Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation
aa r X i v : . [ m a t h - ph ] O c t Nonlocal gauge equivalence
Nonlocal gauge equivalence: Hirota versus extendedcontinuous Heisenberg and Landau-Lifschitz equation
Julia Cen • , Francisco Correa ◦ and Andreas Fring • • Department of Mathematics, City, University of London,Northampton Square, London EC1V 0HB, UK ◦ Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile,Casilla 567, Valdivia, ChileE-mail: [email protected], [email protected], [email protected]
Abstract:
We exploit the gauge equivalence between the Hirota equation and the ex-tended continuous Heisenberg equation to investigate how nonlocality properties of onesystem are inherited by the other. We provide closed generic expressions for nonlocalmulti-soliton solutions for both systems. By demonstrating that a specific auto-gaugetransformation for the extended continuous Heisenberg equation becomes equivalent toa Darboux transformation, we use the latter to construct the nonlocal multi-soliton so-lutions from which the corresponding nonlocal solutions to the Hirota equation can becomputed directly. We discuss properties and solutions of a nonlocal version of the nonlo-cal extended Landau-Lifschitz equation obtained from the nonlocal extended continuousHeisenberg equation or directly from the nonlocal solutions of the Hirota equation.
1. Introduction
Traditionally the study of solutions to nonlinear wave equations has mainly focused onsystems that involve local fields that all depend on a single point in space-time. However,there are many well-known phenomena in nature related to events that appear to be cor-related to each other even though they are separated in a spacelike or timelike fashion.In quantum mechanics entanglement, see e.g. [1], is a well studied phenomenon that insome settings seems to exclude the possibility of signalling [2] and can be implementedeven for many particle systems [3]. There are, however, also physical phenomena that areof a nonlocal nature that are describable with nonlinear wave equations, such as nonlocalrogue waves [4, 5], weather forecast models in which nonlocality is caused by feedback loops[6, 7], gravitational waves [8], etc.Recently a simple principle was identified [9, 10] that introduces nonlocality into non-linear integrable systems in a systematic and mathematically well-defined manner. This onlocal gauge equivalence is achieved by exploring various versions of PT -symmetry present in the zero curvaturecondition that relates fields in the theory to each other in a nonlocal fashion. One par-ticular type of these new nonlocal nonlinear Schr¨odinger equation has attracted a lot ofattention [11, 12, 13, 14, 15, 16, 17, 18, 19]. These studies were extended to other types ofsystems, such as Fordy-Kulish equations [20], Davey-Stewartson equations [21, 22], Sasa-Satsuma equations [23], Kadomtsev-Petviashvili equations [24] and Korteweg de-Vries sys-tems [25, 26, 27]. Here we will build on a particular case of the various nonlocal versionsof the Hirota equation [28]. In the local case the extension from the nonlinear Schr¨odingerequation to the Hirota equation is suggested by experiments in the high-intensity and shortpulse subpicosecond regime [29, 30] where the accurate description of the former equation[31] breaks down. The nonlocality is known to be implementable by the applications ofvarious variants of PT -symmetry [32] to the equations resulting from an AKNS zero cur-vature construction [33]. On the other hand one may also directly decompose the fieldsin some local systems into field depending of different points in space-time and obtainnonlocal systems in this manner, as described in [25, 26, 27]. Here we explore the possi-bility to exploit the gauge equivalence of two systems and investigate how the nonlocalityproperties of one system is inherited by the other. As concrete systems we investigate thegauge equivalent pair of the extended versions of the continuous limit of the Heisenbergequation (ECHE) [34, 35, 36, 37, 38] and the extended Landau-Lifschitz equations (ELLE)[39, 40]. The local version of the original Landau-Lifschitz equation famously describes theprecession of the magnetization in a solid when subjected to a torque resulting from aneffective external magnetic field. Various extended versions have been proposed, such asfor instance the Landau-Lifshitz-Gilbert equation [41] to take damping into account. Thenonlocal versions of these equation studied here provide further extensions with complexcomponents. We will see how the nonlocality may be incorporated most naturally into apair of auxiliary equations occurring this setting.We will also show how the nonlocality is implemented into two standard types ofsolution procedures for nonlinear systems, Hirota’s direct method and the method of usingDarboux transformations. Similarly as in [28] we find new types of solutions in the nonlocalsetting which have no counterpart in the local case.Our manuscript is organized as follows: In section 2 we introduce our three systems thenonlocal Hirota equation, the extended versions of the continuous limit of the Heisenbergequation and the extended Landau Lifschitz equations, and explain how they are relatedto each other. In section 3 we explain in general how specific choices for the gauge trans-formations can become equivalent to Darboux transformations that when iterated may beused to construct multi-soliton solutions. We then utilize this scheme to derive explicit ex-pressions for the multi-soliton solutions to the continuous limit of the Heisenberg equation.These solutions may then be used to calculate directly solutions to the Hirota equationas explained in section 4. In section 5 we discuss how to obtain nonlocal multi-solitonsolutions to the extended Landau-Lifschitz equation. We provide two alternative ways toachieve this, directly via the nonlocal solutions of the Hirota equation or by implement-ing the nonlocality on some auxiliary equations that emerge in the solution process of thecontinuous limit of the Heisenberg equation. Our conclusions are stated in section 6.– 2 – onlocal gauge equivalence
