
A(F) = urn ce(F’), F’ e F~, PeôT~,
84-6,7, Donetck (1982). [6]R.F. Bikbaev, A. Bobenko and A.R. Its, DokI. Akad. Nauk
(2))
P~ -‘P
SSSR 272 (1983) 1293.
0((A(P)—A(D(~))—K) 0(A(P)—A(D2)—K)
x exp(
—
~-~—
J
M( Q, F) f(Q))~
Q(~)
f(Q) =ln[
1 —r(Q)r(aQ)]
,
where M( Q, F) is the multivalued Cauchy kernel (see
[81 B.A. [7] S.P. Novicov, Dubrovin,Funct. Usp. Mat. Anal. Nauk AppI. 36 8 (1974) (1981) 43. 2, 11. [9]J.D. Fay, Lecture notes in mathematics, Vol. 352. Theta functions on Riemann surfaces (Springer, Berlin, 1973). [10]G. Gardner, G. Green, M. Kruskal and R. Miura, Phys. Rev. Lett. 19 (1965) 1095. [11] R.F. Bikbaev and R.A. Sharipov, Teor. Mat. Phys. (1988), tobepublished. [121 V.E. Zakharov and S.V. Manakov, Zh. Eksp. Teor. Fiz. 71 (1976) 203.
107
PHYSICS LETfERS A
19 December 1988
MAGNETIZATION WAVES IN THE LANDAU-LIFSHITZ MODEL R.F. BIKBAEV and R.A. SHARIPOV Mathematical Institute ofBashkir ScienttfIc Center, Academy of Sciences of the USSR, 450057 Ufa, Tukaeva 50, USSR Received 1 August 1988; accepted for publication 13 October 1988 Communicated by A.P. Fordy
The solutions of the Landau—Lifshitz equation with finite-gap behaviour at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.
1. The Landau—Lifshitz equation [1] describing the dynamics of the magnetization vector S for the one-dimensional ferromagnet of the “light-plane” type can be written in the following form St =SxS~~ +SxJS,
SI =
1, J=diag(O, 0,
2). —
(1)
16w
Lifshitz equation, i.e. the solutions S with the following behaviour as x—~’± ~,
S(x, t)—+S(x, t~ r, Dl(2)
öl(
2)),
x—~~t~co
,
(3)
where S(x, t I F, D, ö) denotes a real smooth g-gap solution of (1) with a phase ô constructed on a base of the hyperelliptic Riemann surface F with a fixed divisor D = P 2~~< Wthe =2 2g+1 of r, 1 + + Pg on it. Given branching points can 2~= define w <21<22<... < one the meromorphic function ~, ~ )...(2~2~g) on F and the pair ...
In refs. 12,3] (1)represented was shownas toabe completely integrable and eq. it was compatibility condition for the pair of linear equations
a~ w= u~i’, a, P= 1”?,
(2)
with 2 x 2 matrices U and V of the form 3
U=
—
i ~
SaWa o~,
—
~
ofinfinity points P,~with Y-~±2~÷ as P—i~P~. The Riemann surface F consists of two sheets: F~(upper sheet) and F_ (lower sheet). It admits of the hyperelliptic involution a, which does interchange sheets, and the antiholomorphic involution r,
a~I 3
2(tP)=~P~,Y(iP)=—,
3
V= 2i ~ w
1 w2w3 W~’sao~.—
a=I
~
[SXSx]’~waaa,
which does not. The boundary
a r’+ is a collection of
a=1
g cycles y~, ‘y~and the cycle ‘y~passing through two infinities Ps,. Let us choose the canonical basis of cycles a, b1, i=l, g on F as it is shown on fig. 1. The finitegap solution S(x, t I f, D, c5) then is given up to a phase shift by explicit formulae in terms of Riemann 0-functions, 2 C s’ C1 C2 C3 C4 s 1 C2 + C3 C4 C3 C2 C1 C4’ C3 C2 C1 C4’ ••~,
where
are the Pauli matrices and w1 = W2 = (02, w3 =2. Soliton-like solutions of (1) are well-known (see refs. [4,5]). The class of periodic and almost periodic wave-like solutions of (1) contains an important subclass of algebro-geometric (or finite-gap) solutions. They were constructed in refs. [5,6]. The study of algebro-geometric solutions for integrable equations was initiated by Novicov in ref. [7], it led to the well-developed theory of finite-gap integration (see review [8]). °a
J~-
In this paper we study the large-time asymptotics for “nearly finite-gap solutions” of the Landau—
...,
— —‘
—
—
—
—
—
—
3— C S
3C2+C4CL —
~3
~2
—
~1
105
Volume 134, number 2
PHYSICS LETFERS A
19 December 1988
iô/2
0(A(2)+Q+4)
et(P)=f~e
)
xexp[i(Qx+Qt2~t)J, e~(P)=f 2 0(A(2)+4) t1~x+Qt2~t)J. (6) xexp[i(Q Multiples~,~ are defined by fixing det e(P) and the condition eI(2 22g+1 ) =exp(iö). 0)/e2( Remark. The torus T 0 is an exceptional real torus in the following sense: the Baker—Akhiezer function e(P, x, t) is a regular bounded function in x, I, when PeôF’~.
I’
//
-
Fig. 1.
