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Non-isospectral deformations of the Heisenberg ferromagnet equation
Volume 149, number 2,3
PHYSICS LETTERS A
17 September 1990
Non-isospectral deformations of the Heisenberg ferromagnet equation Jan Cie~1iñski PhysicsInstitute, BialystokBranchof Warsaw University, ul. Lipowa 41, 15-424 Bialystok, Poland Received 30 August 1989; revised manuscript received 20 April 1990; accepted for publication 11 July 1990 Communicated by A.P. Fordy
We consider spin models of the form S,=yS,,+SA ~ They can be interpreted as one-dimensional reductions of the three-dimensional classical, continuous, inhomogeneous Heisenberg ferromagnet. We isolate integrable subcases corresponding to non-isospectral (in x and t) deformations ofthe standard, quadratic in the spectral parameter, su (2) AKNS linear problem.
In the last years many results were obtained con-
bility of one-dimensional (i.e. depending on one pa-
cerning the classical, continuous, one-dimensional, inhomogeneous Heisenberg ferromagnet equation
rameter) configurations of the three-dimensional classical, continuous, inhomogeneous Heisenberg ferromagnet model,
(1) where S=S(x, t)eE3, 5-5=1, a comma denotes differentiation, A denotes vector product in E3, a dot denotes a scalar product in E3 and the coupling function f=f(x, 1) eR is given. Two different geometric approaches to the model (1) were developed [1 —4]. In the case of the coupling functionflinear in x eq. (1) turns out to be integrable by the inverse scattering method [5] and the Darboux—Bäcklund transformation is known [6]. The spectral paramS,,=SA(f5,~),~,
S, =SA [(fS~)~+ (fS,~),~+ (f5,~),~], (3) where S=S(x, y, z, t), f=f(x, y, z, t) and x, y, z are Cartesian coordinates in F~.To obtain one-dimensional reductions of (3) we introduce some (t-dependent) curvilinear coordinates q q2, q3 in P3 (q”=q’~(x,y, z, t), k= 1, 2, 3). Eq. (3) assumes the form ~,
5~= ~~7=S A (f.,fggu5 1)~ q’~S1, (4) Vg where g=det(g~),g~=r~r~ are components of the 3), metric (r= (x, y, z) istoa {g,~}, position in (P ),~ {gIJ} is tensor the inverse matrix and,vector finally means differentiation with respect to q. We use the —
eter of the corresponding linear problem has to depend on t when f~x~O. Cylindrically symmetricferromagnet configurations twodimensional Heisenberg areofthe described by the equation 5,
5 A (Srr + r — ‘Sr),
(2)
where S=S(r, t)eE3 and 55=1. It was proved to be integrable and the spectral parameter depends on t [7]. One can use in that case a slightly different linear problem with spectral parameter depending on rand t [81. The aim ofthe present paper is to combine and to generalize ideas connected with eqs. (1) and (2). To be more specific: we are going to study the integra-
Einstein summation convention in formula (4), of course. The assumption S=S(u, I), where u is one of our coordinates (let us take for example u q’), leads us to the following equation,
s,t=sA(fg~suu+-1-s,~~ (fsJ~~~).k) k= I
(5)
— ~
“
~U’
0375-9601/90/$ 03.50 © 1990— Elsevier Science Publishers B.V. (North-Holland)
139
Volume 149, number 2,3
PHYSICS LETTERS A
where ~ = g’1 etc. In general the coefficients in formula (5) depend on q2, q3 and, as a consequence, eq. (5) has a trivial set of solutions (S= const). For the generic choice of coordinates q’ it is impossible to find even a single function ffor which these coefficients depend on u=q’ only. Eq. (5) becomes a non-trivial one-dimensional spin model if there exist some functions a, /3 and y such that (6a)
fguu=a(u,t),
~J~k~t
(f%fgguk)~=/3(u, 1),
17 September 1990
Which coordinate systems generate one-dimensional reductions of eq. (3)? For the moment, these questions are left open. We will try to carry out a detailed analysis of eqs. (6) in the future. Let us proceed to the main subject of this paper. We will study the integrability of (7) using a connection between this equation and a generalization of the nonlinear Schrodinger equation. The homogeneous, one-dimensional Heisenberg ferromagnet equation,
(6b)
(8)
S,SAS~~, —
u, =y(u, t)
.