2. Nonlocal gauge equivalence
Many integrable systems are related to each other by means of gauge transformations, oftenin an unexpected way. Such type of correspondences can be exploited to gain insight intoeither system from its gauge partner, for instance by transforming solutions of one systemto solutions of the other. Often this process can only be carried out in one direction. Onemay also consider auto-gauge transformation from a system to itself, which when iteratedcan be used to generate new types of solutions, similar to Darboux or auto-B¨acklundtransformations. In general, we consider here two systems whose auxiliary functions Ψ and Ψ are related to each other by means of a gauge field operator G as Ψ = G Ψ .Formally the system can be cast into two gauge equivalent zero curvature conditions forthe two sets of two operators, U , V and U , V , together with their equivalent two linearfirst order differential equations involving the auxiliary function Ψ and Ψ ∂ t U i − ∂ x V i + [ U i , V i ] = 0 ⇔ Ψ i,t = V i Ψ i , Ψ i,x = U i Ψ i i = 1 , . (2.1)Given the transformation from Ψ to Ψ , the operators U , V and U , V are related as U = GU G − + G x G − , and V = GV G − + G t G − . (2.2)The relations (2.1) and (2.2) are entirely generic providing a connection between two typesof integrable systems, assuming the invertible gauge transformation map G exists. Specificsystems are obtained by concrete choices of the two sets of two operators U , V and U , V . Let us first specify the system 1, by taking U and V to be of the form U = A + λA , V = B + λB + λ B + λ B , (2.3)where A = q ( x, t ) r ( x, t ) 0 ! , A = − i i ! = − iσ , (2.4) B = iα (cid:2) σ ( A ) x − σ A (cid:3) + β (cid:2) A + ( A ) x A − A ( A ) x − ( A ) xx (cid:3) , (2.5) B = 2 αA + 2 iβσ (cid:2) ( A ) x − A (cid:3) , (2.6) B = 4 βA − iασ , (2.7) B = − iβσ , (2.8)with σ i , i = 1 , , λ the spectral parameter and α, β arereal constants. Using the explicit expressions (2.3)-(2.8) the zero curvature condition (2.1)becomes equivalent to the Hirota system [42, 28] for the fields q ( x, t ) and r ( x, t ) as q t − iαq xx + 2 iαq r + β [ q xxx − qrq x ] = 0 , (2.9) r t + iαr xx − iαqr + β ( r xxx − qrr x ) = 0 . (2.10)– 3 – onlocal gauge equivalence These equations may be viewed with q ( x, t ) and r ( x, t ) as entirely independent functions,but most commonly one imposes the relation r ( x, t ) = q ∗ ( x, t ), such that the two equationsbecome their mutual conjugates and are therefore essentially reduced to one equation only- the Hirota equation. Recently [28] alternative possibilities that exploit PT -symmetryhave been proposed, such as taking r ( x, t ) = κq ∗ ( − x, t ), r ( x, t ) = κq ∗ ( x, − t ), r ( x, t ) = κq ∗ ( − x, − t ), r ( x, t ) = κq ( − x, t ), r ( x, t ) = κq ( x, − t ) or r ( x, t ) = κq ( − x, − t ) with κ ∈ R and a suitable adaptation of the parameters α and β . As for these type of choices theequations contain fields that depend simultaneously on x and − x , and/or t and − t , theyare referred to as nonlocal. These type of novel variants of integrable systems are the mainfocus of this manuscript. It was shown in [28] that the different versions display quitedistinct and varied behaviour and therefore deserve to be investigated in their own right.However, in what follows we will exclusively focus on the complex parity extended versioncorresponding to the choice r ( x, t ) = κq ∗ ( − x, t ) together with β = iδ , δ ∈ R , and refer toit as the nonlocal case throughout the manuscript. The treatment of the other cases goesalong the same lines, but will not be discussed here. Having committed to a fixed form of the system 1, we elaborate next on a more preciseform of the system 2 following from that concrete choice. Employing the expansion (2.3),we obtain from (2.2) the expressions U = − iλG − σ G, V = λG − B G + λ G − B G + λ G − B G (2.11)together with G x = A G, and G t = B G. (2.12)With given A and B , it is the solution for these two equations in (2.12) that determinesthe precise form of the gauge map G for a particular set of models.An interesting and universally applied equation emerges when we use the gauge field G to define a new field operator S := G − σ G . (2.13)The following properties follow directly from above S = 1 , S x = 2 G − σ A G, SS x = − S x S = 2 G − A G, [ S, S xx ] = 2 (cid:0) SS xx + S x (cid:1) . (2.14)Next we notice that instead of expressing the operators U and V in terms of the gaugefield G , one can express them entirely in terms of the operator S as U = − iλS, V = α (cid:0) λSS x − λ iS (cid:1) + β (cid:20) λ (cid:18) i SS x + iS xx (cid:19) + λ SS x − λ iS (cid:21) . (2.15)Using this variant we evaluate the zero curvature condition (2.1) with (2.11) and theidentities (2.14), to obtain the equation of motion for the S -operator S t = iα (cid:0) S x + SS xx (cid:1) − β (cid:20) (cid:0) SS x (cid:1) x + S xxx (cid:21) (2.16)= i α [ S, S xx ] − β (cid:0) S