C~=0[n, 0](Q+A+z), C, = —0[n, 0](Q+4—z), C3O(Q+~1+z), C4=0(Q+A—z), n = ~(1, 0
0)
O(A(2)+4)
t)
,
2. In order to construct a scattering theory for S(x, of the form (3) let us define the vectorial Jost
The change of phase ö is equivalent to the rotation ofthe vector S around the third coordinate axis. The vector 4 is connected with the divisor by the Abel map A: div(F)—~Jac(F),A1(P)=f~’0w,,PelT, according to the formula 4= —A(D)—Kwhere Kis the vector of Riemann constants. Real solutions S(x, t) corresponding to real divisors are determined by the restrictions
functions ~(P) and 9’(P) solving (2) and having asymptotics ~(P)—~e~(P, D1 , ô1) as x—+ +cc,
A(D)_A(rD)=A(2o
~(P)= ~‘(P)a(P)+ Y’(aP)b(P), PeÔF+.
+22g+1
—P~—P;)=0.
(4)
Vector Q=i ( V ~~x+ V (2 ~t) is composed of two vectors v (1) and V (2) the vectors of b-periods of two normalized abelian differentials of the second kind withthe only poles at infinities P~having main parts of the form 2) ±42d2+.... ~ Q( Vector zeJac(F) is equal toA(P~)where the path of integration y is shown on fig. i.
~P(P)—*e~(P, D2, ô2)
The functions ck(P), W(P) are bounded with each other by scattering data a(P), b(P),
i
106
inIlrPraPI
w(F)
2xi a(22g+I)+b(22g÷I)\
~
on which the 0-function O(A (2~)+ Q+ 4) does not vanish [9]. The main instrument in constructing finite-gap solutions is the matrix Baker—Akhiezer function e(P) = (e~(F), e + (oF)) solving (2). 4, efl eqs. is given, Thetofirst column of it,f(x, e + (F) up scalar multiples t), =I =t(e 1,j 2, by
j
A(D 2D1)
(5)
(7)
In this paper we study the non-soliton case, i.e. a (F) ~ 0, if Fe F~. Starting from (7) we obtain a scattering theory for (1), (3) most similar to that of ref. [10] for the fast-decreasing case. The only difference consists in the existence of relations between the asymptotic divisors D1, D2, phases ô1, ô2 and scattering data a(P), b(P),
The reality condition (4) defines 2g disjoint real toriT~,v==0,...,2~_1inJacJ(F).Wechooseonly one of them: torus T0 with Re[4+A(20)]=0
as x—~—cx.
— ~2
=
—
in
(
a(2~)—b(2~) )‘
(8)
where r(P) = b(P) Ia(P) is the reflection coefficient. It should be pointed out that for or choice of divisors D1, D2 (i.e. torus T~)1 —r(P)r(aP) is a real and positive function on ar’~. For the integral asymptotical analysis of functions (1), (3) similar we use a singular equation for Jost to that of ref. [11]. Our method is a generalisation
Volume 134, number 2
PHYSICS LETTERS A
of the asymptotical construction of ref. [121. The final result of our investigation is the following: the main term of the asymptotics for S(x, t) as t—~+ cx is given by the finite-gap solution S(x,
t)
=S(x,
t~D(~),ö(~))+e(~~, t)
~ t)=o(l) , (9) with the phase ö(~)and divisor D(~)depending on the “slow variable” ~=x/t according to
19 December 1988
ref. [9]). The function F(P) is given by ?(P) = r(F), FE Q ( ~) 0, P~Q (~). The value of the rest term of asymptotics (9) depends on 2), whetherinP0e thear’+ second or not. f=o(t_N), In the first VN>0. case ~(~=O(t” The scattering problem studied here describes the in=
teraction of two magnetization waves with the same spectrum F. After finishing all the “transition pro-
A (D(~)) =A (D
—
2)
J
1 2xi
cesses two interacting waves” consolidate into one asymptotical wave (9) with slowly changing phases.
ln[l—r(F)r(aP)]w(F),
2 ( ~)
References
)~
“A~(22g+i)l+?(22g+I)\
A(20)
l—?(2~)
(10)
[1] L.D. Landau and E.M. Lifshitz, Phys. Z. Sowjetunion 8 (1935) 153.
Here the path of integration Q(~)is a part of the contour a r’÷ which is situated to the left of the unique stationary point F0 ( ~) (see fig. 1), (Qt I Q (F0) = 0. The function A(F) is given by
[2]A.E. Borovik, JETP Lett. 28 (1978) 629. [31E.K. Skluanin, preprint LOMI E-3-79, Leningrad (1979). [4] A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetization non-linear waves (Naukova Dumka, Kiev, 1983). [5] R.F. Bikbaev, A.I. Bobenko and A.R. Its, preprint Don FTI-
A(F) = urn ce(F’), F’ e F~, PeôT~,
84-6,7, Donetck (1982). [6]R.F. Bikbaev, A. Bobenko and A.R. Its, DokI. Akad. Nauk
(2))
P~ -‘P
SSSR 272 (1983) 1293.
0((A(P)—A(D(~))—K) 0(A(P)—A(D2)—K)
x exp(
—
~-~—
J
M( Q, F) f(Q))~
Q(~)
f(Q) =ln[
1 —r(Q)r(aQ)]
,
where M( Q, F) is the multivalued Cauchy kernel (see
[81 B.A. [7] S.P. Novicov, Dubrovin,Funct. Usp. Mat. Anal. Nauk AppI. 36 8 (1974) (1981) 43. 2, 11. [9]J.D. Fay, Lecture notes in mathematics, Vol. 352. Theta functions on Riemann surfaces (Springer, Berlin, 1973). [10]G. Gardner, G. Green, M. Kruskal and R. Miura, Phys. Rev. Lett. 19 (1965) 1095. [11] R.F. Bikbaev and R.A. Sharipov, Teor. Mat. Phys. (1988), tobepublished. [121 V.E. Zakharov and S.V. Manakov, Zh. Eksp. Teor. Fiz. 71 (1976) 203.
107