On canonical variables in integrable models of magnets

ON CANONICAL V ARIABLES IN INTEGRABLE MODELS OF MA GNETS E.SH.GUTSHABASH Institute Resear h for Ph ysis, St.-P etersburg State Univ ersit y , Russia, e-mail: gutshabEG2097.spb.edu 1. INTR ODUCTION The most general form ulation of phenomenologial mo dels of magnets (or spin sys- tems) whi h inludes all the kno wn ompletely in tegrable ones, has the follo wing form: S t = F 0 ( x, y , S , S x , S y , S xx , S y y , S xy , J , u, u x , u y , u xy , α 2 ) , (1 . 1) u xx + α 2 u y y = R 0 ( S , S x , S y ) , where S = S ( x, y , t ) is the magnetization v etor, F 0 ( , ) is a v etor-funtion, u = u ( x, y ) is an auxiliary eld, R 0 ( , ) is a salar funtion, J is a set of onstan ts  haraterizing the magnet, and α 2 = ± 1 . The funtion F 0 usually tak es the form: F 0 = S ∧ δ F ef f δ S + F 1 , (1 . 2) where F ef f is the funtional of the rystal's free energy (throughout the pap er the sym b ol δ /δ stands for the v ariational deriv ativ e). The rst term in the righ t hand side w as suggested b y Landau and Lifshits [1℄ to desrib e the ex hange in terations. The represen tation (1.1)-(1.2) is often inon v enien t for solving problems. One w ould lik e to deal with more tratable forms of the equations (1.1), whi h, in turn, requires in tro dution of new dep enden t v ariables. Apparen tly , su h a v ariable, the stereographi pro jetion, has b een used for the rst time in pap er [2℄ to desrib e the instan ton solutions in the t w o-dimensional O (3) σ -mo del (the 2 D stationary Heisen b erg ferromagnet). Later it w as exploited in v arious situations, see, e. g. [3-5℄. In the presen t pap er w e sho w on examples of three mo dels - the deformed Heisen- b erg, the Landau - Lifshits, and the Ishimori magnets - that it is helpful to in tro due the orresp onding anonial v ariables. In partiular, they allo w to simplify signian tly ertain alulations, as ompared to the usage of the S v ariable, and, more substan tially , to larify a set of questions imp ortan t b oth from ph ysial and mathematial viewp oin ts. Another argumen t in their fa v or is that in these v ariables the mo dels t in a lass of mo dels admitting a dieren tial-geometri in terpretation in tensiv ely studied reen tly [6℄. The mo del of deformed Heisen b erg magnet w as suggested in [7℄ where also an exat solution of it for the ase of trivial ba kground w as obtained b y the in v erse sattering metho d, and the onserv ation la ws w ere alulated. In doing so, it w as sho wn that 1 p erturbations lo alized in the spae are spreaded, that is, the solutions are instable. The gauge equiv alene of this mo del and the nonlinear S hr odinger equation with an in tegral nonlinearit y w as established in [8℄ and [9℄. The matrix Darb oux transform metho d w as applied in the pap er [10℄, where exat solutions of the mo del where alulated on the ba kground of new spiral-logarithmi strutures. The Landau - Lifshits equation is a sub jet of v ast studies. In partiular, the Lax rep- resen tation and onserv ation la ws for it in the ompletely anisotropi ase ha v e rst b een obtained in [11℄, soliton solutions w ere found b y the dressing metho d in [12℄. Hamilton asp ets of the equation w ere analyzed in detail in reen t pap er [13℄. The Ishimori magnet w as also onsidered in man y pap ers. In partiular, series of exat solutions w ere obtained in [14℄ b y the in v erse sattering and ¯ ∂ - dressing metho ds, the Darb oux transform w as applied to it in [15℄ and [16℄ (in [16℄ - on the ba kground of spiral strutures). Notie also the imp ortan t pap er [5℄, where the gauge equiv alene of the Ishimori-I I and Da vy-Stew artson-I I mo dels w as established. The struture of this pap er is as follo ws. In setion 2 w e onsider the deformed Heisen- b erg magnet mo del, dene the anonial v ariables and analyze stabilit y of the solutions. In setion 3 the Landau - Lifshits equation is obtained in terms of the stereographi pro jetion and a stationary v ersion of this equation is studied. Finally , in setion 4 w e dene t w o pairs of anonial v ariables for the Ishimori mo del, re-write the mo del and the Hamiltonian in these v ariables, and alulate the Hamiltonian on some of the simplest kno wn solutions. This is preeded b y a disussion of the ph ysial in terpretation of the mo del. The App endix on tains a Lax pair for an "extended" system of the deformed Heisen b erg magnet mo del. 2. DEF ORMED HEISENBER G MA GNET EQUA TION a). Canoni al variables. Let us onsider the deformed Heisen b erg magnet equation [7℄ 1 : S t = S ∧ S xx + 1 x S ∧ S x . (2 . 1) Here x = p x 2 1 + x 2 2 > 0 , x 1 , x 2 are the Cartesian o ordinates on the plane, S ( x, t ) = ( S 1 , S 2 , S 3 ) , | S | = 1 . The phase spae for this equation is generated b y initial data ( S 1 , S 2 , S 3 ) sub jet to the onstrain t | S | = 1 . The P oisson bra k ets of the anonial v ariables S i in the mo del satisfy the standard relations: { S i ( x ) , S j ( y ) } = − ε ij k S k ( x ) δ ( x − y ) , i, j, k = 1 , 2 . 3 , (2 . 2) where ǫ ij k is the fully an tisymmetri third rank tensor. F or an y t w o funtionals F , G w e then ha v e { F , G } = − Z ∞ 0 ǫ ij k S k δ F δ S i ( x ) δ G δ S j ( x ) dx. (2 . 