x + S [ S, S xxx ] (cid:1) . (2.17)– 4 – onlocal gauge equivalence For β = 0 this equation reduces to the well-known continuous limit of the Heisenberg spinchain [34, 35, 36, 37, 38] and for β = 0 to the first member of the corresponding hierarchy[43]. We refer to this equation as the extended continuous Heisenberg (ECH) equation.The equation (2.16) is rather universal as it also emerges for other types of integrablehigher order equations of nonlinear Schr¨odinger type, such as the modified Korteweg-deVries equation [44, 45] or the Sasa–Satsuma equation [44, 46]. The distinction towardsspecific models of this general type is obtained by specifying the gauge map G .Given the above gauge correspondence one may now obtain solutions to the nonlinearequations of a member of the nonlinear Schr¨odinger hierarchy from the equations of motionof the corresponding member the continuous Heisenberg hierarchy, or vice versa. Forinstance, given a solution q ( x, t ) and r ( x, t ) to the Hirota equations (2.9), (2.10) onemay use equation (2.12) to construct the gauge field operator G and subsequently simplycompute S , that solves (2.16) by construction, by means of the relation (2.13). In reverse,from a solution S to (2.16) we may construct G by (2.13) and subsequently q ( x, t ) and r ( x, t ) from (2.12). We elaborate below on the details of this correspondence. Equation (2.16) possesses an interesting and well known vector variant with many physicalapplications that arises when decomposing S in the standard fashion as S = s · σ , where σ is a vector whose entries are Pauli matrices σ =( σ , σ , σ ). Then the equation of motion(2.16) becomes equivalent to an extended version of the Landau-Lifschitz equation s t = − α s × s xx − β ( s x · s x ) s x + β s × ( s × s xxx ) . (2.18)The ELL equations (2.18) is easily derived from (2.16) when using the standard identity σ i σ j = δ ij I + iε ijk σ k with ε ijk denoting the Levi-Civita tensor. Since S = 1, we imme-diately obtain that s is a unit vector s · s = 1. For β →
3. Nonlocal multi-solitons for the ECHE from Darboux transformations
Let us now explain how to solve the above nonlinear systems and construct their nonlocalmulti-soliton solutions by means of repeated gauge transformations. The key idea is thatthe gauge is chosen in such a way that the transformation becomes equivalent to a Darbouxtransformation. We start with the extended continuous Heisenberg equation and introducefor convenience T := − iS , ψ := Ψ , U := U and V := V so that the spectral problem in(2.1) for system 2 with (2.15) reads ψ x = U ψ = λT ψ, ψ t = V ψ = X k =0 λ − k V ( k ) ψ (3.1)where V (0) = 4 βT, V (1) = 2 αT − βT T x , V (2) = 32 βT T x − αT T x − βT xx . (3.2)– 5 – onlocal gauge equivalence Next we carry out another gauge transformation ˆ G on the system (3.1) relating the eigen-states ψ to new eigenstates ˆ ψ as ˆ ψ = ˆ Gψ , so that similarly to the relations in (2.2) weobtain a new spectral problem withˆ ψ x = ˆ U ˆ ψ, ˆ ψ t = ˆ V ˆ ψ, (3.3)in which the new operators ˆ U , ˆ V are related to the original U , V asˆ U = ˆ GU ˆ G − + ˆ G x ˆ G − , and ˆ V = ˆ GV ˆ G − + ˆ G t ˆ G − . (3.4)The key ingredient to achieve the equivalence between the gauge transformation and theDarboux transformation [47, 48] lies in the right choice of the gauge transformation ˆ G .Following essentially [43], we take nowˆ G ( λ ) := − I + λL , with L := H Λ − H − , H := [ ψ ( λ ) , ψ ( λ )] , Λ = diag( λ , λ ) . (3.5)Thus H is taken to be a 2 × ψ ( λ ) and ψ ( λ ) denotingsolution to (3.1) for some specific values of the spectral parameter λ = λ = 0. We noticethat det L = λ − λ − = 0, so that the inverse of L exists. Using (3.1) we then compute thederivatives H x = T H Λ , (3.6) H t = X k =0 V ( k ) H Λ − k , (3.7) L x = T − LT L − , (3.8) L t = X k =0 (cid:16) V ( k ) L k − − LV ( k ) L k − (cid:17) , (3.9)which allows us to evaluate the right hand sides of the equations in (3.4) toˆ U = LU L − , (3.10)ˆ V (0) = LV (0) L − , (3.11)ˆ V (1) = LV (1) L − − V (0) L − + ˆ V (0) L − , (3.12)ˆ V (2) = LV (2) L − − V (1) L − + ˆ V (1) L − . (3.13)The matrix ˆ V is of the same form as V , that is ˆ V = P k =0 λ − k ˆ V ( k ) . Equation (3.10)is equivalent to ˆ SL = LS , which is reminiscent of the intertwining relations employedin Darboux transformations. Since the gauge system is of the same form as the originalequation, this means that if S is also a solution to (2.16), then ˆ S is a solution to the sameequation.We can now iterate this systems like a standard Darboux-Crum transformation [47, 49].Indexing all quantities, we have at each stage the spectral problem ψ ( n − x ( λ ) = U ( n − ( λ ) ψ ( n − ( λ ) , ψ ( n − t ( λ ) = V ( n − ( λ ) ψ ( n − ( λ ) , n ∈ N , (3.14)– 6 – onlocal gauge equivalence which when solved for ψ ( n − ( λ ) allows to define the new quantities H ( n − ( λ n − , λ n ) : = (cid:16) ψ ( n − ( λ n − ) , ψ ( n − ( λ n ) (cid:17) , (3.15)Λ n : = diag( λ n − , λ n ) , (3.16) L ( n ) ( λ n − , λ n ) : = H ( n − ( λ n − , λ n )Λ − n h H ( n − ( λ n − , λ n ) i − , (3.17)where λ i = λ j = 0, i, j ∈ N . By means of the intertwining operator L ( n ) we can now specifythe gauge transformations as ˆ G ( n ) ( λ ) := − I + λL ( n ) , (3.18)so that we can construct the solution to the spectral problem (3.14) at the next level as ψ ( n ) ( λ ) = ˆ G ( n ) ( λ ) ψ ( n − ( λ ) = G ( n ) ( λ ) ψ (0) ( λ ) , (3.19) U ( n ) ( λ ) = L ( n ) U ( n − ( λ ) h L ( n ) i − = L ( n ) U (0) ( λ ) (cid:16) L ( n ) (cid:17) − , (3.20)with L ( n ) := L ( n ) L ( n − . . . L (1) and G ( n ) ( λ ) := ˆ G ( n ) ( λ ) ˆ G ( n − ( λ ) . . . ˆ G (1) ( λ ). Noting thatdet L ( n ) = Y ni =1 λ − i = ( − n Y ni =1 | λ i − | − =: χ n , (3.21)with λ i − = − λ ∗ i , the inverse of L ( n ) is guaranteed to always exist with the restrictionson the λ i as introduced above. Extrapolating from (3.11)-(3.13) there are naturally alsogeneric formulae for V ( n ) ( λ ), but since we are mainly interested in U ( n ) ( λ ) we will notreport them here. It is clear from (3.14)-(3.20) that once the initial spectral probleminvolving U (0) ( λ ), V (0) ( λ ) and ψ (0) ( λ ) has been solved all higher levels follow simply byiteration.We apply this scheme now to construct multi-soliton solutions to equation (3.1) andin particular explain how nonlocality is naturally introduced into these systems. We start by parameterizing a matrix field solution S to the extended continuous Heisenbergequation (2.16) as S = − ω uv ω ! , ω + uv = 1 , (3.22)where the form of S is dictated by (2.13) with the constraint on the entries resulting fromthe first property in (2.14). Substituting this expression into equation (2.16), we identifyfrom the off-diagonal components of this matrix equation the two constraining nonlineardifferential equations u t = iα ( uω x − ωu x ) x − β (cid:2) u xx + 3 / u ( u x v x + ω x ) (cid:3) x , (3.23) v t = − iα ( vω x − ωv x ) x − β (cid:2) v xx + 3 / v ( v x u x + ω x ) (cid:3) x , (3.24)that u , v and ω have two satisfy. The diagonal entries are trivially satisfied when (3.23) and(3.24) hold. Similarly as the equations (2.9) and (2.10), one may treat (3.23) and (3.24)– 7 – onlocal gauge equivalence as independent equations for the functions u ( x, t ) and v ( x, t ), with ω ( x, t ) obtained fromthe constraint in (3.22). However, just as for the Hirota system one could also make thechoice u ( x, t ) = κv ∗ ( x, t ) so that equation (3.24) simply becomes the complex conjugate ofequation (3.23). Likewise we can make the nonlocal choice u ( x, t ) = κv ∗ ( − x, t ) with β = iδ , δ ∈ R , in which case equation (3.24) becomes the complex conjugate parity transformedof equation (3.23). This means for the matrix S that the nonlocality can be imposed as S ( x, t ) = κS † ( − x, t ), which holds with ω ( x, t ) = κω ∗ ( − x, t ).The multi-soliton solutions to the ECH equation are then computed from the Darboux-Crum transformations as explained above. From (3.20) we obtain therefore S n = L ( n ) S (cid:16) L ( n ) (cid:17) − , (3.25)with factors L ( i ) ( λ i − , λ i ) , i = 1 , . . . , n , being evaluated with the appropriate two compo-nent solutions ( ψ ( λ i − ) , ψ ( λ i )) to the spectral problems (3.14) for the iterated U and V operators. As the entire procedure relies on the solutions to the initial spectral problem,it is the choice of the so-called seed functions ψ (0) ( λ i ) = ( ϕ i , φ i ) and their implementationinto the definition H ( n − that will introduce the nonlocality properties. This mechanismis similar to what we observed in [28]. From the iteration procedure we obtain the closedsolutions L ( n )11 = det Ω n det W n , L ( n )12 = det U n det W n , L ( n )21 = det V n det W n , L ( n )22 = det Υ n det W n , (3.26)with (2 n × n )-matrices W n , Ω n , Υ n , U n and V n defined in terms of the seed functioncomponents as( W n ) ij = ( λ n +1 − ji ϕ i λ n +1 − ji φ i , (Ω n ) ij = ( λ j − i ϕ i λ j − ni φ i , (Υ n ) ij = ( λ ji ϕ i j = 1 , . . . , n,λ j − n − i φ i j = n + 1 , . . . , n, (3.27)( U n ) ij = ( λ n − ji ϕ i j = n, . . . , n,λ n − ji φ i j = 1 , . . . , n − , ( V n ) ij = ( λ j − i φ i j = 1 , . . . , n + 1 ,λ j − n − i ϕ i j = n + 2 , . . . , n, (3.28)with i = 1 , . . . , n .Keeping the matrix S in the same functional form as in the parameterization (3.22)at each step of the iteration procedure S n = − ω n u n v n ω n ! , ω n + u n v n = 1 , (3.29)and abbreviating for convenience the entries of the matrix L ( n ) by A n := L ( n )11 , B n := L ( n )12 , C n := L ( n )21 , D n := L ( n )22 we evaluate the entries to the S -matrix as u n = (cid:0) A n u − B n v + 2 A n B n ω (cid:1) /χ n , (3.30) v n = (cid:0) D n v − C n u − C n D n ω (cid:1) /χ n , (3.31) ω n = [ A n C n u − B n D n v + ( A n D n + B n C n ) ω ] /χ n , (3.32)– 8 – onlocal gauge equivalence We also derive the identity( A n ) x D n − B n ( C n ) x = A n ( D n ) x − ( B n ) x C n = 0 , (3.33)that will be crucial below. As mentioned, in order to obtain the nonlocal solutions we needto impose the constraint u n ( x, t ) = κv ∗ n ( − x, t ). Let us now explain how this is achieved bydiscussing the explicit solutions in more detail. We start with a simple constant solution to the ECH equation (2.16) of the general form(3.29) describing the free case S = − ! , ω = 1, u = v = 0 . (3.34)In order to define the matrix operator H as in (3.5), we need to construct the seed solution ψ ( λ ) to the spectral problem (3.1) and evaluate it for two different and nonzero spectralparameters ψ ( λ ) = ( ϕ , φ ) and ψ ( λ ) = ( ϕ , φ ). The first intertwining operator can becomputed directly and acquires the form L (1) = 1 λ λ det H λ ϕ φ − λ ϕ φ ( λ − λ ) ϕ ϕ ( λ − λ ) φ φ λ ϕ φ − λ ϕ φ ! . (3.35)We confirm that this expression can be cast into the form of the generic expression (3.26)with matrices W = λ ϕ λ φ λ ϕ λ φ ! , Ω = ϕ λ φ ϕ λ φ ! , Υ = λ ϕ φ λ ϕ φ ! , (3.36) U = λ ϕ ϕ λ ϕ ϕ ! , V = φ λ φ φ λ φ ! . (3.37)Given the intertwining operator L (1) , we can now calculate the one-soliton solution S directly from (3.30)-(3.32), obtaining u = 2 ϕ ϕ ( λ ϕ φ − λ ϕ φ )( λ − λ ) λ λ ( ϕ φ − ϕ φ ) , (3.38) v = 2 φ φ ( λ ϕ φ − λ ϕ φ )( λ − λ ) λ λ ( ϕ φ − ϕ φ ) , (3.39) ω = 1 − ϕ ϕ φ φ ( λ − λ ) λ λ ( ϕ φ − ϕ φ ) . (3.40)So far, these expressions hold for any solution to the spectral problem. Imposing next thenonlocality condition u ( x, t ) = κv ∗ ( − x, t ) leads for instance to the constraints ϕ ( x, t ) = − κφ ∗ ( − x, t ) , φ ( x, t ) = ϕ ∗ ( − x, t ) , with λ = − λ ∗ =: λ . (3.41)– 9 – onlocal gauge equivalence We can now solve the spectral problem (3.1) with S for ψ ( λ = λ ) to ψ ( λ ) = e ξ λ ( x,t )+ γ e − ξ λ ( x,t )+ γ ! , (3.42)where we introduced the function ξ λ ( x, t ) := iλx + 2 λ ( iα − δλ ) t. (3.43)and the additional constants γ , γ ∈ C to account for boundary conditions. The secondsolution is then simply obtained from the constraint (3.41) to ψ ( λ ∗ ) = − κe − ξ ∗ λ ( − x,t )+ γ ∗ e ξ ∗ λ ( − x,t )+ γ ∗ ! . (3.44)Notice that ψ ( λ ∗ ) is the solution to the parity transformed and conjugated spectral prob-lem (3.1). Given these solutions we are in a position to compute the functions in (3.38)-(3.40) u ( x, t ) = 4 κ Re λ (cid:0) κλe − ξ ∗ λ ( − x,t ) − γ ∗ + γ ∗ − λ ∗ e ξ λ ( x,t )+ γ − γ (cid:1) e γ +2 Re γ | λ | ( e ξ λ ( x,t )+ ξ ∗ λ ( − x,t )+2 Re γ + κe − ξ λ ( x,t ) − ξ ∗ λ ( − x,t )+2 Re γ ) , (3.45) v ( x, t ) = 4 Re λ (cid:0) κλ ∗ e − ξ λ ( x,t ) − γ + γ − λe ξ ∗ λ ( − x,t )+ γ ∗ − γ ∗ (cid:1) e γ +2 Re γ | λ | ( e ξ λ ( x,t )+ ξ ∗ λ ( − x,t )+2 Re γ + κe − ξ λ ( x,t ) − ξ ∗ λ ( − x,t )+2 Re γ ) , (3.46) ω ( x, t ) = 1 − κ (Re λ ) e γ +2 Re γ | λ | ( e ξ λ ( x,t )+ ξ ∗ λ ( − x,t )+2 Re γ + κe − ξ λ ( x,t ) − ξ ∗ λ ( − x,t )+2 Re γ ) . (3.47)We verify that these expressions do indeed satisfy the nonlinear differential equations (3.23)and (3.24) for the component functions of S , together with the locality constraint u ( x, t ) = κv ∗ ( − x, t ) that converts the two equations (3.23) and (3.24) into each other via a paritytransformation and a complex conjugation. The two-soliton solution is obtained in the next iterative step. With L (1) already computedin (3.35), we evaluate the gauge transformation ˆ G (1) and the matrix H (1) ˆ G (1) ( λ ) = − I + λL (1) , H (1) ( λ , λ ) := (cid:16) ˆ G (1) ( λ ) ψ (0) ( λ ) , ˆ G (1) ( λ ) ψ (0) ( λ ) (cid:17) , (3.48)from which we compute L (2) ( λ , λ ) as defined in (3.17). Subsequently we compute thecomplete intertwining operator L (2) = L (2) ( λ , λ ) L (1) ( λ , λ ) with entries given by thegeneral formula (3.26) with explicit matrices W = λ ϕ λ ϕ λ φ λ φ λ ϕ λ ϕ λ φ λ φ λ ϕ λ ϕ λ φ λ φ λ ϕ λ ϕ λ φ λ φ , (3.49)– 10 – onlocal gauge equivalence Ω = ϕ λ ϕ λ φ λ φ ϕ λ ϕ λ φ λ φ ϕ λ ϕ λ φ λ φ ϕ λ ϕ λ φ λ φ , Υ = λ ϕ λ ϕ φ λ φ λ ϕ λ ϕ φ λ φ λ ϕ λ ϕ φ λ φ λ ϕ λ ϕ φ λ φ , (3.50) U = λ φ λ ϕ λ ϕ ϕ λ φ λ ϕ λ ϕ ϕ λ φ λ ϕ λ ϕ ϕ λ φ λ ϕ λ ϕ ϕ , V = φ λ φ λ φ λ ϕ φ λ φ λ φ λ ϕ φ λ φ λ φ λ ϕ φ λ φ λ φ λ ϕ . (3.51)For the nonlocal case we define ψ and ψ as in (3.42) and (3.44). In addition, we use ψ ( λ = ρ ) = ϕ φ ! = e ξ ρ ( x,t )+ γ e − ξ ρ ( x,t )+ γ ! , ψ ( λ = ρ ∗ ) = ϕ φ ! = − κe − ξ ∗ ρ ( − x,t )+ γ ∗ e ξ ∗ ρ ( − x,t )+ γ ∗ ! , (3.52)so that with (3.30)-(3.32) we determine the nonlocal two-soliton solutions as ω = p − u ( x, t ) v ( x, t ) , (3.53) u = 2 ( L − L + L − L ) ( R + R + R + R + R + R ) λ λ λ λ (Γ + Γ + Γ + Γ + Γ + Γ ) ,v = 2 ( K − K + K − K ) ( T + T + T + T + T + T ) λ λ λ λ (Γ + Γ + Γ + Γ + Γ + Γ ) , where we defined the shorthand symbolsΓ ijkl : = ( λ i − λ j )( λ k − λ l ) ϕ i ϕ j φ k φ l , R ijkl = λ k λ l Γ ijkl , T ijkl = λ i λ j Γ ijkl , (3.54) L ijkl : = λ i ( λ j − λ k )( λ j − λ l )( λ k − λ l ) φ i ϕ j ϕ k ϕ l , (3.55) K ijkl : = λ i ( λ j − λ k )( λ j − λ l )( λ k − λ l ) ϕ i φ j φ k φ l , (3.56)Once more we verify that these expressions satisfy the nonlinear differential equations (3.23)and (3.24) for the component functions of S and in addition are nonlocal, i.e. satisfying u ( x, t ) = κv ∗ ( − x, t ), which is required to convert the two equations into each other via aparity transformation and a complex conjugation. n -soliton solutions We proceed further in the same way for the nonlocal multi-soliton solutions for n >
2. Ingeneral, for a nonlocal n -soliton solution we choose the spectral parameters as λ i = − λ ∗ i − = 0 , λ i = λ j i, j = 1 , , . . . , n, (3.57)and the seed functions computed at these values as ψ i − ( λ i − ) = ϕ i − φ i − ! = e ξ λ i − ( x,t )+ γ i − e − ξ λ i − ( x,t )+ γ i ! , (3.58) ψ i ( λ i ) = ϕ i φ i ! = − κe − ξ ∗ λ i − ( − x,t )+ γ ∗ i e ξ ∗ λ i − ( − x,t )+ γ ∗ i − ! . (3.59)We may then apply directly the formulae (3.30)-(3.32) and evaluate u n , v n , and ω n . Wefind the nonlocality property u n ( x, t ) = κv ∗ n ( − x, t ) for all solutions.– 11 – onlocal gauge equivalence
4. Nonlocal solutions to Hirota’s equation from the ECH equation