(6c)
Thus we come to the following family of nonlinear spin systems parametrized by three functions of two variables, S, = ~
+ S A (aS~~ + /3S~),
(7)
where S=S(u, t)eE3, S~S=l, a=a(u, I), /3=/3(u, t) and y= y( u, I). Every one-dimensional configuration of a three-dimensional Heisenberg ferromagnet is described by eq. (7) for some choice of parametric functions a, /3, y. Several examples are in order (a=a(u, t) is an arbitrary function): (1) Cartesian coordinates (x, ~, z), u = x, then f= a, /3= a~,~= 0. (2) Cylindrical coordinates (x= r cos ~, y= r sin ~ z),
(a) u=r, thenf=a, /3=a~+a/u,~ (b) u_-~Ø,then f=ar2, /3=a,~,y=o. (3) Spherical coordinates (x= R sin d cos ~ y = R sin 0 sin 0, z = R cos i9) (a) u=R, thenf=a, /3=a~+2a/u,y=O; (b) u = 0, then f= aR2 /3= a,~+ a ctg u, y= 0; (c) u = 0, then f= aR2 sin2d, 8= a~,~= 0. In case (1) eq. (1) is obtained. Case (2a), the inhomogeneous cylindrical Heisenberg ferromagnet, is a generalization of the model (2). The cases (2b) and (3c) are in fact identical. Many interesting questions can be put forward, for example: Can we find some one-dimensional reduction of the model (3) for any coupling function f=f(x, ~,,
was proved to be gauge equivalent to the nonlinear Schrödinger equation iq, + ~
140
(9)
where q=q(x, t)eC and the asterisk denotes the complex conjugate [91. The corresponding linear problem is isospectral. In the case off= a ( t )x+ b (1) the model(l)isgauge equivalent [5] tothe so-called nonhomogeneous nonlinear Schrodinger equation
iq,, + (J~) ~ + 2fq~q2+ 2q
J
f~qq*dx= 0,
(10)
where q= q (x, 1) cC. The spectral parameter of the corresponding linear problem is t-dependent. In the case of the cylindrical Heisenberg ferromagnet equation (2) the spectral parameter can be a function of both independent variables [8]. In the present paper we assume the most general dependence of )~on x and t. It enables us to obtain all non-isospectral deformations of the nonlinear Schrodinger equation. We are able to construct the Darboux—Bäcklund transformation in such a case [61. It should be stressed that this transformation “adds” a single soliton to any solution. No boundary conditions are imposed. In this specific sense nonisospectral deformations of the nonlinear Schrödinger equation in this paper are called integrable. Moreover Burtsev et al. [81 claim that the inverse scattering method works in the case of A =A(x, t). Consider the su (2) AKNS non-isospectral linear problem
z, t)?
Does an arbitrary choice of a,fl, y correspond to some one-dimensional reduction of (3)?
+ 2q*q2 = 0,
~
=
(~* ~
(11 a)
Volume 149, number 2,3
(IA
—
PHYSICS LETTERS A
B
‘k~_B*
17 September 1990
B=A
1q—~i(A2qF1,),1,/F1,+AA2q,
—IA)
( b)
‘
where P= ø( u, I) a SU (2), 2=2 (u, t) eP, q= q( u, t) cC, the asterisk denotes the complex conjugate, A=A(u, I; 2)eR, B=B(u, I; 2)cC. To obtain nonisospectral deformations of the nonlinear Schrädinger equation we add the following assumptions, 2, (12a) A=A0+A12+A22 B=B 2, (12b)
(19)
where F=F(u, I), H=H(u, I) and Ck=Ck(t) (k=0, 1, 2) are arbitrary functions (F,1, 0) and A1, A2 (in formula (19)) are defined by (18). Finally, solving (16) and (17), we obtain the expression for the spectral parameter 2: 1 = (H+ ~)F1,, (20) where ~= ~ ( t) satisfies the following Riccati equation, .