3) 1 This equation an b e though t of as a ylindrial-symmetri redution of the (2+1)-dimensional non- in tegrable Landau - Lifshits equation, S t = S ∧ △ S . The relation b et w een the latter and the system of oupled nonlinear S hr odinger equations in the dimension (2+1) has b een disussed in detail in [17℄. 2 On taking in to aoun t (2.2)-(2.3), one an represen t the equation (2.1) in the follo wing Hamiltonian form: S t = 1 x { H , S } , (2 . 4) where the Hamiltonian H is giv en b y H = 1 2 Z ∞ 0 x S 2 x dx. (2 . 5) Let us no w dene a new dep enden t omplex-v alued v ariable, w ( x, t ) = S 1 + iS 2 1 − S 3 , (2 . 6) whi h is, at ea h xed momen t of time t , the stereographial pro jetion of the unit sphere on to the omplex plane, w : S 2 → C ∪ {∞} . In terms of this v ariable the equation (2.1) an b e rewritten as iw t = w xx − 2 w 2 x ¯ w 1 + | w | 2 + 1 x w x , (2 . 7) and the P oisson bra k ets orresp onding to (2.2) tak e the form 2 { w ( x ) , w ( y ) } = { ¯ w ( x ) , ¯ w ( y ) } = 0 , { w ( x ) , ¯ w ( y ) } = − i 2 (1 + | w | 2 ) 2 δ ( x − y ) . (2 . 8) The bra k et (2.3) then b eomes { F , G } = − i 2 Z dx (1 + | w ( x ) | 2 ) 2 h δ F δ w ( x ) δ G δ ¯ w ( x ) − δ F δ ¯ w ( x ) δ G δ w ( x ) i , (2 . 9) and the ev olution of the system will b e desrib ed b y the equation iw t = − 1 2 x (1 + | w | 2 ) 2 δ H δ ¯ w ( x ) , (2 . 10) with the Hamiltonian H = 2 Z ∞ 0 x w x ¯ w x (1 + | w | 2 ) 2 dx. (2 . 11) It should b e notied that the follo wing "omplex extension" of the system (2.1) is of in terest of its o wn 3 , ir t = r xx − 2 r 2 x s 1 + r s + 1 x r x , is t = − s xx + 2 s 2 x r 1 + r s − 1 x s x . (2 . 12) F rom (2.8) w e obtain the P oisson bra k ets of v ariables r è s in the form 2 In the deriv ation of (2.8) w e use the relations { S ± ( x ) , S ± ( y ) } = 0 , { S + ( x ) , S 3 ( y ) } = − iS + ( x ) δ ( x − y ) , { S + ( x ) , S − ( y ) } = 2 i S 3 ( x ) δ ( x − y ) , where S ± = S 1 ± i S 2 , and the Leibnits's rule. 3 In absene of the nonlinear omp onen t the seond equation in (2.12) an b e in terpreted as the free Shr odinger equation with an eetiv e mass. It is eviden t then that the rst equation an b e obtained from the seond b y omplex onjugation. 3 { r ( x ) , s ( y ) } = − i (1 + r s ) 2 δ ( x − y ) , { ¯ r ( x ) , ¯ s ( y ) } = i (1 + ¯ r ¯ s ) 2 δ ( x − y ) . (2 . 13) The system (2.12), as w ell as equation (2.7), is ompletely in tegrable (see App endix) and ha v e a Hamiltonian struture with the Hamiltonian H = Z ∞ 0 x r x s x (1 + r s ) 2 dx (2 . 14) and the equations of motion r t = 1 x { H , r } , s t = − 1 x { H , s } , (2 . 15) and an b e onsidered a mo del of the system of t w o oupled deformed Heisen b erg's mag- nets. The P oisson bra k ets (2.13) an b e found from the expression for sympleti t w o-form, Φ = i Z ∞ 0 h dr ∧ ds (1 + r s ) 2 − d ¯ r ∧ d ¯ s (1 + ¯ r ¯ s ) 2 ) i dx, Φ = d ϕ, (2 . 16) where ϕ = − i Z ∞ 0 h ds s (1 + r s ) − d ¯ s ¯ s (1 + ¯ r ¯ s )) i dx, (2 . 17) th us (2.16) and (2.17) agree with the orresp onding expressions obtained in [18℄ for the standard Heisen b erg magnet. Notie also that the equation (2.7) is a bi-Hamiltonian system: i  w ¯ w  t = G 1  δH 1 δw δH 1 δ ¯ w  = G 2  δH 2 δw δH 2 δ ¯ w  , (2 . 18) where H 1 oinides with H giv en b y (2.11), the seond Hamiltonian H 2 reads as H 2 = − i Z ∞ 0 x w x ¯ w − ¯ w x w (1 + | w | 2 ) | w | 2 dx, (2 . 19) and G 1 = G 1 ( w , ¯ w ) , G 2 = G 2 ( w , ¯ w ) are the so-alled Hamiltonian op erators of the form G 1 = 1 x (1 + | w | 2 ) 2  0 − 1 − 1 0  . (2 . 20) An expression for the matrix op erator G 2 an b e obtained from results in pap er [19℄ on the standard Heisen b erg magnet but is to o um b ersome to b e written here. Let us just men tion that its matrix en tries on tain a dieren tial and an in tegral op erator th us rendering it non-lo al. The relations (2.18)-(2.20) mean that the reursion op erator of the equation (2.7) under the assumption that d et G 2 6 = 0 is represen ted in the form R = G 1 G − 1 2 . (2 . 21) 4 Sine (2.7) is a ompletely in tegrable system, it admits innitely man y in tegrals of motion, { I n } ∞ n =1 [7℄, in in v olution, that is, satisfying { I j , I k } = 0 . In turn, this allo ws to obtain hierar hies of the P oisson strutures, I n = RI n − 1 , (2 . 22) and the higher equations of the deformed Heisen b erg magnet ( j = 0 , 1 , ... ; t 0 = t ), iw t j = R j G 2 δ H 2 δ ¯ w . (2 . 23) b). Stability of  ertain solutions of e quation (2.7). The problem of stabilit y of stationary solutions of the equation (2.7) is of in terest sine the equation on tains the indep enden t v ariable x expliitly . T o analyze it, let w = w st + ˜ w . On linearizing (2.7), rst on the trivial ba kground w st = 0 , whi h orresp onds, in terms of the magnetization v etor, to the v etor S = (0 , 0 , 1) , w e obtain: i ˜ w t ( x, t ) = ˜ w xx ( x, t ) + 1 x ˜ w x ( x, t ) . (2 . 24) Supp ose that ( x > 0 ) ˜ w ( x, 0) = ˜ w 0 ( x ) , ˜ w (0 , t ) = ˜ w 1 ( t ) . (2 . 25) Then the equation (2.24) an b e solv ed b y the Laplae transformation in the t v ariable under the additional assumption that | ˜ w ( x, t ) | < M e s 0 t with an M > 0 and s 0 ≥ 0 . Solving the arising equation and p erforming the in v erse transformation w e nd: ¯ w ( x, t ) = 1 2 π i Z a + i ∞ a − i ∞ e p t [ C 0 ( x, p )) J 0 ( p − ip x )] dp, (2 . 26) where J 0 ( . ) is the Bessel funtion, C 0 ( x, p ) = − i Z x 0 e − R x 0 Q ( ξ ) dξ [ Z x 0 ˜ w 0 ( y ) J 0 ( √ − ip y ) e R y 0 Q ( s ) ds dy ] dx, (2 . 