On a systematic approach to defects in classical integrable field theories
aa r X i v : . [ m a t h - ph ] J a n On a systematic approach to defects in classicalintegrable field theories
V. Caudrelier City UniversityCentre for Mathematical ScienceNothampton SquareLONDON EC1V 0HBUK
Abstract
We present an inverse scattering approach to defects in classical integrable fieldtheories. Integrability is proved systematically by constructing the generating func-tion of the infinite set of modified integrals of motion. The contribution of thedefect to all orders is explicitely identified in terms of a defect matrix. The underly-ing geometric picture is that those defects correspond to B¨acklund transformationslocalized at a given point. A classification of defect matrices as well as the corre-sponding defect conditions is performed. The method is applied to a collection ofwell-known integrable models and previous results are recovered (and extended) di-rectly as special cases. Finally, a brief discussion of the classical r -matrix approachin this context shows the relation to inhomogeneous lattice models and the need toresort to lattice regularizations of integrable field theories with defects. PACS numbers: 02.30.Ik, 02.30.Jr, 02.30.Zz, 11.10.Kk, 11.10.Lm E-mail: [email protected] ntroduction
The topic of defects, or impurities, in integrable systems has quite a rich literature, es-pecially for quantum aspects [1, 2, 3, 4, 6, 7, 8, 9], even if quite a lot remains to bedone. Strangely enough, the problem of integrable defects in classical field theories hadreceived less attention. The pioneering paper [10] is worth mentioning for the introduc-tion of a so-called ”spin impurity” in the nonlinear Schr¨odinger equation as a first step totackle the problem on the half-line with integrable boundary conditions. This topic hasbeen revived recently in the series of papers by P. Bowcock, E. Corrigan and C. Zambon[11, 12]. The lagrangian formalism is used in all these papers to described integrable fieldtheories with internal boundary conditions interpreted as the presence of a defect. Thedefect conditions emerge from a local lagrangian density concentrated at some fixed pointand are obtained from a variational argument. The question that is addressed is then:how to select conditions which leave the full theory integrable? The common underlyingphilosophy is to impose that a modified momentum, taking into account the presence ofthe defect, should be a conserved quantity while the breaking of translation invarianceobviously entails that the bulk momentum will not be conserved. It turns out that thisidea allows to pick up certain classes of defect lagrangian. Then, a general argumentfor integrability is based on the construction of modified Lax pair involving a limitingprocedure. It is checked explicitely for a few conserved charges of certain models. Onemust note the nice observation made in each case: this procedure yields frozen B¨acklundtransformations as the defect conditions for the fields.The object of this paper is to unify the results obtained by this case by case approach.We take advantage of the common features that have been observed. To do so, we usethe efficient inverse scattering method formalism (instead of the lagrangian formalism)and implement defect conditions corresponding to frozen B¨acklund transformations. Itis important to note that the role of B¨acklund transformations as a means to generateintegrable boundary-initial value systems solvable by inverse scattering method has beendiscovered and used in [13, 14, 15]. The idea was to fold two copies of the originalintegrable system related by B¨acklund transformations by using compatible reductionson the fields (for example u ( x ) = u ( − x )). Here, we do not fold and use the fact thatB¨acklund transformations have a very nice formulation in the inverse scattering method.They can be encoded in matrices, representing gauge transformations of the underlyingauxiliary problem and, in the present context, giving rise to defect matrices. Thanksto this formulation, we are able to prove systematically the existence of an infinite setof modified conservation laws, ensuring integrability. The main result of this paper isthe explicit identification of the generating function of the defect contributions at allorders, i.e. for any conserved charge, for any integrable evolution equation of the AKNS[16] or KN [17] schemes of the inverse scattering method. This provides an efficientalgorithm to compute the modified conserved quantities, given the defect matrices. Oneof the advantage of the method is that the proof of integrability does not require anymodification of the usual Lax pair formulation