Solitons of Some Nonlinear Sigma-Like Models
aa r X i v : . [ n li n . S I] S e p Submitted to
Symmetry, Integrability and Geometry: Methods and Applications
Solitons of some nonlinear sigma-like models.
V.E. VekslerchikUsikov Institute for Radiophysics and Electronics12, Proskura st., Kharkov, 61085, Ukraine
E-mail: [email protected]
Abstract.
We present a set of differential identities for some class of matrices. Theseidentities are used to derive N-soliton solutions for the Pohlmeyer nonlinear sigma-model,two-dimensional self-dual Yang-Mills equations and some modification of the vector Calapsoequation.
Key words: nonlinear sigma-models, vector Calapso equation, self-dual Yang-Mills equa-tions, explicit solutions, solitons
This paper is a continuation of a cycle work devoted to the derivation of the soliton solutions forvarious integrable models (see [12, 17, 13, 14, 15, 16] and references therein). In all these studieswe exploit the already known fact that soliton solutions of almost all integrable equations possesssimilar structure which can be clearly expressed in terms of some class of matrices [12, 17].In this work we consider the following three models. The first one is the Pohlmeyer nonlinearsigma-model [7, 9], described by the action S = Z dξ dη (cid:26) ( ∂ ξ u ) ( ∂ η ¯ u )1 − u ¯ u ± u ¯ u (cid:27) (1.1)where ∂ ζ stands for ∂/∂ζ , which, depending on whether the variables are real or complex and onthe choice of the involution ¯ u = κu ∗ (with star indicating the complex conjugation and κ = 1)becomes either Getmanov system studied in [4] or the O (3 ,
1) sigma-model discussed in [10, 11].Also we consider a modification of the vector Calapso equation/ ∂ ϕ = f ϕ (1.2)where ϕ is a complex 4-vector, ϕ ∈ C , / ∂ is a two-dimensional Dirac-type operator and f = f (/ ∂ ϕ ).The third equation discussed in this paper, ∂ ξ (cid:0) U − ∂ η U (cid:1) + (cid:2) σ , U − [ σ , U ] (cid:3) = 0 (1.3)where U is a 2 × We start with the so-called ‘almost-intertwining’ matrices [5] that satisfy the ‘rank one condition’[2, 3], ¯LA − AL = | α ih a | , L¯A − ¯A¯L = | ¯ α ih ¯ a | . (2.1)Here, L and ¯L are diagonal constant N × N matrices, | α i and | ¯ α i are constant N -componentcolumns, h a | and h ¯ a | are N -component rows that depend on the coordinates describing themodel.It should be noted that throughout this paper the overbar does not mean the complex con-jugation (which will be indicated by the ∗ -symbol).The ξ - and η -dependence of the matrices A and ¯A that we use in this study is defined by A = C exp (cid:0) ξ L − − η L (cid:1) , ¯A = ¯C exp (cid:0) − ξ ¯L − + η ¯L (cid:1) (2.2)where C and ¯C are constant N × N matrices. Note that C and ¯C are not arbitrary: their structureis determined by (2.1).Now, our task is to calculate derivatives of various combinations of the matrices A and ¯A ,matrices G and ¯G defined by G = (cid:0) A ¯A (cid:1) − , ¯G = (cid:0) ¯AA (cid:1) − , (2.3)rows h a | , h ¯ a | and columns | α i , | ¯ α i . In particular, we are going to derive a closed set of differentialidentities involving the eight functions u = 1 − h ¯ a | GA | ¯ β i , ¯ u = 1 − h a | ¯G ¯A | β i , v = h ¯ a | G | β i , ¯ v = h a | ¯G | ¯ β i (2.4)and w = h ¯ a | G | α i ,w = h a | ¯G | ¯ α i , w = h ¯ b | G | β i ,w = h b | ¯G | ¯ β i (2.5)where h b | = h a | L − , h ¯ b | = h ¯ a | ¯L − , | β i = ¯L − | α i , | ¯ β i = L − | ¯ α i . (2.6)Using straightforward calculations one can obtain the following identities involving the ∂ ξ -derivatives. Proposition 2.1.