27) Q ( x ) = − 2 p − ip (ln J 0 ( p − ip x )) x + 1 x , R e a > 0 , the path of in tegration is an y straigh t line R e p = a > s 0 > 0 , and the in tegral in (2.26) is understo o d in the sense of the prinipal v alue. It is not diult to see that the logarithmi div ergenies arising in the exp onen tials when in tegrating at the lo w er limit in (2.28), anel ea h other. It follo ws from (2.26) that for a xed x the funtion | ˜ w ( x, t ) | gro ws with the t inrease, and, as in [7℄, w e obtain that the solution is unstable 4 : an arbitrary lo alized initial p erturbation of the system an gro w indenitely as the time passes. W e no w pro eed to analyze stabilit y of the stationary state w st = i e iθ ( x ) where θ ( x ) = ln( x ) + θ 0 , θ 0 ∈ R is a onstan t. This solution is an example of a spiral-logarithmi 4 Of ourse, the stabilit y of that linearized "non-autonomous" equation is mean t. 5 struture found in [10℄: S = (sin θ, cos θ , 0) 5 . On linearizing the equation (2.26) on this ba kground, w e ha v e, i ˜ w t ( x, t ) = ˜ w xx ( x, t ) + 1 x ˜ w x − i e − iθ ( x ) x 2 . (2 . 29) This equation only diers from (2.24) b y the presene of a non-homogeneous term. Hene, its general solution is a sum of (2.26) and a partial solution. It follo ws that it will b e unstable as w ell. Notie then, that the equation (2.7) admits a solution p erio di in t of the form w ( x, t ) = W ( x ) e ik t with k a real onstan t, pro vided that the equation 6 W xx − 2 W 2 x ¯ W 1 + | W | 2 + 1 x W x + k W = 0 (2 . 30) has a solution. This suggests that the study of the linearized stabilit y is insuien t. The analysis of the nonlinear stabilit y requires more subtle metho ds [see, e. g. [20℄ and literature ited therein℄. 3. LAND A U-LIFSHITS MA GNET a). Canoni al variables. The fully anisotropi mo del of Landau-Lifshits has the form 7 S t = S ∧ S xx + S ∧ J S , (3 . 1) where J = d iag ( J 1 , J 2 , J 3 ) are diagonal 3 × 3 matries, and J 1 , J 2 , J 3 are parameters of the anisotrop y , J 1 < J 2 < J 3 . The Hamiltonian for (3.1) an b e written in the form, H = 1 2 Z ∞ −∞ ( S 2 x − S J S ) dx, (3 . 2) or, using the v ariable w dened in (2.6), as 8 H = Z ∞ −∞  2( | w | 2 x + α ( w 2 + ¯ w 2 ) − γ | w | 2 ) (1 + | w | 2 ) 2 − β  dx, (3 . 3) where 5 Using (2.11), it is easy to  he k that the Hamiltonian logarithmially div erges on this solution in b oth limits and, th us, requires a regularization. 6 Remo ving the nonlinear term w e obtain here the stationary S hr odinger equation with the Coulom b p oten tial and an eetiv e mass. 7 It is w ell-kno wn [21℄, that this mo del is one of the most general ompletely in tegrable mo dels admitting 2 × 2 -matrix Lax represen tations. 8 W e assume here that w is a slo wly dereasing funtion. In the ase of a dereasing w one should add J 3 = 4 β to the densit y of the Hamiltonian. 6 α = J 2 − J 1 4 , β = J 3 4 , γ = J 3 − J 1 + J 2 2 . (3 . 4) T aking in to aoun t (2.9), from this w e obtain the follo wing equation of motion for Landau-Lifshits magnet mo del 9 , iw t = i { H, w } = − 1 2 (1 + | w | 2 ) 2 δ H δ ¯ w , (3 . 5) or 10 iw t = w xx − 2 ¯ w ( w 2 x + α ) − α w 3 − γ w 1 + | w | 2 − γ w . (3 . 6) Let us onsider an impliation of this form of the equation. Ob vious transformations lead to the follo wing relation whi h on tains the parameter α only , i ( | w | 2 ) t = ( w x ¯ w − w ¯ w x ) x + 2 w 2 ¯ w 2 x − ¯ w 2 w x 2 1 + | w | 2 + 2 α ( w 2 − ¯ w 2 ) . (3 . 7) Letting w = ρe iϕ , ã äå ρ = ρ ( x, t ) , ϕ = ϕ ( x, t ) , ρ, ϕ ∈ R , w e obtain: ( ρ 2 ) t = 2( ρ 2 φ x ) x − 8 ρ 3 ρ x φ x 1 + ρ 2 + 4 αρ 2 sin 2 φ. (3 . 8) Dening the v ariables R = ρ 2 è Q = 2 ρ 2 φ x , w e no w nd the follo wing "onserv ation la w": R t = Q x − 2 Q [ln(1 + R )] x + 4 αR sin( Z x −∞ Q R ) dx. (3 . 9) It is esp eially simple when α = 0 , whi h orresp onds to the anisotrop y of "the easy plan" t yp e. In a w a y similar to (2.18), one an pro due a bi-Hamiltonian struture for (3.6) with H 1 equal to H dened b y (3.3), the Hamiltonian H 2 = Z ∞ −∞ w x ¯ w − ¯ w x w (1 + | w | 2 ) | w | 2 dx, (3 . 10) and the Hamiltonian op erators G 1 = 1 (1 + | w | 2 ) 2  0 − 1 − 1 0  (3 . 11) and G 2 b eing a matrix in tegro-dieren tial op erator [22℄. In terms of the v ariables w è ¯ w the reursion op erator an b e written as follo ws: R = G 1 G − 1 2 . (3 . 12) 9 Notie that, as w ell as in the ase of the deformed Heisen b erg magnet, w e are able to obtain the orresp onding omplex extension (see, [18℄); w e are not going to dw ell on that here. 10 On taking the omplex onjugated equation and negleting the nonlinear omp onen t, one an obtain the nonstationary Shr odinger equation with the p oten tial V = − γ = c onst . 7 It pro dues an hierar h y of the P oisson strutures similar to (2.22) and higher Landau- Lifshits equations similar to (3.5). b). The Disp ersion r elation. Stationary L andau-Lifshits e quation. Linearizing the equation omplex onjugate to (3.6) and  ho osing ¯ w = ¯ w ( x, t ) in the form ¯ w ∼ exp { i ( k x − ω t ) } , w e ha v e, ω = k 2 − γ , (3 . 