for integrable field theories. Another is1hat there is no guess work for finding the defect contributions. They are obtained froma classification of defect matrices.The paper is organized as follows. In Section 1, the general auxiliary problem formal-ism we use is presented. We establish our main results about the infinite set of conservationlaws in the presence of a defect. The generalization to several defects is also explained.In Section 2, the defect matrices are classified within a certain class of gauge transfor-mations of the auxiliary problem. In section 3, we illustrate our systematic method onseveral well-known examples of integrable nonlinear equations. They correspond to allthe classical field theories that have been explored in the lagrangian formalism (with theexception of the affine Toda field theories). For these models, all the previous resultsare recovered (and even generalized) and are extended to higher orders. Section 4 is de-voted to the extension of the method to another inverse scattering method scheme, theKaup-Newell scheme [17], which describes other classes of integrable nonlinear equationsincluding e.g. the derivative nonlinear Schr¨odinger equation. In section 5, we discuss inmore detail the question of integrability of such models with defects. It is argued that ourapproach allows to make a connection between the lagrangian approach and the classical r -matrix formalism. This requires the use of lattice regularizations. Our conclusions andperspectives for future investigations are gathered in the last section. In the AKNS scheme [16], an integrable evolution equation on the line can be formulated asa compatibility condition, or zero curvature condition, of a linear differential problem foran auxiliary wavefunction Ψ( x, t, λ ) involving two 2 × U ( x, t, λ )and V ( x, t, λ ) such that ( Ψ x = U Ψ , Ψ t = V Ψ , (1.1)where the subcripts x and t denote differentiation with respect to these variables. Inthe rest of the paper, we will drop the arguments whenever this is not misleading. Theparameter λ is called spectral parameter. Then, for appropriate choices of U and V , theintegrable system at hand is equivalent to the compatibility condition Ψ xt = Ψ tx givingrise to the so-called zero curvature condition ∀ λ , U t − V x + [ U, V ] = 0 . (1.2)Large classes of integrable nonlinear evolution equations can be described this way amongwhich some of the most famous are the cubic nonlinear Schr¨odinger (NLS), sine/sinh-Gordon (sG), Liouville, Korteweg-de Vries (KdV) or its modified version (mKdV). It2s known that this presentation allows one to construct generically the infinite set ofconservation laws associated to the integrable equation. For self-containedness, we recallthe main ideas. Let us fix U and V to be 2 × U = (cid:18) − iλ qr iλ (cid:19) ≡ − iλσ + W , V = (cid:18) A BC − A (cid:19) , (1.3)where q ( x, t ) and r ( x, t ) are the fields satisfying the evolution equation. In this paper,we will fix the class of solutions to be that of sufficiently smooth [18] decaying fields as | x | → ∞ and the following behaviour is assumed A ( x, t, λ ) → ω ( λ ) , B ( x, t, λ ) , C ( x, t, λ ) → | x | → ∞ . (1.4)The vector-valued function Ψ is split asΨ = (cid:18) Ψ Ψ (cid:19) . (1.5)Let Γ = Ψ Ψ − , then the identification of the infinite set of conservation laws follows froma conservation equation ( q Γ) t = ( B Γ + A ) x , (1.6)and a Ricatti equation for Γ Γ x = 2 iλ Γ + r − q Γ . (1.7)This equation follows directly from the x part of (1.1). The conservation equation isobtained from the t part to get Γ t , from the (12) element of (1.2) to get q t and combiningwith (1.7) and with the (11) element of (1.2) giving A x .Thus, expanding Γ as λ → ∞ Γ = ∞ X n =1 Γ n (2 iλ ) n , (1.8)the conserved quantities read I n = Z ∞−∞ q Γ n dx , n ≥ , (1.9)where Γ = − r , Γ n +1 = Γ nx + q n − X k =1 Γ k Γ n − k , n ≥ . (1.10) The decaying properties are chosen so as to ensure certain analytic properties of the scattering datafor (1.1), see e.g. [16]. Typically, a polynomial decay is sufficient. The only exception is the Liouville equation for which no specific boundary condition is assumed. .2 Implementing defect boundary conditions Generally speaking, a defect in (1 + 1)-dimensional integrable field theories can be viewedas internal boundary conditions on the field and its time and space derivatives at a givenpoint on the line. In other words, one wants to glu together two solutions of the evolutionequation in a specific way and at a particular point. To this end, let us consider anothercopy of the auxiliary problem. We introduce another Lax pair e U , e V defined as in (1.3)and (1.4) with q , r replaced by ˜ q , ˜ r . We consider