Functions u , ¯ u , v , v , ¯ v , w , w defined in (2.4) and (2.5) satisfy the followingset of equations ∂ ξ u = − vw ,∂ ξ ¯ u = ¯ vw , ∂ ξ v = − uw ,∂ ξ ¯ v = ¯ uw , ∂ ξ w = − uv,∂ ξ w = ¯ u ¯ v. (2.7)In a similar way, one can derive the set of ∂ η -identities.olitons of some nonlinear sigma-like models. 3 Proposition 2.2.
Functions u , ¯ u , v , v , ¯ v , w , w defined in (2.4) and (2.5) satisfy the followingset of equations ∂ η u = − ¯ vw ,∂ η ¯ u = vw , ∂ η v = ¯ uw ,∂ η ¯ v = − uw , ∂ η w = ¯ uv,∂ η w = − u ¯ v. (2.8)We do not present here a proof of all of these identities. In appendix A a reader can findexamples of how to obtain some of them, while the rest can be derived in an analogous way.An immediate consequence of these results is that the function I defined as I = u ¯ u + v ¯ v isconstant: ∂ ξ I = ∂ η I = 0. More careful analysis leads to the identity u ¯ u + v ¯ v = 1 (2.9)which will be often used in what follows. It turns out that a derivation of this simple identityis the most cumbersome part of the calculations of this paper. We present a proof of (2.9) inappendix B.System (2.7)–(2.9) is not new. It is closely related to the Ablowitz-Ladik hierarchy [1], whichis not surprising because, as is shown in [17], the bright solitons of the Ablowitz-Ladik hierarchyare built of matrices A and ¯A (2.1) and have the structure of functions defined in (2.4). However,we do not discuss these questions here and consider (2.7)–(2.9) as a closed set of identities whichis used in what follows to construct explicit solutions for the equations which are subject of thispaper. In this section we construct N-soliton solutions for the equations listed in the introduction.Identities (2.7)–(2.9) give us a possibility to do this with very little effort.
Starting from equations (2.7) and (2.8), it is easy to derive the following identities involvingsecond derivatives of the functions u and ¯ uv ¯ v ∂ ξη u + ¯ u ( ∂ ξ u ) ( ∂ η u ) = uv ¯ v , (3.1) v ¯ v ∂ ξη ¯ u + u ( ∂ ξ ¯ u ) ( ∂ η ¯ u ) = ¯ uv ¯ v . (3.2)Noting that v ¯ v = 1 − u ¯ u , we can rewrite equations (3.1) and (3.2) as U − ∂ ξη u + U − ( ∂ ξ u ) ( ∂ η u ) ¯ u = u, (3.3) U − ∂ ξη ¯ u + U − u ( ∂ ξ ¯ u ) ( ∂ η ¯ u ) = ¯ u (3.4)where U = 1 − u ¯ u. (3.5)These equations are nothing but the Euler equations δ S /δu = δ S /δ ¯ u = 0 (3.6)for the action S = Z dξ dη L (3.7) V.E. Vekslerchikwith the Lagrangian given by L = ( ∂ ξ u ) ( ∂ η ¯ u )1 − u ¯ u + u ¯ u. (3.8)In a similar way, one can derive from (2.7) and (2.8) the identities u ¯ u ∂ ξη v + ( ∂ ξ v ) ( ∂ η v ) ¯ v = − u ¯ u v, (3.9) u ¯ u ∂ ξη ¯ v + v ( ∂ ξ ¯ v ) ( ∂ η ¯ v ) = − u ¯ u ¯ v, (3.10)rewrite them as V − ∂ ξη v + V − ( ∂ ξ v ) ( ∂ η v ) ¯ v = − v, (3.11) V − ∂ ξη ¯ v + V − v ( ∂ ξ ¯ v ) ( ∂ η ¯ v ) = − ¯ v (3.12)where V = 1 − v ¯ v (3.13)and note that they correspond to the Lagrangian L = ( ∂ ξ v ) ( ∂ η ¯ v )1 − v ¯ v − v ¯ v (3.14)These calculations can be summarized as follows. Proposition 3.1.