13) whi h giv es a disp ersion relation for the Landau-Lifshits equation whi h is t ypial for magnets with an ex hange in teration [23℄. In our ase the group and phase v elo ities are giv en b y v g = ∂ ω / ∂ k = 2 k , v ph = ω /k = k − γ /k , resp etiv ely (the latter is innite for k = 0 ), implying that there is a disp ersion in the system. The propagation of a magnetization w a v e in this mo del is p ossible under the ondition k 2 > γ = J 3 − ( J 1 + J 2 ) / 2 > 0 . Letting w = w ( x − µt ) = w ( ξ ) in (3.6) , where µ = c onst is the v elo it y of a stationary prole w a v e, w e obtain the equation 11 w ξ ξ + iµw ξ − 2 ¯ w ( w 2 ξ + α ) − α w 3 − γ w 1 + | w | 2 − γ w = 0 . (3 . 14) F rom this it is not diult to obtain that ( w ξ ¯ w − ¯ w ξ w ) ξ + iµ ( | w | 2 ) ξ − 2 w 2 ξ ¯ w 2 − ¯ w 2 ξ w 2 1 + | w | 2 + 2 α ( w 2 − ¯ w 2 ) = 0 . (3 . 15) Letting w ( ξ ) = ρ e iφ , where ρ = ρ ( ξ ) , φ = φ ( ξ ) , ρ ∈ R + , φ ∈ R , w e then ha v e: 2( ρ 2 ) ξ φ ξ + 2 ρ 2 φ ξ ξ + µ ( ρ 2 ) ξ − 8 ρ 3 ρ ξ φ ξ 1 + ρ 2 + 4 α sin 2 φ = 0 . (3 . 16) Let µ 6 = 0 . Then, ob viously , ρ = c onst, φ = π n/ 2 , n = 0 , ± 1 , ± 2 , ± 3 , satisfy (3.16). F or φ = φ 0 = c onst w e obtain, ρ 2 = ρ 2 0 − (4 α/µ ) sin(2 φ 0 ) ξ , where ρ 0 = c onst ; ïðè ρ = ˜ ρ 0 = c onst (3.16) is redued to the equation of the p endulum: φ ξ ξ + ( 2 α/ ˜ ρ 0 ) sin(2 φ ) = 0 (the existene of other solutions remains an op en problem). Let no w µ = 0 , then from (3.16) it follo ws that ( ρ 2 ) ξ ρ 2 − 4 ρρ ξ 1 + ρ 2 = C 1 , φ ξ ξ + C 1 φ ξ + 2 α sin 2 φ = 0 , (3 . 17) where C 1 is arbitrary onstan t. The rst of these equations an easily b e in tegrated: ρ 1 , 2 ( ξ ) = 1 2 (1 ± p 1 − 4 e − 2( C 1 ξ + C 2 ) ) e C 1 ξ + C 2 , (3 . 18) where C 2 is another arbitrary onstan t (w e assume that e − 2( C 1 ξ + C 2 ) < 1 / 4) , and the seond equation, whi h oinides with the one of the p endulum with the frition 12 , admits, in partiular, solutions of the form φ = π n/ 2 , n = 0 , ± 1 , ± 2 , ± 3 . Th us, solutions of the stationary Landau - Lifshits equation ha v e fairly non-trivial struture in the generi (fully anisotropi) ase. Their further study ould bring a solution 11 Stationary equations of another form for the Landau-Lifshits hierar h y w ere onsidered from the viewp oin t of the Lie-algebrai approa h in [24℄. 12 In the partial ase C 1 = 0 this equation, an ob viously b e in tegrated in terms of the ellipti funtions. 8 to an imp ortan t problem in the theory of dynamial systems - that of onstrution of the phase graph for the equations (3.14) and (3.6) 13 . The same applies to the deformed Heisen b erg magnet from the previous setion. 4. ISHIMORI MA GNET a). Physi al and ge ometri al interpr etations. The Ishimori magnet mo del in terms of the magnetization v etor has the form: S t = S ∧ ( S xx + α 2 S y y ) + u y S x + u x S y , (4 . 1) u xx − α 2 u y y = − 2 α 2 S ( S x ∧ S y ) , (4 . 2) where S ( x, y , t ) = ( S 1 , S 2 , S 3 ) is a three dimensional v etor, | S | = 1 , u = u ( x, y , t ) is an auxiliary salar real-v alued eld, and the parameter α 2 tak es v alues ± 1 . The system is alled the Ishimori-I magnet (MI-I) in the ase α 2 = 1 , the Ishimori-I I magnet (MI-I I) in the ase α 2 = − 1 . Mathematially , ea h of these ases orresp onds to dieren t t yp es of the equations (4.1) and (4.2). The top ologial  harge of the mo del (4.1)-(4.2), Q T = 1 4 π Z R 2 Z S ( S x ∧ S y ) dx dy , (4 . 3) is in v arian t under the ev olution of the system. Sine the homotop y group of the unit 2-sphere π 2 ( ˜ S 2 ) oinides with the group Z of in tegers, the n um b er Q T m ust b e in teger. A ording to (4.3), the salar funtion u = u ( x, y , t ) is related to the densit y of the top o- logial  harge pro dution. The deriv ativ es u x , u y in (4.1) pla y role of frition o eien ts. Th us, (4.1) an b e in terpreted as an equation of fored (b y the frition p o w er) preession of the magnetization v etor, and the system (4.1)-(4.2) is self-onsisten t. F rom the ph ysial viewp oin t, it is easy to see that there is a non-lo al in teration in this system, on top of a lo al (ex hange) one. The me hanism of the former is unlear. Nev ertheless, the study of su h systems is justied sine stable lo alized t w o-dimensional magneti strutures are observ ed in exp erimen ts. An argumen t in fa v or of this assertion is the ab o v e-men tioned gauge equiv alene of the MI-I I mo del and the DS-I I mo del, whi h desrib es quasi-mono  hromati w a v es on the uid surfae [5℄, and also a link found in [26℄ b et w een the MI-I mo del and the nonlinear S hr odinger equation with magneti eld. Also helpful is another, h ydro dynamial, in terpretation of the mo del (4.1)-(4.2). Namely , let u y = − v 1 , u x = v 2 , hene v ( x, y ) = ( v 1 , v 2 ) is the v elo it y eld of a uid. Then the MI mo del an b e rewritten as follo ws: S t + v 1 S x − v 2 S y = S ∧ ( S xx + α 2 S y y ) , (4 . 4) 13 Phase graphs of the equation (3.1) in the ase of partial anisotrop y ha v e b een studied in [25℄. Phase graphs in the fully anisotropi ase ha v e apparen tly not b een onsidered y et. 9 v 2 x + α 2 v 1 y = − 2 α 2 S ( S x ∧ S y ) . If w e dene the stream funtion of the o w, v 1 = − χ 1 y , v 2 = χ 1 x , then the equation (4.2) with α 2 = − 1 (the MI-I I mo del) implies the P oisson equation χ 1 xx + χ 1 yy = 2 S ( S x ∧ S y ) , (4 . 