the analog of (1.1) for e Ψ( x, t, λ ) = L ( x, t, λ )Ψ( x, t, λ ) . (1.11)The matrix valued function L ( x, t, λ ) satisfies the following partial differential equationsfor any x and t , L x = e U L − LU , (1.12) L t = e V L − LV . (1.13)In this paper, we want to think of this matrix as generating the defect conditions at aspecific point, x say. Following the terminology of [11, 12], L is called the defect matrix .We present a classification of the simplest nontrivial such matrices in the next section.Now we turn to the general contruction of the generating function of the infinite setof modified conservation laws due to the presence of a defect. This is the main result ofthis paper and establishes integrability for any nonlinear integrable equation of the AKNSscheme with a defect realizing a frozen B¨acklund transformation. For particular models(NLS, sG, Liouville, KdV and mKdV), it proves to all orders the results of [11, 12] aboutthe defect contribution and gives an explicit form for it. In addition, this is done withoutresorting to a modified Lax pair formalism involving a complicated limiting procedureto construct the conserved charges. To illustrate this, we will discuss those particularexamples in Section 3.To fix ideas, we choose a point x ∈ R and we suppose that the auxiliary problem(1.1) exists for x > x while the one for e U and e V exists for x < x . We also assume thatthe two systems are connected by the relations (1.12) and (1.13) at x = x . Then, thefollowing holds Proposition 1.1
The generating function for the integral of motions reads I ( λ ) = I leftbulk ( λ ) + I rightbulk ( λ ) + I defect ( λ ) , (1.14) where I leftbulk ( λ ) = Z x −∞ ˜ q e Γ dx , (1.15) I rightbulk ( λ ) = Z ∞ x q Γ dx , (1.16) I defect ( λ ) = − ln( L + L Γ) | x = x , (1.17) and L ij ’s are the entries of the defect matrix L . q Γ) t = ( B Γ + A ) x , ∀ x > x , (1.18) (cid:16) ˜ q e Γ (cid:17) t = (cid:16) e B e Γ + e A (cid:17) x , ∀ x < x , (1.19)where e Γ is defined from e Ψ as in the previous section, e Γ = e Ψ e Ψ − . From this and therapid decay of the fields, we get ∂ t Z ∞ x q Γ dx + ∂ t Z x −∞ ˜ q e Γ dx = (cid:16) e B e Γ + e A − ( B Γ + A ) (cid:17) | x = x (1.20)The crucial point now is that the right-hand-side is a total time derivative of a quantityevaluated at x = x : it is the contribution of the defect to the conserved quantities as wenow show. From (1.11) we get e Γ = ( L + L Γ)( L + L Γ) − . Then, using (1.13) at x = x to eliminate e A and e B , one gets (cid:16) e B e Γ + e A − ( B Γ + A ) (cid:17) | x = x (1.21)= (cid:8) ∂ t L + ∂ t L Γ + L ( C − A Γ − B Γ ) (cid:9) ( L + L Γ) − | x = x . (1.22)The final step consists in noting that the t part of (1.1) implies another Ricatti equationΓ t = C − A Γ − B Γ , (1.23)so that ∂ t Z ∞ x q Γ dx + ∂ t Z x −∞ ˜ q e Γ dx = ( L + L Γ) t ( L + L Γ) | x = x , (1.24)Therefore, ∂ t I ( λ ) = 0 . (1.25)Note that this holds in all generality for any integrable evolution equation in the AKNSscheme with decaying fields. The latter can be relaxed for some models, and we discussin detail below the Liouville equation for which this does not hold. It is quite straightforward to repeat the general argument for the construction of conser-vations laws in the case of several defects. Suppose we have N + 1 auxiliary problemsconnected two by two at points x , . . . , x N by matrices L , . . . , L N . Then, it is easy to seethat the contribution of these N defects is the sum of the contributions from each defect.5ndeed, using x = −∞ , x N +1 = + ∞ and otherwise obvious notations, the generatingfunction for the integral of motion reads I ( λ ) = N +1 X j =1 I jbulk ( λ ) + N X j =1 I jdefect ( λ ) , (1.26) I jbulk ( λ ) = Z x j x j − q j Γ j dx , j = 1 , . . . , N + 1 , (1.27) I jdefect ( λ ) = ln(( L j ) + ( L j ) Γ j ) | x = x , j = 1 , . . . , N . (1.28) In this section, we derive a large class of defect matrices satisfying (1.12) and (1.13)together with the associated conditions they entail on the fields: the B¨acklund transfor-mations. The latter will become the defect conditions when imposed at x = x . The matrix L preserves the zero curvature condition as is easily seen by writing L xt = L tx ,( ∀ λ , U t − V x + [ U, V ] = 0) ⇔ ( ∀ λ , e U t − e V x + [ e U , e V ] = 0) . (2.1)In other words, if q , r are solutions of the evolution equation described by ( U, V ) then˜ q , ˜ r are solutions of the evolution equation described by ( e U , e V ) under the transformationinduced by L and vice versa. This is just the usual definition of a B¨acklund transformationand this shows the connection with the idea of frozen B¨acklund transformations discussedabove. In other words, we look for defect matrices in