Functions u , ¯ u , v , v and ¯ v defined in (2.4) satisfy the Euler equations forthe Lagrangian L = ( ∂ ξ w ) ( ∂ η ¯ w )1 − w ¯ w ± w ¯ w (3.15)Thus, functions defined in (2.4) provide solutions for the field equations for the Pohlmeyernonlinear sigma-model. The elementary consequence of the equations (2.7) and (2.8) is the fact that the four-vector ϕ defined by ϕ = ( w , w , w , w ) T (3.16)obeys the identity ∂ ξη ϕ = f ϕ (3.17)where f = v ¯ v − u ¯ u. (3.18)Moreover, one can express the function f in terms of w , w , w and w by noting that( ∂ ξ w ) ( ∂ ξ w ) + ( ∂ η w ) ( ∂ η w ) = − u ¯ uv ¯ v (3.19)which, together with (2.9), leads to( ∂ ξ w ) ( ∂ ξ w ) + ( ∂ η w ) ( ∂ η w ) = (cid:0) f − (cid:1) . (3.20)olitons of some nonlinear sigma-like models. 5This means that (3.17) can be presented as a closed equation for the vector ϕ .After introducing the Dirac operator by/ ∂ = ∂ ξ ∂ η − ∂ ξ − ∂ η , (3.21)noting that ∂ ξη = − / ∂ and rewriting the identity (3.20) in terms of ϕ as f = 1 − (/ ∂ ϕ ) T γ (/ ∂ ϕ ) (3.22)where γ = , (3.23)one can present this result as Proposition 3.2.
Vector ϕ given by (3.16) where functions w , w , w and w are defined in(2.5), is a solution for the equations/ ∂ ϕ = f ϕ (3.24) with f = p − ( / ∂ ϕ ) T γ ( / ∂ ϕ ) . (3.25)Rewriting this statement in terms of the vector φ , φ = / ∂ ϕ , (3.26)one can easily obtain the following corollary Proposition 3.3.
Vector φ given by φ = (¯ u ¯ v, ¯ uv, uv, u ¯ v ) T (3.27) where functions u , ¯ u , v and ¯ v are defined in (2.4) satisfies equation/ ∂ / ∂ φ p − φ T γ φ = φ (3.28) which describes some vector variant of the sigma model discussed above. To derive the soliton solutions for the two-dimensional self-dual SU (2) Yang-Mills-like equationsconsider the matrix U = (cid:18) u − ¯ vv ¯ u (cid:19) , (3.29) V.E. Vekslerchikwhich, due to (2.9), belongs to SL (2 , C ), det U = 1. Equations (2.7) and (2.8) imply ∂ ξ U = (cid:18) − vw − ¯ uw − uw ¯ vw (cid:19) , ∂ η U = (cid:18) − ¯ vw uw ¯ uw vw (cid:19) . (3.30)After differentiating these identities once more and using, again, (2.7) and (2.8), one can arrive,after some simple calculations, at ∂ ξη U − ( ∂ ξ U ) U − ( ∂ η U ) = (cid:18) uv ¯ v + u ¯ u ¯ v − u ¯ uv ¯ uv ¯ v (cid:19) (3.31)and U − ( ∂ ξη U ) − U − ( ∂ ξ U ) U − ( ∂ η U ) = (cid:18) u ¯ v − uv (cid:19) . (3.32)It easy to demonstrate that the right-hand side of the last equation can be rewritten as (cid:18) u ¯ v − uv (cid:19) = − (cid:2) σ , U − [ σ , U ] (cid:3) (3.33)To summarize, we have derived the following result. Proposition 3.4.