5) that is, the stationary (the time t is a parameter here) v ortiit y equation with a soure in the righ t hand side of the magnitude prop ortional to the densit y of the top ologial  harge pro dution (details on the equation of planar h ydro dynamial v ortex an b e found in [27℄). Let ˜ F ( x, y , t ) = 2 S ( S x ∧ S y ) . On taking one of the expressions of the form ± e ± χ 1 , e χ 1 − e − 2 χ 1 , ± sinh χ 1 , ± cosh χ 1 , ± sin χ 1 , ± cos χ 1 , for ˜ F ( x, y , t ) , w e obtain a losed ompletely in tegrable equation of ellipti t yp e for the funtion χ 1 . The solution of an appropriate b oundary-v alue problem for this equation m ust satisfy the additional ondition 1 8 π Z R 2 Z △ χ 1 ( x, y ) dx dy = N 0 , N 0 ∈ Z , (4 . 6 a ) or ( r = p x 2 + y 2 ) lim r →∞ 1 8 π I ( χ 1 x dy − χ 1 y dx ) = N 0 . (4 . 6 b ) b). New  anoni al variables. Let us no w onsider another anonial v ariables. First, w e pass from the v ariable S to new v ariables p è q ( p, q ∈ R ) in (4.1)-(4.2), setting [28℄: S 3 ( x, y , t ) = p ( x, y , t ) , S + ( x, y , t ) = p 1 − p 2 ( x, y , t ) e iq ( x,y, t ) . (4 . 7) Expressions for P oisson bra k ets of the quan tities p è q follo w diretly from (2.2), on taking in to aoun t that the problem is t w o-dimensional, { p ( r ) , q ( r ′ ) } = δ ( r − r ′ ) , { p ( r ) , p ( r ′ ) } = { q ( r ) , q ( r ′ ) } = 0 , r = ( x, y ) , (4 . 8) and then for the an y t w o funtionals F and G one an obtain: { F , G } = Z R 2 Z [ δ F δ p ( r ) δ G δ q ( r ) − δ F δ q ( r ) δ G δ p ( r ) ] dx dy . (4 . 9) In terms of this v ariables the MI mo del (4.1)-(4.2) an b e rewritten as a Hamiltonian system, q t = δ H δ p = − p xx + α 2 p y y 1 − p 2 − p ( p 2 x + α 2 p 2 y ) (1 − p 2 ) 2 − p ( q 2 x + α 2 q 2 y ) + u y q x + u x q y , p t = − δ H δ q = (1 − p 2 )( q xx + α 2 q y y ) − 2 p ( p x q x + α 2 p y q y ) + u y p x + u x p y , (4 . 10) u xx − α 2 u y y = − 2 α 2 ( p y q x − p x q y ) , 10 and for the top ologial  harge w e will ha v e: Q T = 1 4 π Z R 2 Z ( p y q x − p x q y ) dx dy . (4 . 11) Here the Hamiltonian H has the form 14 : H = H 1 + H 2 , H 1 = 1 2 Z R 2 Z [ p 2 x + α 2 p 2 y 1 − p 2 + (1 − p 2 )( q 2 x + α 2 q 2 y )] dxdy , (4 . 12) H 2 = 1 4 Z R 2 Z [ α 2 A 2 + B 2 ] dx dy , where A = u x , B = − α 2 u y , so that A x + B y = 2 α 2 ( p x q y − p y q x ) ; in this ase it an b e tak e plae the onditions: δ A δ p = C δ y ( y − y ′ ) δ ( x − x ′ ) , δ B δ p = D δ ( y − y ′ ) δ x ( x − x ′ ) , (4 . 13) δ A δ q = E δ y ( y − y ′ ) δ ( x − x ′ ) , δ B δ q = F δ ( y − y ′ ) δ x ( x − x ′ ) , where C , D , E , F are some funtions. Letting D = C, F = E and taking in to aoun t (4.10), w e obtain the follo wing relations on the funtions C = C ( x, y , t ) and E = E ( x, y , t ) (the sym b ol <, > refers to the salar pro dut in R 2 , and T stands for the transp osition): < ∇ u, ( 1 2 α 2 q y + C y , 1 2 α 2 q x − C x ) T > = 0 , (4 . 14) < ∇ u, ( − 1 2 α 2 p y + E y , − 1 2 α 2 p x − E x ) T > = 0 , from this w e nd: C ( x, y , t ) = C 0 ( u ( x, y )) + 1 2 α 2 Z s ( u x q y + u y q x ) ds, (4 . 15) E ( x, y , t ) = E 0 ( u ( x, y )) + 1 2 α 2 Z s ( u x p y + u y p x ) ds, where C 0 , E 0 are arbitrary funtionals, and the in tegration go es along the  harateristi s of the equations (4.14). Assuming that E 0 = C 0 , w e see that the funtional C 0 m ust ob ey an additional ondition: 14 P ap er [29℄ on tains an expression for the Hamiltonian of the so-alled mo died MI dieren t from (4.1) b y the sign in the last but one term. Th us, the Hamiltonian for the mo del (4.1)-(4.2) seems to ha v e b een obtained here for the rst time, b oth in the q , p and w, ¯ w v ariables, the latter b eing dened b elo w. Also, in on trast with the mo died mo del, it is easy to see that it is imp ossible to dene the Clebs h v ariables in our ase. 11 δ C 0 δ u ( δ u δ q − δ u δ p ) = 2 α 2 u xy . (4 . 16) Sine u xy 6 = 0 in the generi ase, from this it follo ws that one more ondition is neessary: δ C 0 /δ u 6 = 0 (if, of ourse, at this δ u/δ q 6 = δ u / δ p ). No w let us pass to the v ariable w in (4.1)-(4.2), dened in (2.6) (assuming that w = w ( x, y , t ) ) 15 : iw t = w xx + α 2 w y y − 2 ¯ w ( w 2 x + α 2 w 2 y ) 1 + | w | 2 + i ( u x w y + u y w x ) , (4 . 17) u xx − α 2 u y y = 4 iα 2 w x ¯ w y − ¯ w x w y (1 + | w | 2 ) 2 . Then for the top ologial  harge w e obtain Q T = − i 2 π Z R 2 Z w x ¯ w y − ¯ w x w y (1 + | w | 2 ) 2 dx dy . (4 . 18) The non-v anishing of the P oisson bra k et for the anonial v ariables w ( x, y ) , ¯ w ( x, y ) omes along as in (2.8): { w ( x, y ) , ¯ w ( x ′ , y ′ ) } = − i 2 (1 + | w | 2 ) 2 δ ( r − r ′ ) , r = ( x, y ) . (4 . 19) This allo ws to rewrite (4.1)-(4.2) in a transparen tly Hamiltonian form: iw t = − 1 2 (1 + | w | 2 ) 2 δ H δ ¯ w . (4 . 20) Here H is the Hamiltonian of the form H = H 1 + H 2 , H 1 = 2 Z R 2 Z w x ¯ w x + α 2 w y ¯ w y (1 + | w | 2 ) 2 dx dy , (4 . 21) H 2 = 1 4 Z R 2 Z [ α 2 u 2 x + u 2 y ] dx dy , and w e assume in the ourse of the deriv ation of the equations for the mo del that the follo wing onditions, analogous to (4.13), are satised: δ u x δ ¯ w = − 4 iw x α 2 (1 + | w | 2 ) 2 δ ( x − x ′ ) δ ( y − y ′ ) , δ u y δ ¯ w = − 4 iw y (1 + | w | 2 ) 2 δ ( x − x ′ ) δ ( y − y ′ ) . (4 . 22) Clearly , all three represen tations of the MI mo del, (4.1)-(4.2), (4.10) è (4.17) are equiv alen t. 15 The reetion ( w, ¯ w ) → ( p, q ) an b y giv en b y relations q = − arctan( i ( w − ¯ w ) / ( w + ¯ w )) , p = ( | w | 2 − 1 ) / (1 + | w | 2 ) . 