0+B12+B22
2 = ~
h2k
(13a)
~,,=C
(1 3b)
2. (21) The general solution to this equation can be cxpressed as
2+C 0i~
k=0
2~
A~.=
Pk
wher:Ak=Ak(u, t)eP, A 2(u, t)~0,Bk=Bk(u, t)eC, t)eP and NEI\1. The integrability conditions ~,= ~ and A~ considered as identities with respect to 2, are equivalent to the following nonlinear system,
~=~o _ZeXP(2
J
C0(t’ )pio(t’ ) dt’)
hk=hk(u, t)cR, Pk=Pk(U,
j —
z
x exp(2
j Co ( t’)
J
iq,—iB01,—iB1p0+2A0q=0,
(l4a)
A01,+A1p0 —h0 +i(Boq*_B~q)=0,
(l4b)
i (B,~+ B~Pk + 2B2 Pk_ 1) = 2 (Ak q— Bk_I),
(1 5a)
where ij~is any particular solution to (21) and z is a constant. Therefore the spectral parameter A is a function depending on u, t and z.
(15b)
Formulas (l8)—(20) define the elements of the linear problem (11). It can be used to integrate (in the sense of the Darboux—Bäcklund transformation
=h~+j(B7~q_B~q*) k~I
tk1,—Pktm
,,~,
L m~rtmpk_m+I
Co(t”)?io(t”)dt”)dt’]~
(22)
‘~k—,n+1Pm)
I
at least) the nonlinear system (14). This system of l~~s C1
two differential tions (q and A0) equations can be rewritten for twoasunknown an integro-diffunc-
where k= 1, 2, 3, Ak=Bk=0 for k> 2 and hk=pk=Ofork>N. (The casel=const is discussed for example in ref. [10].) We obtained the general solution to eqs. (15) and, as a result, the formulas (12) and (13) assume the following form,
ferential (in general) equation for q. The case H(u, t)=0, F(u, t)=u, C0=—2a, C~=2b,C2=0 corresponds to eq. (10). In particular, for a=0 and b= 1 we obtain the nonlinear Schrodinger equation (9). Now we are going to show that (14) is gaugeequivalent to (7) for some a,fl, y. Consider the following gauge transformation of the linear system (11):
—0 —
...,
(16)
2+C 2,t:F~(H~+CoH
2)+A(F,a,/F,1,_2CoH)
~=G’~,
.
.
(23)
where G=~(u,t; 2~)is a solution to the linear systern (11) with A given by 2~=(H+t~o)F,1,.Then
A=A0 +2[F, —2H(C0F—C~)]/F1, (18)
~1,=2(2—10)Sc~,
(24a) 141
Volume 149, number 2,3
PHYSICS LETTERS A
=2(1—20)[A~+A2(2+A0)]S~
~,
—A2(A—20)[S,S1,]~,
(24b)
17 September 1990
transformed into (1) by a change of coordinates if these exists v = v (u, t) such that (28a)
whereAs, A2 are given by (18) and S=~iG_1(~
—
~)G.
The integrability conditions ~ following nonlinear equation,
(25) =
~
(e~’/a),,=(ye7a)1,. (28b) It turns out that the system (28) is satisfied by a,fl, y given by (27) and v given by
give us the e~=C1
C0F exp(/ 2 J~ocodt). F1, —
(29)
S,,=(A1 +220A2)S1, ~[S,A2S1,1, + (A2,1, +Pi A2 )S,1,] (26) where square brackets denote the commutator. Taking into account the3 well isomorphism (the known Killing—Cartan formbein tween su (2) and to F the scalar product in F3 and the su (2) corresponds commutator correspond to the vector product in F3) one can see that eq. (26) is identical with (7), where —
In other words, all non-isospectral deformations of the Heisenberg ferrornagnet equation can be transformed into (1) by a change of variables. Therefore we did(as notfar obtain new integrable spin systerns as theessentially Darboux—Bäcklund transformation is concerned). Our approach gives the possibility to find three-dimensional coupling functions f=f(x, y, z, t) for which we can generate solutions of (3). These so-
a = (C 1
—
C0F) /2 p2
/3= (aF,1,),JF1,,
S
(27a)
lutions are characterized by some prescribed sym-
(27b)
metry (given by the coordinate system used). Example. Consider an orthogonal and t-independent coordinate system. One can prove, by compar-
,~
(27c) Thus we come to the conclusion that for a,/3, y given by (27) eq. (7) is integrable. In that case eq. (7) will be called the non-isospectral deformation of the Heisenberg ferromagnet equation. In formulas (27) C0=C0(t), C~=C1(t),F=F(u, 1) and ~ = ~o ( t) can be considered as arbitrary functions (in the last case it isaconsequence of eq. (21) C2 uniquely determined Theischoice F( u, t) = u andby,~= 0 corresponds to the inhomogeneous Heisenberg ferromagnet equation (l)withf(x,t)=C 2, ~=0, C1(t)—xC0(t),andthechoice F(u, t)=u 1=0, C0=—8 corresponds to the cylindrical Heisenberg ferromagnet model (2). It is not difficult to notice that by the the cylindrical change of coordinates r2 = 2x one can transform ferromagnet equation (2) into eq. (1) with f= 2x. There is an important difference between the initialvalue problems of (1) and (2). The first is a full-line problem while the second is a half-line problem. This means there is no one-to-one correspondence between these two initial valueproblems. However, this difference is not relevant in the construction of the Darboux—Bäcklund transformation. The direct calculations show that eq. (7) can be —
142
,~.