The matrix (3.29) with the functions u , ¯ u , v and ¯ v defined in (2.4) satisfiesthe two-dimensional self-dual Yang-Mills-like equations ∂ ξ (cid:0) U − ∂ η U (cid:1) + (cid:2) σ , U − [ σ , U ] (cid:3) = 0 . (3.34)It is easy to see that if one starts from the system (2.7)–(2.9), then to derive soliton solutionsfor equations (3.15), (3.24) with (3.25) or (3.34) becomes a rather easy task. Till now, we have not raised the questions related to the complex or Hermitian conjugation:the functions u and ¯ u , v and ¯ v as well as w , w , w and w have been treated as independent.However, in physical applications of the models discussed in this paper these questions are veryimportant. Thus, in this section we study the properties of the described in the previous sectionsolutions from this viewpoint.Our first problem is to determine conditions which ensure the following identity: ¯A = κ A ∗ , κ = ± ∗ stands for the complex conjugation.Analyzing the compatibility of equations (2.1) and equations (2.2), which determine the ξ - and η -dependence, with the complex conjugation on can distinguish two important cases ¯L = ( L ∗ ) ± . In what follows, we consider these cases separately and see how the involutionmodifies equations (3.15), (3.24) with (3.25), (3.34) and their solitons.olitons of some nonlinear sigma-like models. 7 In this case the diagonal matrices L and ¯L , the rows h a | and h ¯ a | and the columns | α i and | ¯ α i arerelated by ¯L = L ∗ , h ¯ a | = h a | ∗ , | ¯ α i = κ | α i ∗ (4.2)while the variables ξ and η should satisfy conditions ξ = − ξ ∗ and η = − η ∗ . After introducingreal variables t and x by ξ = i ( t + x ) , η = i ( t − x ) (4.3)the row h a | and the matrix A can be presented as h a | = h c | exp [ i Θ ( t, x ) ] , A = C exp [ i Θ ( t, x ) ] . (4.4)Here h c | is an arbitrary constant row, C is the constant matrix given by C = α j c k L ∗ j − L k ! Nj,k =1 (4.5)where c j , α j and L j are the components of h c | , | α i and L , h c | = ( c , ... , c N ) , | α i = ( α , ... , α N ) T , L = diag ( L , ... , L N ) , (4.6)which play the role of the constant parameters of the N-soliton solutions , and Θ ( t, x ) = t (cid:0) L − − L (cid:1) + x (cid:0) L − + L (cid:1) . (4.7)The restrictions (4.2) lead to the following relations between the functions which are defined in(2.4) and (2.5):¯ u = u ∗ , ¯ v = κv ∗ , w = κw ∗ , w = κw ∗ . (4.8)Now we can reformulate some of the results presented in the previous section. Proposition 4.1.
Functions u and v defined in (2.4) together with (4.1), (4.2) and (4.4)–(4.7)satisfy the Euler equations for the Lagrangians L u = ( ∂ µ u ) ( ∂ µ u ∗ )1 − | u | − | u | (4.9) L v = ( ∂ µ v ) ( ∂ µ v ∗ )1 − κ | v | + 4 | v | (4.10) correspondingly. Here ∂ µ = ( ∂ t , ∂ x ) , ∂ µ = ( ∂ t , − ∂ x ) and summation over µ is understood. Considering the Calapso equation (proposition 3.2), it should be noted that due to the sym-metry (4.8) we can rewrite it as an equation for the C vectors ψ = √ ( w , w ) T . After theredefinition of the Dirac operator,/ ∂ = (cid:18) ∂ t − ∂ x ∂ t + ∂ x (cid:19) (4.11)one can easily verify that(/ ∂ψ ) † (/ ∂ψ ) = 4 | u | | v | (4.12)which leads to the following result. Strictly speaking, of 3 N parameters (4.6), only 2 N are essential. By simle matrix transformation some ofthem can be eliminated by redefining the other. Thus, one can put, say, all α j = 1. V.E. Vekslerchik
Proposition 4.2.
Vector ψ = 1 √ (cid:18) w w (cid:19) . (4.13) where functions w and w are defined in (2.5) together with (4.1), (4.2) and (4.4)–(4.7) is asolution for the Calapso-like equation/ ∂ ψ = 4 f ψ (4.14) with f = q − κ ( / ∂ψ ) † ( / ∂ψ ) . (4.15) In this case the diagonal matrices L and ¯L , the rows h a | and h ¯ a | and the columns | α i and | ¯ α i arerelated by ¯L = (cid:0) L − (cid:1) ∗ , h ¯ a | = (cid:0) h a | L − (cid:1) ∗ , | ¯ α i = − κ L ( | α i ) ∗ (4.16)and η = ξ ∗ . After introducing real variables x and y by ξ = x + iy, η = x − iy. (4.17)the row h a | and the matrix A can be presented as h a | = h c | exp [ Θ ( x, y ) ] , A = C exp [ Θ ( x, y ) ] . (4.18)Here h c | is an arbitrary constant row, C is the constant matrix given by C = L ∗ j α j c k − L ∗ j L k ! Nj,k =1 (4.19)where c j , α j and L j are defined in (4.6) and Θ ( x, y ) = x (cid:0) L − − L (cid:1) + iy (cid:0) L − + L (cid:1) . (4.20)One can show that in this case the functions u and ¯ u are real,Im u = Im ¯ u = 0 , (4.21)while other functions defined in (2.4) and (2.5) are related by¯ v = − κv ∗ , w = − κw ∗ , w = − κw ∗ . (4.22)Now we give a few examples of the ‘physical’ forms of the results presented in section 3. Proposition 4.3.