the class of matrices realizingB¨acklund transformations between two nonlinear integrable evolution equations. Notethat the evolution equations need not be the same in general. If they are, the terminologyauto-B¨acklund transformation is usually used. Such matrices are sometimes referred toas Darboux matrices (see e.g. [19]). Even if a lot is known on these matrices, we proceedwith their derivation in the form needed for this paper. We adopt a pedestrian methodwhich does not require any previous knowledge of their theory. In particular, no referenceto the wavefunction of the auxiliary problem or to a special Riemann problem is needed(which are usually the methods encountered in the literature).Let us establish some general facts about L . First, there is some freedom in its normal-ization coming from the invariance of the zero-curvature condition under the transforma-tion ( U, V ) → ( M − U M, M − V M ) for any invertible matrix M independent of x and t .This also obviously preserves the tracelessness property. In particular, left multiplicationof L by M − amounts to apply this transformation to ( e U , e V ) while right multiplicationby M applies it to ( U, V ). Then, we have the6 roposition 2.1
The determinant of L is independent of x and t det L ( x, t, λ ) = f ( λ ) . (2.2)Proof: The result follows from the Jacobi formula(det L ( x, t, λ )) x = det L ( x, t, λ )Tr( e U − U ) , (2.3)(det L ( x, t, λ )) t = det L ( x, t, λ )Tr( e V − V ) , (2.4)and the tracelessness of U, e U , V, e V .At this stage, it is hard to go further without specifying U, e U , V, e V a bit more. Let ussimply note that given U, e U , V, e V and initial-boundary values for the fields, the integrationof (1.12) and (1.13) gives for instance (the path of integration being irrelevant) L ( x, t, λ ) = L ( x , t , λ ) + Z xx ( e U L − LU ) | τ = t dy + Z tt ( e V L − LV ) | y = x dτ . (2.5)The formal iteration of the previous equation suggests that, in general, L has a compli-cated Laurent series structure as a function of λ . In the following, we will assume that L has only a finite number of terms and, recalling that it is defined up to a scalar functionin λ , we will look for a solution of the form L ( x, t, λ ) = N X n =0 L ( n ) ( x, t ) λ − n . (2.6)Actually, we shall consider the case N = 1 which we study in detail. For convenience, wealso restrict our attention to auto-B¨acklund matrices. We comment later on on the factthat this is not necessary in our approach, one of the crucial ingredient being simply thatthe evolution equations have the same dispersion relation (see (2.18) below). The defect matrix is of the form L ( x, t, λ ) = L (0) ( x, t ) + L (1) ( x, t ) λ − . (2.7)The defining relation (1.12) is equivalent to0 = (cid:2) L (0) , σ (cid:3) , (2.8) L (0) x = i (cid:2) L (1) , σ (cid:3) + f W L (0) − L (0) W , (2.9) L (1) x = f W L (1) − L (1) W . (2.10) Note that it is implicitly assumed that both L (0) and L (1) are not trivial since otherwise, the B¨acklundtransformation is essentially the trivial one f W = W . V , e V are polynomials in λ with coefficients V ( j ) ( x, t ), e V ( j ) ( x, t ), j = 0 , · · · , N , equation(1.13) is equivalent to L (1) t = e V (0) L (1) − L (1) V (0) , (2.11) L (0) t = e V (1) L (1) − L (1) V (1) + e V (0) L (0) − L (0) V (0) , (2.12)0 = e V (2) L (1) − L (1) V (2) + e V (1) L (0) − L (0) V (1) , (2.13)... (2.14)0 = e V ( N ) L (0) − L (0) V ( N ) . (2.15)If V , e V are polynomials in λ − , the equations are the same under the exchange L (0) ↔ L (1) .Let us make a few remarks. First, when we have found the matrix L , equations (2.10)and (2.11) will give the x and t parts of the corresponding B¨acklund transformations forthe fields. Then, in traditional approaches, equation (2.9) is used to construct new solitonsolutions, f W , from given solutions W and the knowledge of the B¨acklund transformation.We will not discuss this last step in this paper and refer the reader to the vast literatureon the subject (see e.g. [20] and references therein).We now proceed with the statement of the general results of this section. Proposition 2.2 Defect matrix