Let us now demonstrate how to obtain nonlocal solutions for the Hirota equation fromthose of the extended continuous Heisenberg equation. With S being parameterized as in(3.22) we solve for this purpose equation (2.13) for GG = a a ω +1 v c c ω − v ! , (4.1)where the functions a ( x, t ) and c ( x, t ) remain unknown at this point. They can be de-termined when substituting G into the equations (2.12). Solving the first equation for q ( x, t ) and r ( x, t ) we find q ( x, t ) = µ ( t )2 (cid:18) v x v + ωv x − ω x vv (cid:19) exp (cid:20)Z x ω ( s, t ) v s ( s, t ) − ω s ( s, t ) v ( s, t ) v ( s, t ) ds (cid:21) , (4.2) r ( x, t ) = 12 µ ( t ) (cid:18) v x v − ωv x − ω x vv (cid:19) exp (cid:20) − Z x ω ( s, t ) v s ( s, t ) − ω s ( s, t ) v ( s, t ) v ( s, t ) ds (cid:21) , (4.3)where µ ( t ) is an arbitrary function of t at this stage. Notice that the integral represen-tations (4.2) and (4.3) are valid for any solution to the ECH equation (2.16). Next wedemonstrate how to solve these integrals. Using the expression in (3.30)-(3.32), with sup-pressed subscripts n and S chosen as in (3.22), we can re-express the terms in (4.2) and(4.3) via the components of the intertwining operator L as ωv x − ω x vv = − A x D − BC x AD − BC + ∂ x ln (cid:20) CD ( AD − BC ) (cid:21) = ∂ x ln (cid:18) CD (cid:19) , (4.4) v x v = DC AC x − A x CAD − BC − CD BD x − B x DAD − BC = C x C + D x D , (4.5)where we used the property (3.33). With these relations the integral representations (4.2),(4.3) simplify to q n ( x, t ) = cµ n ( C n ) x D n = cµ n (cid:18) (det V n ) x det Υ n − (det W n ) x det V n det W n det Υ n (cid:19) , (4.6) r n ( x, t ) = 1 cµ n ( D n ) x C n = 1 cµ n (cid:18) (det Υ n ) x det V n − (det W n ) x det Υ n det W n det V n (cid:19) , (4.7)where c is an integration constant. Thus, we have now obtained a simple relation betweenthe spectral problem of the extended continuous Heisenberg equation and the solutions tothe Hirota equation. It appears that this is a novel relation even for the local scenario.The nonlocality property of the solutions to the ECH equation is then naturally inheritedby the solutions to the Hirota equation. Using the nonlocal choices for the seed functionsas specified in (3.58) and (3.59) we may compute directly the right hand sides in (4.6) and(4.7). Crucially these solutions satisfy the nonlocality property r n ( x, t ) = κc µ n q ∗ n ( − x, t ) . (4.8)We discuss this in more detail for the one-soliton solution for which the more explicitexpressions are less lengthy. – 12 – onlocal gauge equivalence Reading off the entries C and D from the L (1) -operator in (3.35), the one-soliton solutionin (4.6) and (4.7) acquires the form q ( x, t ) = cµ ( λ − λ ) φ [ ϕ ( φ ) x − φ ( ϕ ) x ] + φ [ φ ( ϕ ) x − ϕ ( φ ) x ]( ϕ φ − ϕ φ )( λ ϕ φ − λ ϕ φ ) , (4.9) r ( x, t ) = 1 cµ φ φ [ ϕ ( ϕ ) x − ϕ ( ϕ ) x ] − ϕ ϕ [ φ ( φ ) x + φ ( φ ) x ] φ φ ( ϕ φ − ϕ φ ) . (4.10)Specifying the solutions to the spectral problem as in (3.42) and (3.44) with λ = λ , λ = − λ ∗ and c = − q ( x, t ) = 4 iµ Re λ e − ξ ( x,t )+ ξ ∗ ( − x,t )+ γ + γ ∗ e ξ ( x,t )+ ξ ∗ ( − x,t )+ γ + γ ∗ + κe − ξ ( x,t ) − ξ ∗ ( − x,t )+ γ + γ ∗ , (4.11) r ( x, t ) = − iκ Re λµ e γ + γ ∗ e ξ ∗ ( − x,t )+ γ + γ ∗ + κe − ξ ( x,t )+ γ + γ ∗ . (4.12)We verify that (4.11) and (4.12) are solution to the nonlocal Hirota equation (2.9) and(2.10) for any constant value of µ . The nonlocality property inherited from the extendedcontinuous Heisenberg equation is r ( x, t ) = κµ q ∗ ( − x, t ) . (4.13)Thus for µ = 1 the nonlocality property between q and r becomes the same as the onebetween the functions v and u . Clearly, we may proceed in the same manner and use(4.6) and (4.7) to calculate directly the nonlocal n -soliton solutions to the to the Hirotaequation from the spectral problem of the nonlocal ECH equation.
5. Nonlocal solutions to the extended Landau-Lifschitz equation
Given the nonlocal solutions to the ECH equation (2.16), it is now also straightforwardto construct nonlocal solutions to the ELL equation (2.18) from them simply by using therepresentation S n = s n · σ with S n taken to be in the parameterization (3.29). Suppressingthe index n , a direct expansion then yields s = 12 ( u + v ) , s = i u − v ) , s = − ω = ±√ − uv. (5.1)For the local choice u ( x, t ) = v ∗ ( x, t ) these function are evidently real s ( x, t ) = Re u, s = − Im u, s = ± q − | u | . (5.2)Thus, since s is a real unit vector function and s · s = 1, its endpoint traces out a curveon the unit sphere, as demonstrated with an example of a one-soliton solution in figure 1below. However, for the nonlocal choice u ( x, t ) = κv ∗ ( − x, t ), the vector function s is no– 13 – onlocal gauge equivalence longer real so that we may decompose it into s = m + i l , where now m and l are realvalued vector functions. From the relation s · s = 1 it follows directly that m − l = 1 andthat these vector functions are orthogonal to each other m · l = 0. The nonlocal extendedLandau Lifschitz equation (2.18) then becomes a set of coupled equations for the real valuedvector functions m and lm t = α ( l × l xx − m × m xx ) + 32 δ [( m x · m x ) m x + 2 ( l x · m x ) m x − ( l x · l x ) l x ] (5.3)+ δ [ l × ( l × l xxx ) − m × ( l × m xxx ) − m × ( m × l xxx ) − l × ( m × m xxx )] , l t = − α ( l × m xx + m × l xx ) + 32 δ [( l x · l x ) m x + 2 ( l x · m x ) l x − ( m x · m x ) m x ] (5.4)+ δ [ m × ( m × m xxx ) − l × ( m × l xxx ) − l × ( l × m xxx ) − m × ( l × l xxx )] , Given s , the real component entries of the new vectors are trivially obtained from (5.1) to m i = ( s i + s ∗ i ) / l i = i ( s ∗ i − s i ) / m ( x, t ) = 14 (cid:2) u ( x, t ) + v ( x, t ) + κv ( − x, t ) + κ − u ( − x, t ) (cid:3) , (5.5) m ( x, t ) = i (cid:2) u ( x, t ) − v ( x, t ) − κv ( − x, t ) + κ − u ( − x, t ) (cid:3) , (5.6) m ( x, t ) = −