ing (6) with (27), that such a system generates an integrable one-dimensional reduction of the threedimensional Heisenberg ferromagnet equation (3) iff ~ is factorised in the following way, .,~/~Di(q1)D2(q2,q3,t), (30) where D1 and D2 are some functions (q’=u). The 2, q3, t), corresponding coupling functions f_—f( q given by to the integrable case, is u, then
f~guur~(u)(at+bw
JDi(v)dv)~
(31)
where are be arbitrary The dependence 2, a, q3 bmay hidden functions. only in g’~. onBelow q we present the list of “integrable” coupling functions corresponding to the simplest coordinate systems. (1) Cartesian coordinates, f=ax+b,
(32a)
(2) Cylindrical coordinates (r2=x2+y2, tg Ø=y/ x), (a)
f=a+b/r2,
(32b)
Volume 149, number 2,3
PHYSICS LETTERS A
17 September 1990
(32c) (b) f= (aØ+ b ) r2, (3) Spherical coordinates (R2=x2+y2+z2, cosd =z/R),
This work was partially supported by the Polish Ministry of National Education (Research Project CPBP 01.03).
(a) f=a/R+b/R4,
(32d)
References
(b) f=(a+bcos0)R2/sin2O.
(32e)
[1] M. Lakshmanan and R.K. Bullough, Phys. Lett. A 80 (1980) 287.
Summary. We have shown that the one-dimensional inhomogeneous Heisenberg ferromagnet equation (1) with f linear in x is the universal integrable (in the sense of the Darboux—Bäcklund transformation) one-dimensional reduction of the three-dimensional inhomogeneous Heisenberg ferromagnet. Any other reduction of (3) can be obtamed from (1) by a change of variables. The initial value problems are, in general, different, which should be taken into account in applying the inverse scattering method. The second result of this paper is a method to generate solutions to the three-dimensional inhomogeneous Heisenberg ferromagnet (3) for some coupling functions f=f(x, y, z, t). Examples of such functions are given by (31) and (32).
[21 R. Balakrishnan, J. Phys. C 15 (1982) LI 305. [3] A. Sym and W. Wesselius, Phys. Lett. A 120 (1987) 183.
[41J. Cie~1iñski,A. Sym and W. Wesselius, Twente University memorandum No. 789. On the geometry of the inhomogeneous Heisenberg ferromagnet: non-integrable
case (May 1989), to be published. 117. [6]J. Cie~1iñski,Geometric method of integration of the [5lM.LakshmananandS.Ganesan,PhysicaAl32(1985)
inhomogeneous Heisenberg chain, in: Proc. Workshop on The nonlinear evolution equation: integrabiity and spectral methods (Como, 1988). [7] A.V. Mikhailov and A.I. Yaremtschuk, JETP Lett. 36 (1982) 78. [81S.P. Burtsev, V.E. Zakharov and A.V. Mikhailov, Theor. Math. Phys. 70 (1987) 323.
[9] V.E. Zakharov and L.A. Takhtajan, Theor. Math. Phys. 38 (1979) 26.