The defect matrix L has the following general form L = + λ − n α + ± (cid:2) α − − a a (cid:3) / o a a n α + ∓ (cid:2) α − − a a (cid:3) / o , (2.16) where a = − i q − q ) , a = i r − r ) , (2.17) and α ± ∈ C are the ( x, t -independent) parameters of the defect. Proof: Equation (2.8) implies that L (0) in (2.7) is diagonal and then, equation (2.9) showsthat the diagonal elements do not depend on x . Therefore, we can consider equation(2.12) as | x | → ∞ . Recall that we have V, e V → ω ( λ ) σ , | x | → ∞ . (2.18)Writing ω ( λ ) = N X n =0 ω ( n ) λ n and denoting L (1) ∞ = lim | x |→∞ L (1) ( x, t ), we get L (0) t = ω (1) (cid:2) σ , L (1) ∞ (cid:3) + ω (0) (cid:2) σ , L (0) (cid:3) (2.19)= − iω (1) lim | x |→∞ ( f W L (0) − L (0) W ) (2.20)= 0 , (2.21)8here we have used equation (2.8) and (2.9) in the second equality and the fact that weconsider decaying fields in the last equality. The proof is similar if V , e V are polynomialsin λ − . So, as explained above, we can left multiply with (cid:0) L (0) (cid:1) − and work with L ′ = (cid:0) L (0) (cid:1) − L . We drop the ′ in the following but remember that L (0) should now be 1I inall the equations (2.8-2.15). Next, denote L (1) = (cid:18) a a a a (cid:19) , (2.22)and α , α its eigenvalues. Then, equation (2.9) gives immediately a = − i (˜ q − q ), a = i (˜ r − r ). Now, the elements a and a are easily computed from a a − a a = α α and a + a = α + α , (2.23)and introducing α ± = α ± α . Finally, we need to show that α ± is independent of x and t .Let ℓ and ℓ be the eigenvalues of L . It is enough to prove that ℓ and ℓ are independentof x and t . In turn, it is sufficient to prove that ℓ ℓ and ℓ + ℓ are independent of x and t . From proposition 2.1, we already know that ℓ ℓ = f ( λ ). Next, we prove ℓ + ℓ = g ( λ ).Suppose ℓ + ℓ = g ( x, t, λ ), then, using (1.12) and (2.9) g x = Tr h L ( f W − W ) i (2.24)= λ − Tr h L (1) ( f W − W ) i (2.25)= iλ Tr (cid:2) L (1) (cid:2) σ , L (1) (cid:3)(cid:3) (2.26)= 0 . (2.27)Now, g t = Tr h L ( e V − V ) i can be evaluated as x → ∞ for which we know that e V − V → g t = 0.The B¨acklund transformations associated with the matrix L read: • For the x part, a x = ˜ qa − ra , (2.28) a x = ˜ qa − qa , (2.29) a x = ˜ ra − ra , (2.30) a x = ˜ ra − qa , (2.31) • For the t part if V , e V are polynomials in λ , a t = ( e A (0) − A (0) ) a + e B (0) a − C (0) a , (2.32) a t = ( e A (0) + A (0) ) a + e B (0) a − B (0) a , (2.33) a t = − ( e A (0) + A (0) ) a + e C (0) a − C (0) a , (2.34) a t = − ( e A (0) − A (0) ) a + e C (0) a − B (0) a , (2.35)9 For the t part if V , e V are polynomials in λ − , a t = ( e A (1) − A (1) ) d + B (0) a − C (0) a , (2.36) a t = 2 A (0) a + B (0) ( a − a ) + d e B (1) − d B (1) , (2.37) a t = − A (0) a + C (0) ( a − a ) + d e C (1) − d C (1) , (2.38) a t = − ( e A (1) − A (1) ) a + C (0) a − B (0) a , (2.39)where a , a , a , a are as in Proposition 2.2. At this stage, it seems that there is anoverdetermination since there are four equations for each part whereas only two of eachtype are needed (the x and t transforms relating ˜ q and q and those relating ˜ r and r ). Itturns out that half of them are indeed redundant. Proposition 2.3 B¨acklund transformations
The B¨acklund transformations corresponding to L are given by the equations for a and a in (2.29, 2.30) and (2.33, 2.34) or (2.37, 2.38). The remaining equations for a and a can be deduced from them. Proof: We know that the eigenvalues of L (1) are constant so( a a ) x = ( a a ) x , a x + a x = 0 . (2.40)From this and the equations (2.29, 2.30) for a and a , we deduce a x ( a − a ) = a (˜ qa − ra ) + a (˜ ra − qa ) . (2.41)Now, using (˜ q − q ) a + (˜ r − r ) a = 0,( a x − ˜ qa + ra )( a − a ) = 0 . (2.42)The possibility a = a must be rejected in general since together with a x + a x = 0 itwould imply that a and a are independent of x . Thus, we obtain the equation for a and hence for a .The proof for the t part is similar. Useful identities in getting the result are obtainedfrom Tr L t = Tr h L ( e V − V ) i = 0 , (2.43)and expanding in powers of λ or λ − .We finish this general discussion by making a connection with Darboux matrices (seee.g. [19]). Suppose that α = α then we can define P = 1 α − α ( L (1) − α ) , (2.44)10nd multiply L ( λ ) by λλ + α (since it is defined up to a function of λ ) to get L ( λ ) = 1I + α − α λ + α P . (2.45)The important fact is that P is a projector ( P = P can be checked directly from the defi-nition (2.44)). The form (2.45) for L is usually encountered where the so-called B¨acklund-Darboux transformations are used to generate multi-soliton solutions from a given one.This form is also useful to exhibit the inverse of the B¨acklund matrix L − ( λ ) = 1I − α − α λ + α P . (2.46)
In this section, our results are applied on a variety of examples. We are able to reproducethe results of [11, 12] in a very simple way. For the NLS equation, we obtain a moregeneral result corresponding to a B¨acklund transformation with two real parameters (asit should, see e.g. [21]) instead of one. For the Korteweg-de Vries (KdV) and modifiedKdV equations, we obtain the defect contribution directly in terms of the original fieldsand reproduce the lagrangian approach expressions in terms of ”potential” fields. For eachmodel, the defect conditions are given and consistently reproduces the associated well-known Backl¨und transformations, but taken at x = x here, as expected by construction.We gather the examples in three classes according to certain symmetry considerationsyielding information on the defect parameters α ± . q = u , r = ǫu ∗ , ǫ = ± , u complex scalar field Let us introduce K = (cid:18) ǫ (cid:19) . (3.1)We have the following symmetries (we drop x and t ) U ∗ ( λ ∗ ) = KU ( λ ) K − , e U ∗ ( λ ∗ ) = K e U ( λ ) K − , (3.2)and we assume that V and e V have the same properties. Therefore, we can look for theB¨acklund matrix L such that L ∗ ( λ ∗ ) = KL ( λ ) K − . This implies α = α ∗ so α + ∈ R and α − ∈ i R . Then, it can be shown that the remaining conditions imply that the nontrivialB¨acklund matrix reads L ( λ ) = 1I + λ − n α + ± i p β + ǫ | ˜ u − u | o − i (˜ u − u ) i ǫ (˜ u ∗ − u ∗ ) n α + ∓ i p β + ǫ | ˜ u − u | o , (3.3)11here β = iα − ∈ R . For this class, the defect matrix, and hence the defect conditionsare parametrized by two arbitrary real numbers α + and β . In the case ǫ = −
1, we seethat the transformation is such that | ˜ u − u | ≤ β . We must take into account the factthat r = ǫq ∗ in the discussion of the integrals of motion. In particular, it turns out thatto generate real integral of the motion (real classical observables), one has to consider thefollowing combination I sym ( λ ) = i ( I ( λ ) − I ∗ ( λ ∗ )) . (3.4)So in practice, we will compute the contribution of the defect as I symdefect ( λ ) = − i (ln( L + L Γ) − ln( L + L Γ) ∗ ) | x = x . (3.5)Now we expand (3.4) in powers of λ − up to order 3 to illustrate the method. Forconvenience, we define Ω ǫ = p β + ǫ | ˜ u − u | . At order λ − , we find that the modifiedconserved density reads Z x −∞ | ˜ u | dx + Z ∞ x | u | dx ∓ ǫ Ω ǫ | x = x , (3.6)where the last term is the explicit defect contribution. Similarly, at order λ − , the modifiedconserved momentum is Z x −∞ i (˜ u ˜ u ∗ x − ˜ u ∗ ˜ u x ) dx + Z ∞ x i ( uu ∗ x − u ∗ u x ) dx − i [( u ∗ ˜ u − u ˜ u ∗ ) ∓ ǫβ Ω ǫ ] | x = x . (3.7)Finally, the modified conserved energy is found to be Z x −∞ ( | ˜ u x | + ǫ | ˜ u | ) dx + Z ∞ x ( | u x | + ǫ | u | ) dx + h ∓ Ω ǫ ( | ˜ u | + | u | ) ∓ ǫ ǫ (3 β − Ω ǫ ) − iβ (˜ uu ∗ − u ˜ u ∗ ) i | x = x . (3.8)These results hold for any member of class I so in particular they hold for the cubicfocusing ( ǫ = −
1) or defocusing ( ǫ = 1) nonlinear Schr¨odinger equation for the complexscalar field u iu t + u xx = ǫ | u | u , (3.9)and similarly for ˜ u . Indeed, this equation is obtained in the AKNS scheme by taking A ( λ ) = − iλ + i | u | , B ( λ ) = ǫC ∗ ( λ ∗ ) = 2 λ + iu x , (3.10)and similarly for ˜ u .Now the corresponding defect conditions can be derived from the general B¨acklundtransformations given in the previous section. From the symmetry a = ǫa ∗ , we need only12onsider (2.29) and (2.33). Note that the x part of the defect conditions is the same forall the models in class I. Here, for NLS, we have at x = x (˜ u − u ) x = iα + (˜ u − u ) ± (˜ u + u ) p β + ǫ | ˜ u − u | , (3.11)(˜ u − u ) t = − α + (˜ u − u ) x ± i (˜ u + u ) x p β + ǫ | ˜ u − u | + i (˜ u − u )( | u | + | ˜ u | ) . (3.12)Setting ǫ = − − sign in (3.6), (3.7), (3.8), we note that we have toimpose further β = 0 to recover the results of [12] (where the notation Ω = p α − | ˜ u − u | is used). This is due to the fact that the lagrangian the authors took for the defect (the B functional in their notations) is not the most general one. It corresponds to particularB¨acklund transformations with β = 0. q = u , r = ǫu , , ǫ = ± , u real scalar field This class is a subclass of class I with u ∗ = u (and ˜ u ∗ = ˜ u ). this immediately implies α + = 0 . (3.13)The defect matrix for this class reads L ( λ ) = 1I + λ − (cid:18) ± i p α + ǫ (˜ u − u ) − i (˜ u − u ) i ǫ (˜ u − u ) ∓ i p α + ǫ (˜ u − u ) (cid:19) . (3.14)So we can simply use the result of the previous class, setting u ∗ = u and ˜ u ∗ = ˜ u (and thus,without symetrizing). We exhibit the first orders for a few examples, taking advantage ofspecific forms of the defect matrix in each case. The modified Korteweg-de Vries equation equation u t − ǫu u x + u xxx = 0 , (3.15)is obtained in the AKNS scheme by taking A ( λ ) = − iλ − iλǫu , B ( λ ) = ǫC ∗ ( λ ∗ ) = 4 λ u + 2 iλu x − u xx + 2 ǫu , (3.16)and similarly for ˜ u . The modified conserved density reads Z x −∞ ˜ u dx + Z ∞ x u dx ∓ ǫ p α + ǫ (˜ u − u ) | x = x . (3.17)The modified conserved momentum is Z x −∞ ˜ u ˜ u x dx + Z ∞ x uu x dx −