12 Notie also that, one an dene a "omplex extension" of the system (4.17) analogous to the ones ab o v e. Letting formally w 1 = ¯ w , one obtains, iw t = w xx + α 2 w y y − 2 w 1 ( w 2 x + α 2 w 2 y ) (1 + w w 1 ) 2 + i ( u x w y + u y w x ) , iw 1 t = − ( w 1 xx + α 2 w 1 yy ) + 2 w ( w 2 1 x + α 2 w 2 1 y ) (1 + w w 1 ) 2 + i ( u x w 1 y + u y w 1 x ) , (4 . 23) u xx − α 2 u y y = 4 iα 2 w x w 1 y − w 1 x w y (1 + w w 1 ) 2 . This system an b e in terpreted as a mo del of t w o oupled Ishimori magnets. Non trivial P oisson bra k ets follo w from (4.19): { w ( x, y ) , w 1 ( x ′ , y ′ ) } = − i 2 (1 + w w 1 ) 2 δ ( r − r ′ ) , { ¯ w ( x, y ) , ¯ w 1 ( x ′ , y ′ ) } = i 2 (1 + ¯ w ¯ w 1 ) 2 δ ( r − r ′ ) , (4 . 24) and the "top ologial  harge" of this mo del is 16 Q T = − i 2 π Z R 2 Z w x w 1 y − w 1 x w y (1 + w w 1 ) 2 dx dy . (4 . 25) The equations of motion (4.23) are Hamiltonian: iw t = − 1 2 (1 + w w 1 ) 2 δ H δ w 1 , iw 1 t = 1 2 (1 + w w 1 ) 2 δ H δ w , (4 . 26) where H = H 1 + H 2 , H 1 = 2 Z R 2 Z w x w 1 x + α 2 w y w 1 y (1 + w w 1 ) 2 dx dy , (4 . 27) H 2 = 1 4 Z R 2 Z [ α 2 u 2 x + u 2 y ] dx dy , and w e supp ose that δ u x δ w 1 = − 4 iw y α 2 (1 + w w 1 ) 2 , δ u y δ w 1 = − 4 iw x (1 + w w 1 ) 2 , (4 . 27 a ) δ u x δ w = 4 iw 1 y α 2 (1 + w w 1 ) 2 , δ u y δ w = 4 iw 1 x (1 + w w 1 ) 2 . Returning to (4.17), w e in tro due the omplex o ordinates z = x + iy , ¯ z = x − iy , so that ∂ z = 1 / 2( ∂ x − i∂ y ) , ∂ ¯ z = 1 / 2( ∂ x + i∂ y ) , dx dy = ( i/ 2 ) dz ∧ d ¯ z and rewrite (4.18) and (4.21) in terms of these v ariables. i). Let α 2 = 1 , that is, the MI-I mo del is onsidered. In this ase w e obtain: 16 In general, w 1 6 = ¯ w , th us Q T an b e non-in teger and ev en omplex. Su h a situation, inluding an in terpretation of the quan tit y Q T , should b e onsidered separately . 13 iw t = 4 w z ¯ z − 8 w z w ¯ z 1 + | w | 2 ¯ w − 2( w z u z − w ¯ z u ¯ z ) , (4 . 28) u z z + u ¯ z ¯ z = 4 w z ¯ w ¯ z − w ¯ z ¯ w z (1 + | w | 2 ) 2 . The top ologial  harge is giv en b y Q T = i 2 π Z Z w z ¯ w ¯ z − w ¯ z ¯ w z (1 + | w | 2 ) 2 dz ∧ d ¯ z , (4 . 29) and the Hamiltonian is H = 2 i Z Z w z ¯ w ¯ z + w ¯ z ¯ w z (1 + | w | 2 ) 2 dz ∧ d ¯ z + i 2 Z Z u z u ¯ z dz ∧ d ¯ z . (4 . 30) ii). Let α 2 = − 1 . W e then ha v e the MI-I I mo del, iw t = 2( w z z + w ¯ z ¯ z ) − 4 w 2 z + w 2 ¯ z 1 + | w | 2 ¯ w − 2( w z u z − w ¯ z u ¯ z ) , (4 . 31) u z ¯ z = − 2 w z ¯ w ¯ z − w ¯ z ¯ w z (1 + | w | 2 ) 2 . The expression for the top ologial  harge oinides with (4.29), and for the Hamiltonian w e ha v e: H = 2 i Z Z w z ¯ w z + w ¯ z ¯ w ¯ z (1 + | w | 2 ) 2 dz ∧ d ¯ z − i 4 Z Z ( u 2 z + u 2 ¯ z ) dz ∧ d ¯ z . (4 . 32) Notie also that the Hamiltonian of the MI-I magnet and its top ologial  harge are related, as follo ws from (4.29) and (4.30), b y the inequalit y of Bogomol'n yi, whi h is a lo w er estimate for the Hamiltonian taking in to aoun t all the dynamial ongurations. Namely , H ≥ 4 π Q T . (4 . 33) Comparing (4.29) and (4.32) one an see that for the MI-I I mo del su h an estimate do es not exist. ). Hamiltonians and top olo gi al har ges for some of the simplest solutions. 17 Equations (4.1)-(4.2) an b e in terpreted as the ompatibilit y onditions for the follo w- ing o v erdetermined matrix systems on the funtion Ψ = Ψ( x, y , t ) : Ψ y = 1 α S Ψ x , (4 . 34) Ψ t = − 2 iS Ψ xx + Q Ψ x , (4 . 35) 17 Similar alulations w ere giv en in [29℄ for the ase of the mo died MI and ertain other systems. 14 where Q = u y I + α 3 u x S + iα S y S − iS x , Ψ = Ψ( x, y , t ) ∈ M at (2 , C ) , S = P 3 i =1 S i σ i , σ i are the standard P auli matries, I is the unit 2 × 2 matrix. By the denition, the S matries ha v e the prop erties: S = S ∗ , S 2 = I , det S = − 1 , S p S = 0 (the asterisk stands for the Hermitian onjugation). F or a future referene, let us pro vide an expression for the S matries in terms of the w v ariable: S = | w | 1 − 1 1+ | w | 2 2 ¯ w 1+ | w | 2 2 w 1+ | w | 2 − | w | 2 − 1 1+ | w | 2 ! , (4 . 36) and onsider some of the simplest examples of alulations. 1. Let α 2 = − 1 in (4.17), that is, the MI-I I mo del is onsidered. As w as sho wn in [26℄, the onditions w ¯ z = 0 , w z = 0 , (4 . 37) are then ompatible with (4.31). The rst and seond onditions here mean the presene of instan ton and an ti-instan ton setors, resp etiv ely , in the MI-I I mo del. Consider the instan ton setor assuming that w ( z ) = (( z − z 0 ) /λ ) n [30℄ 18 , where n ∈ Z + , λ ∈ C (the z 0 and λ  haraterize, resp etiv ely , the p osition and size of the instan ton). A alulation b y the relation (4.39) giv es: Q T = i 2 π Z Z | w z | 2 (1 + | w | 2 ) 2 dz ∧ d ¯ z = n, (4 . 38) and H 1 = 0 b y (4.32). T o nd H 2 , one has to kno w the funtion u . Using the seond equation in (4.31) with w ¯ z = 0 and returning to the Cartesian o ordinates, w e obtain ( z 0 = x 0 + iy 0 ), △ u = − 8 λ 2 n [( x − x 0 ) 2 + ( y − y 0 ) 2 ] n [ λ 2 n + [( x − x 0 ) 2 + ( y − y 0 ) 2 ] n ] 2 , (4 . 39) whi h implies that 19 u ( x, y ) = − 8 λ 2 n Z R 2 Z G 0 ( x − x ′ , y − y ′ ) [( x ′ − x 0 ) 2 + ( y ′ − y 0 ) 2 ] n [ λ 2 n + [( x ′ − x 0 ) 2 + ( y ′ − y 0 ) 2 ] n ] 2 dx ′ dy ′ , (4 . 40) where G 0 ( x, y ) = (1 / 2 π ) ln( x 2 + y 2 ) is the Green funtion of the t w o-dimensional Laplae op erator. Th us, the energy of the instan ton solution on the formal lev el is giv en b y 20 H = 1 4 Z R 2 Z ( u 2 y − u 2 x ) dx dy . (4 . 41) The whole instan ton setor is then split in to disjoin t lasses ea h orresp onding to the relev an t v alue of the Q T quan tit y . 