Function v defined in (2.4) together with (4.1), (4.16) and (4.18)–(4.20)satisfies the Euler equations for the Lagrangian L = ( ∇ v, ∇ v ∗ )1 + κ | v | − | v | (4.23) where ∇ is the gradient operator, ∇ = ( ∂ x , ∂ y ) T . olitons of some nonlinear sigma-like models. 9As in the ‘Minkowski’ case, we rewrite the Calapso equation as an equation for the C vectors ψ = √ ( w , w ) T . Defining the two-dimensional Dirac operator by/ ∂ = (cid:18) ∂ x − i∂ y ∂ x + i∂ y (cid:19) (4.24)one can obtain that (/ ∂ψ ) † σ (/ ∂ψ ) = − u ¯ u | v | which leads to the following result. Proposition 4.4.
Vector ψ = 1 √ (cid:18) w w (cid:19) (4.25) with functions w and w defined in (2.5) together with (4.1), (4.16) and (4.18)–(4.20) is asolution for the Calapso-like equation/ ∂ ψ = − f ψ (4.26) where/ ∂ = σ ∂ x + σ ∂ y , (4.27) σ i are the Pauli matrices and f = q − κ ( / ∂ψ ) † σ ( / ∂ψ ) . (4.28)Considering the self-dual Yang-Mills equation (see proposition 3.4), we restrict ourselves withthe κ = 1 case. Proposition 4.5.
The matrix U = (cid:18) u v ∗ v ¯ u (cid:19) , (4.29) with the functions u , ¯ u and v defined in (2.4), (4.18)–(4.20) with κ = 1 and ¯A = A ∗ is aHermitian solution, U † = U (4.30) for the equation ∂∂ξ (cid:18) U − ∂∂ξ ∗ U (cid:19) + (cid:2) σ , U − [ σ , U ] (cid:3) = 0 . (4.31) First, we want to give a comment about the interrelations between the models discussed in thispaper. As one can easily see, all models considered here are closely related to the auxiliarysystem (2.7), (2.8). At the same time, it has been shown in [10] that the sigma-model fromsection 3.1 can be described in terms of the Ablowitz-Ladik hierarchy. This indicates that bothvector Calapso equation (3.24), (3.25) and matrix Yang-Mills-type equation (3.34) can also be‘embedded’ into the Ablowitz-Ladik hierarchy which is usually associated with the evolutionaryequations like the discrete nonlinear Schr¨odinger or modified KdV equations.0 V.E. VekslerchikNext, we would like to point a reader’s attention to one of the advantages of the directapproach. If we were trying to derive soliton solutions for the different cases of, say, the generalPohlmeyer sigma-model [7] (the so-called O (4) version of [4] and the O (3 , O (4) and O (3 ,
1) versionson an equal footing: the only difference is the restriction on the constant parameters given by(4.2) and (4.16). Moreover, we have simultaneously obtained both the so-called dark solitons ofthe model with the Lagrangian (4.9) and the bright solitons of the model (4.10) which, again,needs separate consideration in the framework of the inverse scattering transform.Finally, we would like to note that three models considered in this paper are far from beingthe only ones whose solutions can be ‘extracted’ from the rather simple system (2.7)–(2.9). Wehope that the studies presented here can be successfully continued to find other soliton modelswith possible physical applications.
A Derivation of (2.7).