12 [ ω ( x, t ) + ω ( − x, t )] (5.7) l ( x, t ) = i (cid:2) − u ( x, t ) − v ( x, t ) + κv ( − x, t ) + κ − u ( − x, t ) (cid:3) , (5.8) l ( x, t ) = 14 (cid:2) u ( x, t ) − v ( x, t ) + κv ( − x, t ) − κ − u ( − x, t ) (cid:3) , (5.9) l ( x, t ) = i ω ( x, t ) − ω ( − x, t )] . (5.10)Clearly despite the fact that s · s = 1, the real and imaginary components no longer traceout a curve on the unit sphere.When solving the ECH equation directly we have implemented to nonlocality throughthe compatibility relations between the auxiliary equations (3.23) and (3.24), which wasthen inherited by s . We may also attempt to implement the nonlocality from the Hirotasystem directly into S and therefore s . For this purpose we make use of the fact that sofar the gauge operator G , that relates the spectral problem of the Hirota system to thespectral problem of the ECH equation has been left completely generic and the entries ofthe matrix A are constrained by the equations of motion (2.9) and (2.10).As commented above, when specifying the ( A ) -entry to r ( x, t ) = κq ∗ ( x, t ) with κ = ±
1, the two equations (2.9) and (2.10) reduce to standard local Hirota equation[42, 28]. The first equation in (2.12) then implies that G = G ∗ and G = κG ∗ ,reducing the four equations resulting from each matrix entry to the two equations a x = κb ∗ u, b x = b ∗ u, (5.11)where we used the more compact notation G =: a , G =: b . Having specified the gaugetransformation G , we may compute the matrix S directly from its defining relation (2.13)– 14 – onlocal gauge equivalence so that the components of the vector s become in this case s = a ∗ b − κab ∗ | a | − κ | b | , s = i a ∗ b + κab ∗ | a | − κ | b | , s = | a | + κ | b | | a | − κ | b | . (5.12)Hence for the choice κ = − s is real valued.For the nonlocal choice r ( x, t ) = κq ∗ ( − x, t ) the first equation in (2.12) implies that G = ˜ G ∗ and G = − κ ˜ G ∗ . We adopt here the notation from [28] and suppress theexplicit dependence on ( x, t ), indicating the functional dependence on ( − x, t ) by a tilde,i.e. ˜ q := q ( − x, t ), ˜ G ∗ := G ∗ ( − x, t ), etc. The first equation in (2.12) then reduces to thetwo equations a x = − κ ˜ b ∗ u, b x = ˜ a ∗ u, (5.13)so that in this case the components of the vector s become s = ˜ a ∗ b − κa ˜ b ∗ a ˜ a ∗ − κb ˜ b ∗ , s = i ˜ ab + κa ˜ b ∗ a ˜ a ∗ − κb ˜ b ∗ , s = i a ˜ a ∗ + κb ˜ b ∗ a ˜ a ∗ − κb ˜ b ∗ , (5.14)which are solutions to the nonlocal extended Landau-Lifschitz equations (5.3), (5.4). We will now discuss some concrete soliton solutions obtained as outlined in the previoussection. We start by solving the constraint (5.11) for a local solution first that determinesthe gauge transformation G . Making the additional assumption b = cq ( x, t ) for someconstant c , equation (5.11) becomes a x = κc | q | , a = c (ln q ∗ ) x . (5.15)The compatibility between the two equations in (5.15) implies that | q | = κ (ln q ∗ ) xx ⇔ D x f · f = − κ | g | , (ln g ) xx = 0, (5.16)with q = g/f , g ∈ C , f ∈ R . We notice that the first relation in (5.16), following directlyfrom (5.15), corresponds to one of the bilinear equations into which the Hirota equationcan be converted with an additional constraint. Here D x denotes a Hirota derivative, see[28] for more details. Evidently the additional constraint is not satisfied by all solutions tothe Hirota equation. The second equation in (2.12) is then satisfied trivially for solutionsof (5.16). We have therefore obtained a solution for the gauge field operator in the form G = c (ln q ∗ ) x qκq ∗ (ln q ) x ! . (5.17)From the definition of S and its decomposition the components of s are computed to s ( x, t ) = 1 + 2 κ | q | q x q ∗ x − κ | q | , s ( x, t ) = | q | ( q x − κq ∗ x ) q x q ∗ x − κ | q | , s ( x, t ) = i | q | ( q x + κq ∗ x ) q x q ∗ x − κ | q | . (5.18)– 15 – onlocal gauge equivalence It is trivial to verify that s · s = 1. Thus for any solution to the Hirota equation, with theadditional constraint as specified in (5.16), the vector s constitutes a solution to the ELLequation as given in (2.18).One such solution we may employ is for instance the local one-soliton solution obtainedin [28] q ( x, t ) = ( µ + µ ∗ ) exp[ γ + µx + µ t ( iα − βµ )]( µ + µ ∗ ) + exp[ γ + γ ∗ + iαt ( µ − µ ∗ ) − βt ( µ + µ ∗ ) + x ( µ + µ ∗ )] . (5.19)We briefly discuss some of the key characteristic behaviours of s for various choices ofthe parameters. When β = 0, the solutions correspond to the one-soliton solutions of thenonlinear Schr¨odinger equation. For real parameters µ we obtain the well known periodicsolutions to the ELL equation as seen in the left panel of figure 1. However, when the shiftparameters µ is taken to be complex we obtain decaying solutions tending towards a fixedpoint. Figure 1:
Local solutions to the extended Landau Lifschitz equation (2.18) from a gauge equivalentone-soliton solution (5.19) of the nonlinear Schr¨odinger equation for different initial values x ,complex shift γ = 0 . i . α = 0 . β = 0. In the left panel the spectral parameter is real µ = 0 . µ = 0 . i . When taking β = 0, that is the solutions to the Hirota equation even for real values µ the bahaviour of the trajectories is drastically changed even for small values of β , as theybecome more knotty and convoluted as seen in the left panel of figure 2. Complex valuesof µ are once more decaying solutions tending towards a fixed point.Next we discuss the nonlocal solutions obtained by solving (5.11), by making the sameadditional assumption as for the construction of the local solutions b = cq ( x, t ) for somereal constant c . In this case equation (5.13) becomes a x = − κcq ˜ q ∗ , a = c (ln ˜ q ∗ ) x . (5.20)Now the compatibility between the two equations in (5.20) implies that κq ˜ q ∗ = − (ln ˜ q ∗ ) xx ⇔ D x f · f = κgh , (ln h ) xx = 0 . (5.21)– 16 – onlocal gauge equivalence Figure 2:
Local solutions to the extended Landau Lifschitz equation (2.18) from a gauge equivalentone-soliton solution (5.19) of the Hirota for a fixed value of x , complex shift γ = 0 . i . α = 0 . β = 0 .