[101M. Ablowitz and H. Segur, Solitons and the scattering transform (SIAM, Philadelphia, 1981).
inverse
143
PHYSICS LETTERS A
17 September 1990
Non-isospectral deformations of the Heisenberg ferromagnet equation Jan Cie~1iñski PhysicsInstitute, BialystokBranchof Warsaw University, ul. Lipowa 41, 15-424 Bialystok, Poland Received 30 August 1989; revised manuscript received 20 April 1990; accepted for publication 11 July 1990 Communicated by A.P. Fordy
We consider spin models of the form S,=yS,,+SA ~ They can be interpreted as one-dimensional reductions of the three-dimensional classical, continuous, inhomogeneous Heisenberg ferromagnet. We isolate integrable subcases corresponding to non-isospectral (in x and t) deformations ofthe standard, quadratic in the spectral parameter, su (2) AKNS linear problem.
In the last years many results were obtained con-
bility of one-dimensional (i.e. depending on one pa-
cerning the classical, continuous, one-dimensional, inhomogeneous Heisenberg ferromagnet equation
rameter) configurations of the three-dimensional classical, continuous, inhomogeneous Heisenberg ferromagnet model,
(1) where S=S(x, t)eE3, 5-5=1, a comma denotes differentiation, A denotes vector product in E3, a dot denotes a scalar product in E3 and the coupling function f=f(x, 1) eR is given. Two different geometric approaches to the model (1) were developed [1 —4]. In the case of the coupling functionflinear in x eq. (1) turns out to be integrable by the inverse scattering method [5] and the Darboux—Bäcklund transformation is known [6]. The spectral paramS,,=SA(f5,~),~,
S, =SA [(fS~)~+ (fS,~),~+ (f5,~),~], (3) where S=S(x, y, z, t), f=f(x, y, z, t) and x, y, z are Cartesian coordinates in F~.To obtain one-dimensional reductions of (3) we introduce some (t-dependent) curvilinear coordinates q q2, q3 in P3 (q”=q’~(x,y, z, t), k= 1, 2, 3). Eq. (3) assumes the form ~,
5~= ~~7=S A (f.,fggu5 1)~ q’~S1, (4) Vg where g=det(g~),g~=r~r~ are components of the 3), metric (r= (x, y, z) istoa {g,~}, position in (P ),~ {gIJ} is tensor the inverse matrix and,vector finally means differentiation with respect to q. We use the —
eter of the corresponding linear problem has to depend on t when f~x~O. Cylindrically symmetricferromagnet configurations twodimensional Heisenberg areofthe described by the equation 5,
5 A (Srr + r — ‘Sr),
(2)
where S=S(r, t)eE3 and 55=1. It was proved to be integrable and the spectral parameter depends on t [7]. One can use in that case a slightly different linear problem with spectral parameter depending on rand t [81. The aim ofthe present paper is to combine and to generalize ideas connected with eqs. (1) and (2). To be more specific: we are going to study the integra-
Einstein summation convention in formula (4), of course. The assumption S=S(u, I), where u is one of our coordinates (let us take for example u q’), leads us to the following equation,
s,t=sA(fg~suu+-1-s,~~ (fsJ~~~).k) k= I
(5)
— ~
“
~U’
0375-9601/90/$ 03.50 © 1990— Elsevier Science Publishers B.V. (North-Holland)
139
Volume 149, number 2,3
PHYSICS LETTERS A
where ~ = g’1 etc. In general the coefficients in formula (5) depend on q2, q3 and, as a consequence, eq. (5) has a trivial set of solutions (S= const). For the generic choice of coordinates q’ it is impossible to find even a single function ffor which these coefficients depend on u=q’ only. Eq. (5) becomes a non-trivial one-dimensional spin model if there exist some functions a, /3 and y such that (6a)
fguu=a(u,t),
~J~k~t
(f%fgguk)~=/3(u, 1),
17 September 1990
Which coordinate systems generate one-dimensional reductions of eq. (3)? For the moment, these questions are left open. We will try to carry out a detailed analysis of eqs. (6) in the future. Let us proceed to the main subject of this paper. We will study the integrability of (7) using a connection between this equation and a generalization of the nonlinear Schrodinger equation. The homogeneous, one-dimensional Heisenberg ferromagnet equation,
(6b)
(8)
S,SAS~~, —
u, =y(u, t)
.