18 The  hoie of a more general solution, sa y , in the form of the Bela vin-P oly ak o v instan ton (linear- frational funtion with omplex p oles) [2℄, unfortunately , signian tly ompliates the alulations. 19 The follo wing in tegral an b e simplied b y a  hange of v ariables and subsequen t on tour in tegration, but the remaining in tegral, apparen tly , annot b e alulated expliitly . 20 Ob viously , the Hamiltonian is p ositiv e in the domain where | u y | > | u x | . 15 2. Let us onsider the MI-I mo del ( α 2 = 1 ) and sho w that instan ton solutions exist in there as w ell 21 , 22 . Indeed, for w ¯ z = 0 the system (4.28) is redued to the follo wing one, w t = − 2 u z w z , u z z + u ¯ z ¯ z = 4 w z ¯ w ¯ z (1 + | w | 2 ) 2 . (4 . 42) Dieren tiating the rst relation in ¯ z , w e obtain that u z ¯ z = 0 , whene the ompatibilit y is a hiev ed if u xx = 4 w z ¯ w ¯ z (1 + | w | 2 ) 2 , (4 . 43) or u ( x, y , t ) = 4 Z x −∞ dx ′ Z x ′ −∞ dx ′′ | w z | 2 (1 + | w | 2 ) 2 + f 0 ( y , t ) x + c 1 , (4 . 44) where c 1 is an arbitrary onstan t, and f 0 ( ., . ) is an arbitrary funtion. F or the instan ton solution w ( z ) of the same form as in the previous ase the n um b er Q T = 0 and the funtion u ( x, y , t ) = 4 n 2 λ 2 n Z x −∞ dx ′ Z x ′ −∞ dx ′′ [( x ′′ − x 0 ) 2 + ( y − y 0 ) 2 ] n − 1 [ λ 2 n + ( | x ′′ − x 0 | 2 + | y − y 0 | 2 ) n ] 2 + f 0 ( y , t ) x + c 1 . (4 . 45) The expression for the Hamiltonian tak es the form, H = 4 π n + 1 4 Z R 2 Z ( u 2 x + u 2 y ) dx dy . (4 . 46) 3. Let us alulate the top ologial  harge and the Hamiltonian for a solution of the form of a spiral struture. Namely , let S = (0 , sin Φ 1 , cos Φ 1 ) , where Φ 1 = δ 0 t + α 0 x + β 0 y + γ 0 , α 0 , β 0 , γ 0 , δ 0 ∈ R are parameters, that is, the solution is a t w o-dimensional spiral struture [16℄; then, aording to (2.6) (see also (4.36)), w ( z , ¯ z ) = i ta n(Φ 1 / 2) , Φ 1 = δ 0 t + αz + ¯ α ¯ z + γ 0 , α = α 0 / 2 + β 0 / (2 i ) . It follo ws from (4.29) that Q T = 0 . T o determine the funtion u = u ( x, y , t ) , one has to substitute the funtion w ( z , ¯ z ) ) in the equations (4.28), (4.31), whi h giv es t w o linear equations for u . Assuming their ompatibilit y and in tegrating, w e nd ( α 2 = − 1 )[16℄: u ( x, y ) = g 0 ( y + β 0 α 0 x ) + Z s g 1 ( y ( s ′ ) + β 0 α 0 x ( s ′ ) , t ) ds ′ , (4 . 47) where g 0 , g 1 are arbitrary funtions su h that g 0 is onstan t on the  harateristi y + ( β 0 /α 0 ) x = c onst , and s is the  harateristi tak en to b e the in tegration path. Similarly , for α 2 = 1 w e ha v e: 21 This is not surprising, alb eit apparen tly w en t unnotied in the literature, giv en that in the "stati limit" the MI-I mo del turns in to the ellipti v ersion of the nonlinear O (3) σ -mo del for whi h the instan ton solutions w ere onstruted initially . Notie also that the mo del w as solv ed b y the in v erse sattering metho d in [31℄-[32℄. 22 F rom the viewp oin t of the higher-dimensional in v erse sattering metho d and the dressing pro edures for solutions, the  harateristi v ariables ξ = ( y − x ) / 2 and η = ( x + y ) / 2 [14℄, [15℄ are more natural than z and ¯ z . 16 u ( x, y ) = g 2 ( y − β 0 α 0 x ) + Z s 1 g 3 ( y ( s ′ ) − β 0 α 0 x ( s ′ ) , t ) ds ′ , (4 . 48) where g 2 , g 3 are arbitrary funtions, and g 2 is onstan t on the  harateristi s 1 y − ( β 0 /α 0 ) x = c onst . The substitution w = w ( z , ¯ z ) in (4.30) and (4.32) (in b oth MI-I and MI-I I ases) leads to div ergene of the Hamiltonian H 1 , and, therefore, that of the Hamiltonian H as a whole, sine the funtional H 2 is nite. 4. As w as rst sho wn in [33℄ (see also [14℄, [16℄), in the reetionless setion of the MI-I I mo del the system (4.34)-(4.35) an b e written in the form ˜ Ψ ¯ z = 0 , ˜ Ψ t + 2 i ˜ Ψ z z = 0 . (4 . 49) In turn, the latter system has w ell-kno wn p olynomial solutions desribing v ortex states, ( ˜ Ψ = { ˜ Ψ ij } , i, j = 1 , 2 , ˜ Ψ 22 = ¯ ˜ Ψ 11 , ˜ Ψ 12 = − ¯ ˜ Ψ 21 )[33℄: ˜ Ψ 11 ( z , t ) = N 1 X j =0 X m +2 n = j a j m ! n ! ( − 1 2 z ) m ( − 1 2 it ) n , (4 . 50) ˜ Ψ 21 ( z , t ) = M 1 X j =0 X m +2 n = j b j m ! n ! ( − 1 2 z ) m ( − 1 2 it ) n , where N 1 is an in teger, M 1 = N 1 − 1 , a j , b j are omplex n um b ers, and the inner summations run o v er all m, n ≥ 0 su h that m + 2 n = j . In partiular, in this simplest ase N 1 = 1 it follo ws that ˜ Ψ 11 = a 0 + a 1 z , ˜ Ψ 21 = b 0 . (4 . 51) Let us emplo y no w a dressing (sa y , the Darb oux dressing [16℄) relation for the matrix S 23 , ˜ S = ˜ Ψ S (1) ˜ Ψ − 1 , where S (1) is the initial solution of the system (4.1)-(4.2), assuming that S (1) = σ 3 . This leads to the so-alled "one-lump" stationary solution, whi h w e write here in terms of the stereographi pro jetion, w ( z , ¯ z ) = ¯ b 0 ¯ a 1 ( ¯ z − ¯ z 0 ) + ¯ d 0 , (4 . 52) where d 0 = a 0 − a 1 z 0 , z 0 is the o ordinate of the v ortex en ter on the omplex plan. It is kno wn [33℄, that Q T = 1 for su h a solution. Let us alulate the funtion u = u ( x, y ) . In the ase under onsideration the seond equation in (4.31) is redued to the follo wing, u z ¯ z = 2 | a 1 | 2 | b 0 | 2 ( | a 1 ( z − z 0 ) + d 0 | 2 + | b 0 | 2 ) 2 . (4 . 53) F rom this w e nd 23 Its struture is iden tial, at least, for all the mo dels of magnets treated here and all kno wn metho ds of their solutions. 