1. In the left panel the spectral parameter is real µ = 0 . µ = 0 . i . with q = g/f , f, g ∈ C and h = 2 f ˜ g ∗ / ˜ f ∗ . Once again the first relation on the right handside in (5.21) occurs in the bilinearisation of the nonlocal Hirota equation, see section 4.1[28]. However, as for the local case in (5.16) the second relation is an additional constraintthat is not automatically satisfied by all solutions. We have therefore obtained a solutionfor the nonlocal gauge field operator in the form G = c (ln ˜ q ∗ ) x q ˜ q ∗ (ln q ) x ! , (5.22)so that the matrix S can be computed directly from its defining relation (2.13). Using theexpansion for S in terms of the components of s we compute s ( x, t ) = q ˜ q ∗ x − ˜ q ∗ q x ˜ q ∗ q − ˜ q ∗ x q x , s ( x, t ) = i q ˜ q ∗ x + ˜ q ∗ q x ˜ q ∗ q − ˜ q ∗ x q x , s ( x, t ) = − q ˜ q ∗ + q x ˜ q ∗ x ˜ q ∗ q − ˜ q ∗ x q x . (5.23)It is trivial to verify that s · s = 1. Thus for any solution to the nonlocal Hirota equation,with the additional constraint as specified in (5.21), the complex valued vector s constitutesa solution to the ELLE as given in (2.18).One such solution one may employ is the nonlocal one-soliton solution obtained in [28] q ( x, t ) = ( µ − µ ∗ ) exp [ γ + µ ( x + iµt ( α − δµ ))]( µ − µ ∗ ) + exp [ γ + γ ∗ + it ( α ( µ − µ ∗ ) + δ ( µ ∗ − µ )) + x ( µ − µ ∗ )] . (5.24)Let us analyze how m and l behave in this case. As expected, the trajectories will no stayon the unit sphere. However, for certain choices of the parameters it is possible to obtainwell localized closed three dimensional trajectories that trace out curves with fixed pointsat t = ±∞ as seen for an example in figure 3. Thus the nonlocal nature of the solutionsto the Hirota equation has apparently disappeared in the setting of the extended Landau– 17 – onlocal gauge equivalence Lifschitz equation. However, not all solutions are of this type as some of them are nowunbounded.
Figure 3:
Nonlocal solutions to the extended Landau Lifschitz equation (5.3) and (5.4) from agauge equivalent one-soliton solution (5.24) of the Hirota equation for a fixed value of x , vanishingcomplex shift γ = 0, µ = i . α = 1 . δ = 0 . While the computation of the solutions to the ELL equation is straightforward when com-puting G directly with some additional constraints, not all Hirota solutions obey them. Letus therefore use the two-soliton solution (3.53) in the representation (5.5)-(5.10) to studythe nonlocal two-soliton solutions to the ELL equation. The two-soliton structure is bestrevealed when plotting it for fixed time over space. In figure 4 we show each component of m and l separately, displaying clearly two distinct one-soliton structures. - -
10 10 20 x - -
55m m m m - -
10 10 20 x - l l Figure 4:
Nonlocal two-soliton solutions to the extended Landau Lifschitz equation for fixed time t = 3 as a function of with parameters α = 1 . δ = 0 . κ = 3, λ = 0 . − i . ρ = 0 . i . γ = i . γ = i . γ = − i . γ = i . – 18 – onlocal gauge equivalence
6. Conclusions
We discussed two different types of local/nonlocal gauge transformations: The first ofthem, G, relates the auxiliary functions in the two spectral problems of the local/nonlocalextended continuous limit of the Heisenberg equation to the local/nonlocal Hirota equation.The explicit form of the gauge functions can be used to establish a concrete relation betweensolutions of one system to the other. This concrete map when applied to solutions worksmost efficiently in one direction from the spectral problem the local/nonlocal extendedcontinuous Heisenberg equation to solutions of the (nonlocal) Hirota equation, as statedexplicitly in (4.6) and (4.7). This map is not easily invertible and instead we used (5.16)and (5.21) to provide an alternative. While we demonstrated that the (5.16) and (5.21)are equivalent to equations that emerge in the bilinearization process, they also requirean additional constraint that is not satisfied by all solutions, so that not all solutionsare obtainable in this manner. The second type of gauge transformation, ˆ G , is an auto-gauge transformation that relates the auxiliary functions in the spectral problem of the(nonlocal) extended continuous Heisenberg equation to itself. This gauge transformationcan be interpreted as a Darboux transformation and allows to construct a new solutionfrom a known one. In an analogous fashion to Darboux-Crum transformations, it can beiterated to produce multi-soliton solutions.The nonlocality can be implemented separately in the two systems by applying aparity complex conjugation map to different sets of equations. For the Hirota system it ismost naturally applied to the pair of equations (2.9) and (2.10), resulting from the zerocurvature formulation as discussed in [28]. For the extended version of continuous limit ofthe Heisenberg equation it is most obviously applied to its component version (3.23) and(3.24). The two versions of nonlocality in the two systems were shown to be related to eachby means of the gauge transformation G as demonstrated by (4.13). As demonstrated, onemay, however, also map components of the gauge transformation matrix to each other ina consistent manner. At the level of the spectral problem the nonlocality is implementedvia mapping components of the seed functions consistently to each other.Various issues are worthy of further exploration. It is well known [50, 51, 35, 52] thatthe Landau Lifschitz equation, i.e. (2.18) for β = 0, admits a geometric interpretationthat directly relates the curvature and torsion of a vector field to any solution of thenonlinear Schr¨odinger equation when expressed in form of the Hasimoto map [50]. In [53]we demonstrate that these relations and interpretations can be extended to the extendednonlocal versions of this equation. Naturally it would also be interesting to explore thebehaviour of the systems arising from the other types of PT -conjugation as discussed in[28]. Acknowledgments:
JC is supported by a City, University of London Research Fellow-ship. FC was partially supported by Fondecyt grant 1171475 and would like to thank theDepartment of Mathematics at City, University of London and the Departamento de F´ısicaTe´orica, ´Atomica y ´Optica at the Universidad de Valladolid for kind hospitality. AF wouldlike to thank the Instituto de Ciencias F´ısicas y Matem´aticas at the Universidad Australde Chile for kind hospitality and Fondecyt for financial support.– 19 – onlocal gauge equivalence
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