(6c)
Thus we come to the following family of nonlinear spin systems parametrized by three functions of two variables, S, = ~
+ S A (aS~~ + /3S~),
(7)
where S=S(u, t)eE3, S~S=l, a=a(u, I), /3=/3(u, t) and y= y( u, I). Every one-dimensional configuration of a three-dimensional Heisenberg ferromagnet is described by eq. (7) for some choice of parametric functions a, /3, y. Several examples are in order (a=a(u, t) is an arbitrary function): (1) Cartesian coordinates (x, ~, z), u = x, then f= a, /3= a~,~= 0. (2) Cylindrical coordinates (x= r cos ~, y= r sin ~ z),
(a) u=r, thenf=a, /3=a~+a/u,~ (b) u_-~Ø,then f=ar2, /3=a,~,y=o. (3) Spherical coordinates (x= R sin d cos ~ y = R sin 0 sin 0, z = R cos i9) (a) u=R, thenf=a, /3=a~+2a/u,y=O; (b) u = 0, then f= aR2 /3= a,~+ a ctg u, y= 0; (c) u = 0, then f= aR2 sin2d, 8= a~,~= 0. In case (1) eq. (1) is obtained. Case (2a), the inhomogeneous cylindrical Heisenberg ferromagnet, is a generalization of the model (2). The cases (2b) and (3c) are in fact identical. Many interesting questions can be put forward, for example: Can we find some one-dimensional reduction of the model (3) for any coupling function f=f(x, ~,,
was proved to be gauge equivalent to the nonlinear Schrödinger equation iq, + ~
140
(9)
where q=q(x, t)eC and the asterisk denotes the complex conjugate [91. The corresponding linear problem is isospectral. In the case off= a ( t )x+ b (1) the model(l)isgauge equivalent [5] tothe so-called nonhomogeneous nonlinear Schrodinger equation
iq,, + (J~) ~ + 2fq~q2+ 2q
J
f~qq*dx= 0,
(10)
where q= q (x, 1) cC. The spectral parameter of the corresponding linear problem is t-dependent. In the case of the cylindrical Heisenberg ferromagnet equation (2) the spectral parameter can be a function of both independent variables [8]. In the present paper we assume the most general dependence of )~on x and t. It enables us to obtain all non-isospectral deformations of the nonlinear Schrodinger equation. We are able to construct the Darboux—Bäcklund transformation in such a case [61. It should be stressed that this transformation “adds” a single soliton to any solution. No boundary conditions are imposed. In this specific sense nonisospectral deformations of the nonlinear Schrödinger equation in this paper are called integrable. Moreover Burtsev et al. [81 claim that the inverse scattering method works in the case of A =A(x, t). Consider the su (2) AKNS non-isospectral linear problem
z, t)?
Does an arbitrary choice of a,fl, y correspond to some one-dimensional reduction of (3)?
+ 2q*q2 = 0,
~
=
(~* ~
(11 a)
Volume 149, number 2,3
(IA
—
PHYSICS LETTERS A
B
‘k~_B*
17 September 1990
B=A
1q—~i(A2qF1,),1,/F1,+AA2q,
—IA)
( b)
‘
where P= ø( u, I) a SU (2), 2=2 (u, t) eP, q= q( u, t) cC, the asterisk denotes the complex conjugate, A=A(u, I; 2)eR, B=B(u, I; 2)cC. To obtain nonisospectral deformations of the nonlinear Schrädinger equation we add the following assumptions, 2, (12a) A=A0+A12+A22 B=B 2, (12b)
(19)
where F=F(u, I), H=H(u, I) and Ck=Ck(t) (k=0, 1, 2) are arbitrary functions (F,1, 0) and A1, A2 (in formula (19)) are defined by (18). Finally, solving (16) and (17), we obtain the expression for the spectral parameter 2: 1 = (H+ ~)F1,, (20) where ~= ~ ( t) satisfies the following Riccati equation, .