17 u ( x, y ) = 2 | a 1 | 2 | b 0 | 2 Z R 2 Z G 0 ( x − x ′ , y − y ′ ) 1 ( | a 1 ( z ′ − z 0 ) + d 0 | 2 + | b 0 | 2 ) 2 dx ′ dy ′ . (4 . 54) T aking in to aoun t that H 1 = 0 , w e no w obtain from (4.32) the Hamiltonian of the one-lump solution in the form H = Z R 2 Z ( u 2 y − u 2 x ) dx dy . (4 . 55) The left hand side is p ositiv e in the planar domain where | u y | > | u x | in the same w a y as in (4.41). 5. CONCLUSION The results for the Ishimori magnet sho w, in partiular, that the Hamiltonians and top ologial  harges annot alw a ys b e alulated analytially ev en for the simplest so- lutions, and n umeris are required for sp ei Cau h y problems. In this resp et, it is esp eially in teresting, in our opinion, to  he k the h yp othesis [16℄ of p ossibilit y of a phase transition in the mo del whi h in v olv es a  hange of top ology and symmetry prop erties of the system. Conerning the "extended" systems (2.12) and (4.23), w e w ould lik e to p oin t out that if the initial systems are gauge equiv alen t to the nonlinear S hr odinger equation with an in tegral nonlinearit y [8,9℄ (the Da vy-Stew artson-I I mo del [5℄ in the IM-I I ase), then it is in teresting and imp ortan t to nd ob jets gauge equiv alen t to the extended systems. Ov erall, the represen tations onsidered in this pap er an, hop efully , b e useful in study- ing other (1+1) (and more realisti (2+1) and (3+1)) dimensional mo dels of mag- nets, σ -mo dels and  hiral elds, inluding nonin tegrable ases. The author is indebted to P . Kulish for supp ort. APPENDIX W e pro vide here an expression for a Lax pair for the system (2.12). First, dene the funtion D = D ( x, t ) ∈ M at (2 , C ) , x > 0 , of the form D ( x, t ) =  sr − 1 1+ sr 2 s 1+ sr 2 r 1+ sr − sr − 1 1+ sr  . ( A. 1) The matrix D has the follo wing prop erties: D 2 = I , S p D = 0 , d et D = − 1 . Ho w ev er, unlik e the matrix S , it is not Hermitian. A straigh tforw ard alulation sho ws that (2.12) is the ompatibilit y ondition for the follo wing o v erdetermined linear system of equations: 18 Ψ x = − i 2 λD Ψ , Ψ t = ( λ 2 D x D + i 2 λ 2 D )Ψ , ( A. 2) where Ψ = Ψ( x, t, λ ) ∈ M at (2 , C ) , λ = λ ( x, t, u ) , u ∈ C . This means that the parameter u pla ys the role of a "hidden" sp etral parameter, so that the onditions λ x = λ x , λ t = − 2 λ 2 x , ( A. 3) or λ = x 2( t + u ) , ( A. 4) are fullled, and the matries D satisfy the equation iD t = 1 2 [ D , D xx ] − 1 x D x D . ( A. 5 ) Th us, w e ha v e a non-isosp etral deformation of the asso iated linear system (the ase of a single deformed Heisen b erg magnet w as onsidered in [8℄). Notie also that so far the in v erse sattering transform metho d has not b een applied to su h systems at the full sale. REFERENCES 1. L.D.L andau, - Colleted P ap ers, (1969), 128-143, Moso w, Nauk a. 2. A.A.Belavin and A.M.Polyakov, - JETP Letters, 22 (1975), 503-506. 3. A.M.Per elomov, - Usp ekhi Fizi h. Nauk, 134 (1990), 577-609. 4. Ju.M.Izumov and Ju.N.Skriabin, - Statistial me hanis of the magnet- nonordered medium, (1987), Moso w, Nauk a. 5. V.D.Lip ovskii and A.V.Shir okov, - F unt. An. and Appl, 23 (1989), 65. 6. O.I.Mokhov, - Sympleti and P oisson geometry on the spaes of the lo ops of the smo oth manifolds and in tegrable equations (2004), - Insti- tute of the Computers' s resear hes, Moso w-Igevsk. 7. A.V.Mikhailov and A.I.Jar emhuk, - JETP Letters, 36 (1982), 78. 8. S.P.Burtsev, V.E.Zakhar ov and A.V.Mikhailov, - TMP , 70 (1987), 323. 9. K.Porsezian and M.L akshmanan, - J.Math.Ph ys., 32, (1991), 2923. 10. E.Sh.Gutshab ash, - JETP Letters, 73 (2001), 317-319. 11. E.K.Sklyanin, - Preprin t No E-3-79 LOMI (1979), Leningrad. 19 12. A.I.Bob enko, - Zapiski Nau hn y h Seminaro v POMI, 123, (1983), 58. 13. Niam-Ning Huang, Hao Cai and ot., - arxiv:nlin.SI/0509042 . 14. B.G.Konop elhenko, - Solitons in Multidimensions, (1993), W orld Si- en ti. 15. E.Sh.Gutshab ash, - Zapiski Nau hn y h Seminaro v POMI, 17 (2002), 155-168. 16. E.Sh.Gutshab ash, - JETP Letters, 78 (2003), 1257. 17. Qing Ding, - arxiv.math.DG/0504288 . 18. A.N.L eznov and A.V.R azumov, - J.Math.Ph ys., 35 (1994), 1738. 19. E.V.F er ap ontov, - TMP , 91 (1992), 452. 20. V.G.Mahan 'kov, Ju.P.Ryb akov and V.I.Sanuk, - Usp ekhi Fizi h. Nauk, 162 (1994), 121-148. 21. L.A.T akhtadjan and L.D.F adde ev, - Hamiltonian approa h in theory of Solitons (1986), Moso w, Nauk a. 22. E.Bar ouh, A.F okas and V.Pap age or giou, - J.Math.Ph ys., 29 (1988), 2628. 23. E.M.Lifshits and L.P.Pitaevskii, - Statistial Ph ysis, part 2 (1978), Moso w, Nauk a. 24. V.G.Bariakhtar, E.D.Belokolos and P.I.Golo d, - In: Mo dern Problems in Magnetism, (1986), 30, Kiev, Nauk o v a dumk a. 25. V.M.Ele onskii, N.N.Kir ova and N.E.Kulagin, - JETP , 79 (1980), 321. 26. V.G.Mikhalev, - Zapiski Nau hn y h Seminaro v POMI, 189 (1991), 75. 27. E.Sh.Gutshab ash, - Zapiski Nau hn y h Seminaro v POMI, 19, (2006), 119-133. 28. H.C. F o ge dby, - J.Ph ys.A: Math.Gen., 13 (1980), 1467-1499. 29. L.Martina, G.Pr olo, G.Soliani and L.Solombrino, - Ph ys. Rev.B, 49 (1994), 12915. 30. R.R ajar aman, - Solitons and Instan tons (1982), North-Holland Pub- lishing Compan y , Amsterdam-New Y ork-Oxford. 31. E.Sh.Gutshab ash and V.D.Lip ovskii, - TMP , 90 (1992), 175. 32. G.G.V arzugin, E.Sh.Gutshab ash and V.D.Lip ovskii, - TMP , 104 (1995), 513. 33. Y.Ishimori, - Progr.T eor.Ph ys, 72 (1984), 33. 20