Consider, for example, the function v . Calculating the derivative of G , ∂ ξ G = GA | ¯ β ih ¯ b | G (A.1)and using the derivative of h ¯ a | , ∂ ξ h ¯ a | = −h ¯ a | ¯L − = −h ¯ b | (A.2)one can derive ∂ ξ v = ( ∂ ξ h ¯ a | ) G | β i + h ¯ a | ( ∂ ξ G ) | β i = −h ¯ a | ¯L − G | β i + h ¯ a | GA | ¯ β ih ¯ b | G | β i = −h ¯ b | G | β i + h ¯ b | G | β ih ¯ a | GA | ¯ β i = − uw (A.3)Calculations involving the functions u and ¯ u are slightly more complicated. Say, to calculatethe derivative ∂ ξ u we, first, rewrite (A.1) in an equivalent form, ∂ ξ G = ¯L − G − G¯L − − G | β ih b | ¯G ¯A (A.4)which leads to ∂ ξ GA = ¯L − GA − G¯L − A − G | β ih b | ¯G ¯AA + GAL − = ¯L − GA − G | β ih b | ¯G ¯AA + G | β ih b | = ¯L − GA + G | β ih b | ¯G (A.5)and ∂ ξ u = − ( ∂ ξ h ¯ a | ) GA | ¯ β i − h ¯ a | ( ∂ ξ GA ) | ¯ β i = −h ¯ a | G | β ih b | ¯G | ¯ β i = − vw (A.6)olitons of some nonlinear sigma-like models. 11 B Proof of (2.9).
Consider the following ( N + 2) × ( N + 2) determinant: D = det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ¯ a | G A | ¯ β i N | β ih a | ¯ β i h a | ¯G ¯A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B.1)Calculating this determinant using the Jacobi identity leads to D = D D − D D (B.2)with D = det (cid:12)(cid:12)(cid:12)(cid:12) h ¯ a | GA | ¯ β i N (cid:12)(cid:12)(cid:12)(cid:12) = 1 − h ¯ a | GA | ¯ β i = u, (B.3) D = det (cid:12)(cid:12)(cid:12)(cid:12) N | β ih a | ¯G ¯A (cid:12)(cid:12)(cid:12)(cid:12) = 1 − h a | ¯G ¯A | β i = ¯ u, (B.4) D = det (cid:12)(cid:12)(cid:12)(cid:12) h ¯ a | G N | β i (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:12)(cid:12)(cid:12)(cid:12) −h ¯ a | G | β i N | β i (cid:12)(cid:12)(cid:12)(cid:12) = ( − ) N +1 v, (B.5) D = det (cid:12)(cid:12)(cid:12)(cid:12) A | ¯ β i N h a | ¯ β i h a | ¯G ¯A (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:12)(cid:12)(cid:12)(cid:12) N d h a | ¯G¯A (cid:12)(cid:12)(cid:12)(cid:12) = ( − ) N d (B.6)where d = h a | ¯ β i − h a | ¯G ¯AA | ¯ β i = h a | (cid:0) N − ¯G¯AA (cid:1) | ¯ β i = h a | ¯G | ¯ β i = ¯ v, (B.7)or D = u ¯ u + v ¯ v. (B.8)On the other hand, D = det (cid:12)(cid:12)(cid:12)(cid:12) h ¯ a | GA | ¯ β i − | β ih ¯ a | ¯ β i N − | β ih a | ¯G ¯A (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:12)(cid:12)(cid:12)(cid:12) h ¯ a | GB | ¯ β i N − | β ih a | ¯G ¯A (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:12)(cid:12) N − | β ih a | ¯G ¯A − B | ¯ β ih ¯ a | G (cid:12)(cid:12) = det G · det (cid:12)(cid:12) N + B ¯B (cid:12)(cid:12) (B.9)where B = A − | β ih a | , ¯B = ¯A − | ¯ β ih ¯ a | . (B.10)It follows from the Sylvester equations (2.1) that B = ¯L − AL , ¯B = L − ¯A¯L (B.11)which leads todet (cid:12)(cid:12) N + B ¯B (cid:12)(cid:12) = det (cid:12)(cid:12) N + ¯L − A ¯A¯L (cid:12)(cid:12) = det (cid:12)(cid:12) N + A ¯A (cid:12)(cid:12) = (det G ) − (B.12)which, together with (B.9), implies D = 1 . (B.13)Comparing this result with (B.8) one arrives at the identity (2.9).2 V.E. Vekslerchik References [1] Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations,
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