0+B12+B22
2 = ~
h2k
(13a)
~,,=C
(1 3b)
2. (21) The general solution to this equation can be cxpressed as
2+C 0i~
k=0
2~
A~.=
Pk
wher:Ak=Ak(u, t)eP, A 2(u, t)~0,Bk=Bk(u, t)eC, t)eP and NEI\1. The integrability conditions ~,= ~ and A~ considered as identities with respect to 2, are equivalent to the following nonlinear system,
~=~o _ZeXP(2
J
C0(t’ )pio(t’ ) dt’)
hk=hk(u, t)cR, Pk=Pk(U,
j —
z
x exp(2
j Co ( t’)
J
iq,—iB01,—iB1p0+2A0q=0,
(l4a)
A01,+A1p0 —h0 +i(Boq*_B~q)=0,
(l4b)
i (B,~+ B~Pk + 2B2 Pk_ 1) = 2 (Ak q— Bk_I),
(1 5a)
where ij~is any particular solution to (21) and z is a constant. Therefore the spectral parameter A is a function depending on u, t and z.
(15b)
Formulas (l8)—(20) define the elements of the linear problem (11). It can be used to integrate (in the sense of the Darboux—Bäcklund transformation
=h~+j(B7~q_B~q*) k~I
tk1,—Pktm
,,~,
L m~rtmpk_m+I
Co(t”)?io(t”)dt”)dt’]~
(22)
‘~k—,n+1Pm)
I
at least) the nonlinear system (14). This system of l~~s C1
two differential tions (q and A0) equations can be rewritten for twoasunknown an integro-diffunc-
where k= 1, 2, 3, Ak=Bk=0 for k> 2 and hk=pk=Ofork>N. (The casel=const is discussed for example in ref. [10].) We obtained the general solution to eqs. (15) and, as a result, the formulas (12) and (13) assume the following form,
ferential (in general) equation for q. The case H(u, t)=0, F(u, t)=u, C0=—2a, C~=2b,C2=0 corresponds to eq. (10). In particular, for a=0 and b= 1 we obtain the nonlinear Schrodinger equation (9). Now we are going to show that (14) is gaugeequivalent to (7) for some a,fl, y. Consider the following gauge transformation of the linear system (11):
—0 —
...,
(16)
2+C 2,t:F~(H~+CoH
2)+A(F,a,/F,1,_2CoH)
~=G’~,
.
.
(23)
where G=~(u,t; 2~)is a solution to the linear systern (11) with A given by 2~=(H+t~o)F,1,.Then
A=A0 +2[F, —2H(C0F—C~)]/F1, (18)
~1,=2(2—10)Sc~,
(24a) 141
Volume 149, number 2,3
PHYSICS LETTERS A
=2(1—20)[A~+A2(2+A0)]S~
~,
—A2(A—20)[S,S1,]~,
(24b)
17 September 1990
transformed into (1) by a change of coordinates if these exists v = v (u, t) such that (28a)
whereAs, A2 are given by (18) and S=~iG_1(~
—
~)G.
The integrability conditions ~ following nonlinear equation,
(25) =
~
(e~’/a),,=(ye7a)1,. (28b) It turns out that the system (28) is satisfied by a,fl, y given by (27) and v given by
give us the e~=C1
C0F exp(/ 2 J~ocodt). F1, —
(29)
S,,=(A1 +220A2)S1, ~[S,A2S1,1, + (A2,1, +Pi A2 )S,1,] (26) where square brackets denote the commutator. Taking into account the3 well isomorphism (the known Killing—Cartan formbein tween su (2) and to F the scalar product in F3 and the su (2) corresponds commutator correspond to the vector product in F3) one can see that eq. (26) is identical with (7), where —
In other words, all non-isospectral deformations of the Heisenberg ferrornagnet equation can be transformed into (1) by a change of variables. Therefore we did(as notfar obtain new integrable spin systerns as theessentially Darboux—Bäcklund transformation is concerned). Our approach gives the possibility to find three-dimensional coupling functions f=f(x, y, z, t) for which we can generate solutions of (3). These so-
a = (C 1
—
C0F) /2 p2
/3= (aF,1,),JF1,,
S
(27a)
lutions are characterized by some prescribed sym-
(27b)
metry (given by the coordinate system used). Example. Consider an orthogonal and t-independent coordinate system. One can prove, by compar-
,~
(27c) Thus we come to the conclusion that for a,/3, y given by (27) eq. (7) is integrable. In that case eq. (7) will be called the non-isospectral deformation of the Heisenberg ferromagnet equation. In formulas (27) C0=C0(t), C~=C1(t),F=F(u, 1) and ~ = ~o ( t) can be considered as arbitrary functions (in the last case it isaconsequence of eq. (21) C2 uniquely determined Theischoice F( u, t) = u andby,~= 0 corresponds to the inhomogeneous Heisenberg ferromagnet equation (l)withf(x,t)=C 2, ~=0, C1(t)—xC0(t),andthechoice F(u, t)=u 1=0, C0=—8 corresponds to the cylindrical Heisenberg ferromagnet model (2). It is not difficult to notice that by the the cylindrical change of coordinates r2 = 2x one can transform ferromagnet equation (2) into eq. (1) with f= 2x. There is an important difference between the initialvalue problems of (1) and (2). The first is a full-line problem while the second is a half-line problem. This means there is no one-to-one correspondence between these two initial valueproblems. However, this difference is not relevant in the construction of the Darboux—Bäcklund transformation. The direct calculations show that eq. (7) can be —
142
,~.
ing (6) with (27), that such a system generates an integrable one-dimensional reduction of the threedimensional Heisenberg ferromagnet equation (3) iff ~ is factorised in the following way, .,~/~Di(q1)D2(q2,q3,t), (30) where D1 and D2 are some functions (q’=u). The 2, q3, t), corresponding coupling functions f_—f( q given by to the integrable case, is u, then
f~guur~(u)(at+bw
JDi(v)dv)~
(31)
where are be arbitrary The dependence 2, a, q3 bmay hidden functions. only in g’~. onBelow q we present the list of “integrable” coupling functions corresponding to the simplest coordinate systems. (1) Cartesian coordinates, f=ax+b,
(32a)
(2) Cylindrical coordinates (r2=x2+y2, tg Ø=y/ x), (a)
f=a+b/r2,
(32b)
Volume 149, number 2,3
PHYSICS LETTERS A
17 September 1990
(32c) (b) f= (aØ+ b ) r2, (3) Spherical coordinates (R2=x2+y2+z2, cosd =z/R),
This work was partially supported by the Polish Ministry of National Education (Research Project CPBP 01.03).
(a) f=a/R+b/R4,
(32d)
References
(b) f=(a+bcos0)R2/sin2O.
(32e)
[1] M. Lakshmanan and R.K. Bullough, Phys. Lett. A 80 (1980) 287.
Summary. We have shown that the one-dimensional inhomogeneous Heisenberg ferromagnet equation (1) with f linear in x is the universal integrable (in the sense of the Darboux—Bäcklund transformation) one-dimensional reduction of the three-dimensional inhomogeneous Heisenberg ferromagnet. Any other reduction of (3) can be obtamed from (1) by a change of variables. The initial value problems are, in general, different, which should be taken into account in applying the inverse scattering method. The second result of this paper is a method to generate solutions to the three-dimensional inhomogeneous Heisenberg ferromagnet (3) for some coupling functions f=f(x, y, z, t). Examples of such functions are given by (31) and (32).
[21 R. Balakrishnan, J. Phys. C 15 (1982) LI 305. [3] A. Sym and W. Wesselius, Phys. Lett. A 120 (1987) 183.
[41J. Cie~1iñski,A. Sym and W. Wesselius, Twente University memorandum No. 789. On the geometry of the inhomogeneous Heisenberg ferromagnet: non-integrable
case (May 1989), to be published. 117. [6]J. Cie~1iñski,Geometric method of integration of the [5lM.LakshmananandS.Ganesan,PhysicaAl32(1985)
inhomogeneous Heisenberg chain, in: Proc. Workshop on The nonlinear evolution equation: integrabiity and spectral methods (Como, 1988). [7] A.V. Mikhailov and A.I. Yaremtschuk, JETP Lett. 36 (1982) 78. [81S.P. Burtsev, V.E. Zakharov and A.V. Mikhailov, Theor. Math. Phys. 70 (1987) 323.
[9] V.E. Zakharov and L.A. Takhtajan, Theor. Math. Phys. 38 (1979) 26.
[101M. Ablowitz and H. Segur, Solitons and the scattering transform (SIAM, Philadelphia, 1981).
inverse
143