12 (˜ u − u + α ) | x = x , (3.18)13s can be directly checked by integration by parts (the constant α being irrelevant).Finally, the next order yields Z x −∞ (˜ u x + ǫ ˜ u ) dx + Z ∞ x ( u x + ǫu ) dx ∓ Ω ǫ h (˜ u + u ) − ǫ ǫ i | x = x , (3.19)where here Ω ǫ = p α + ǫ (˜ u − u ) . The corresponding defect conditions read(˜ u − u ) x = ± (˜ u + u ) p α + ǫ (˜ u − u ) , (3.20)(˜ u − u ) t = ± (cid:8) ǫ (˜ u + u ) − (˜ u + u ) xx (cid:9) p α + ǫ (˜ u − u ) . (3.21)It is worth noting that everything is expressed directly in terms of the fields u and ˜ u . Thisshould be compared with the Lagrangian approach of [12] where this was not possible.The use of ”potential” fields p and q such that ˜ u = p x and u = − q x is required in thisformulation. Under this substitution, an alternative form of the defect matrix can bederived L ( λ ) = 1I ± λ − iα (cid:18) cos(˜ v − v ) − sin(˜ v − v ) − sin(˜ v − v ) − cos(˜ v − v ) (cid:19) . (3.22)and we consistently recover their result for the defect contribution to the first conservedquantity (setting ǫ = − ǫ p α + ǫ (˜ u − u ) | x = x becomes α (cos( p − q ) − | x = x , (3.23)(a constant can always be added). This example illustrates the case V , e V polynomials in λ − . The sine-Gordon equation inlight-cone coordinates v xt = sin v , (3.24)is obtained by setting u = − v x , ǫ = − A ( λ ) = i cos v λ , B ( λ ) = i sin v λ , (3.25)and similarly for ˜ v . For this model, the defect matrix takes the nice following form L ( λ ) = 1I ± iα λ (cid:18) cos ˜ v + v − sin ˜ v + v − sin ˜ v + v − cos ˜ v + v (cid:19) , (3.26)where α is a nonzero real parameter. Therefore, the modified conserved momentum reads14 Z x −∞ ˜ v x dx + 14 Z ∞ x v x dx ± α cos ˜ v + v | x = x . (3.27)14his is just the result in [30] for the momentum and energy but expressed here in light-conecoordinates. At the next order, as can be anticipated from (3.18), the bulk contributioncan be integrated by parts and combines nicely with the defect contribution, leaving theconstant − α as the conserved quantity. Finally, the third order conserved quantity is12 Z x −∞ ( − ˜ v xx + ˜ v x dx + 12 Z ∞ x ( − v xx + v x dx ± α v + v v x + ˜ v x v x + v x + 2 α ) | x = x . (3.28)The defect conditions at x = x are given by(˜ v − v ) x = ± α sin ˜ v + v , (3.29)(˜ v + v ) t = ± α sin ˜ v − v . (3.30) This is another example of evolution equation obtained from V and e V polynomials in λ − . We need to discuss this equation in detail since, as mentioned above, the condition ofvanishing fields at infinity and the boundary conditions (1.4) are not applicable. However,it is straightforward to prove that Proposition 2.2 is still valid. Also, it is still possible toprove Proposition 1.1 with a slight modification, as we now show. The Liouville equationin light-cone coordinates for the field vv xt = 2 e v , (3.31)is obtained by taking u = v x , ǫ = 1 , A ( λ ) = ie v λ = − B ( λ ) , (3.32)and similarly for ˜ v . The conservation laws (1.18) and (1.19) still hold since they do notdepend on the boundary conditions. But now, (1.20) becomes ∂ t Z ∞ x q Γ dx + ∂ t Z x −∞ ˜ q e Γ dx = lim x →∞ ( B Γ + A ) − lim x →−∞ ( e B e Γ + e A ) (3.33)+ (cid:16) e B e Γ + e A − ( B Γ + A ) (cid:17) | x = x . (3.34)The last term in the right-hand-side is treated as before. The point is to recast the othertwo terms as time derivatives. For this equation, one has A = C = − B so B Γ + A = A (1 − Γ) and the Ricatti equation (1.23) becomesΓ t = A (1 − Γ) . (3.35)Therefore, since Γ = 1 (this can be seen from (1.7)) B Γ + A = − ∂ t ln(1 − Γ) . (3.36)15he same result holds for e Γ. Therefore, Proposition 1.1 is modified to the following.The generating function for the integral of motions of the Liouville equation reads I ( λ ) = I leftbulk ( λ ) + I rightbulk ( λ ) + I defect ( λ ) , (3.37)where I leftbulk ( λ ) = 12 Z x −∞ ˜ v x e Γ dx − lim x →−∞ ln(1 − e Γ) , (3.38) I rightbulk ( λ ) = 12 Z ∞ x v x Γ dx + lim x →∞ ln(1 − Γ) , (3.39) I defect ( λ ) = − ln( L + L Γ) | x = x , (3.40)and L ij ’s are the entries of the B¨acklund matrix L . The additional contributions essentiallykill terms arising from trivial integration by parts in the integrals of motion.The defect matrix can be written L ( λ ) = 1I + iγ λ e ± ˜ v + v (cid:18) − − (cid:19) , (3.41)where γ is a nonzero real constant. At first order, the modified conserved momentumreads 12 Z x −∞ ˜ v x dx + ˜ v x | −∞ + 12 Z ∞ x v x dx − v x | ∞ − γe ± ˜ v + v | x = x . (3.42)As is now customary, the next order combines nicely to produce( v x − v xx ) | ∞ − (˜ v x − v xx ) | −∞ , (3.43)which can be checked directly to be a constant. Finally, at the third order, we have12 Z x −∞ (cid:18) ˜ v xx + ˜ v x (cid:19) dx − (cid:18) ˜ v xxx − ˜ v x − ˜ v x ˜ v xx (cid:19) | −∞ + 12 Z ∞ x (cid:18) v xx + v x (cid:19) dx + (cid:18) v xxx − v x − v x v xx (cid:19) | ∞ − γ e ± ˜ v + v (˜ v x + v x + ˜ v x v x ) | x = x . (3.44)The defect conditions are given by(˜ v − v ) x = γ e ± ˜ v + v , (3.45)(˜ v + v ) t = ± γ e ∓ ˜ v + v ( e ˜ v − e v ) , (3.46)16 .3 Class III: q = u , r = ǫ , , ǫ = ± , u real scalar field For this class, there is no special symmetry. The derivation of the defect matrix deservedspecial attention for this class. Indeed, if we assume that ( e U , e V ) has the same form as( U, V ), as we have done so far, then an immediate consequence is a = 0 and a = a = α are constant. From this, equation (2.28) implies ˜ u = u i.e. we get the trivial defectconditions. A solution to this problem is to left multiply L by σ or equivalently toconsider ( σ e U σ , σ e V σ ). This amounts to change the sign of the off-diagonal terms.Then, equation (2.30) implies α + = 0. Finally, taking into account the reality of the field,one gets the defect matrix for this class L ( λ ) = 1I + λ − (cid:18) ± i p β + 2 ǫ (˜ u + u ) i (˜ u + u ) − iǫ ∓ i p β + 2 ǫ (˜ u + u ) (cid:19) , (3.47)We apply our method to the Korteweg-de Vries equation u t − ǫuu x + u xxx = 0 , (3.48)which is obtained in the AKNS scheme by taking A ( λ ) = − iλ − iǫλu + ǫu x , (3.49) B ( λ ) = 4 λ u + 2 iλu x + 2 ǫu − u xx , (3.50) C ( λ ) = 4 ǫλ + 8 u , (3.51)and similarly for ˜ u , with appropriate change of signs. We make direct use of (1.14). Thefirst nontrivial order is λ − and for the modified conserved density reads12 Z x −∞ ˜ u dx + 12 Z ∞ x u dx ∓ p β + 2 ǫ (˜ u + u ) (cid:0) ǫ (˜ u + u ) − β (cid:1) | x = x . (3.52)The defect conditions are given by(˜ u + u ) x = ± (˜ u − u ) p β + 2 ǫ (˜ u + u ) , (3.53)(˜ u + u ) t = ± (cid:0) ǫ (˜ u − u ) − (˜ u − u ) xx (cid:1) p β + 2 ǫ (˜ u + u ) . (3.54)Once again, we note that everything is expressed directly in terms of the initial fields u and ˜ u while this was not possible in the lagrangian approach. It is easy to make contactwith the latter by setting u = q x and ˜ u = p x . Then,( p + q ) x = ǫ (cid:2) ( p − q ) − β (cid:3) . (3.55)Taking ǫ = 1 and setting β = − α , we recover the result of [12] for the defect contributionto the density (the momentum in their setting)12 Z x −∞ p x dx + 12 Z ∞ x q x dx , (3.56)that is (cid:18) − α ( p − q ) −
112 ( p − q ) (cid:19) | x = x . (3.57)17 .4 Remarks We will not go into the analysis of the higher N case in (2.6). We simply note that a largeclass of higher N defect matrices is provided by products of N = 1 defect matrices. Thecorresponding defect conditions are then simply compositions of the defect conditionswe derived above. In other words, the defect matrices we constructed have a groupstructure, as can be seen directly from (1.12) and (1.13). Furthermore, Bianchi’s theoremof permutability (see e.g. [19]) implies that this is an abelian group. An importantquestion concerns the existence and properties of those higher N defect matrices whichdo not factorize as products of N = 1 ones. Their study would shed new light on possiblenew defect conditions for the well-known systems we discussed. Also, the question ofdefect matrices preserving integrability but which do not fall at all in the class discussedhere remains entirely open. In this sense, no claim of uniqueness of defect matrices ismade and one should remember that the proposed approach here is sufficient to ensureintegrability in the presence of a defect. The issue of finding necessary defect conditionsfor integrability is not answered. It should also be noted that following the linearizationargument of [11, 12], it appears that the defect conditions constructed here allow for puretransmission only. This has been checked explicitely for all the examples given above. The Kaup-Newell (KN) [17] scheme goes along the same steps as the AKNS scheme toproduce integrable evolution equations with the essential difference that the matrix U involved in the x part of the auxiliary problem (1.1) has the following form U KN = (cid:18) − iλ λqλr iλ (cid:19) = − iλ σ + λW . (4.1)One well-known model obtained in this scheme is the derivative nonlinear Schr¨odingerequation iu t + u xx = 2 ǫ ( | u | u ) x , (4.2)where q = u = ǫr , ǫ = ± V KN should read V KN = (cid:18) − iλ − iǫλ | u | λ u + iλu x + ǫλ | u | u ǫλ u ∗ − iǫλu ∗ x + λ | u | u ∗ iλ + iǫλ | u | (cid:19) . (4.3)It turns out that our method works for this scheme too with the appropriate modifications.We proceed as before by taking two copies of the auxiliary problem related by a matrix M such that M x = e U KN M − M U KN , (4.4) M t = e V KN M − M V KN . (4.5)Then we assume that the two copies are related by M at some point x = x .18 roposition 4.1 The generating function for the integral of motions reads I ( λ ) = I leftbulk ( λ ) + I rightbulk ( λ ) + I defect ( λ ) , (4.6) where I leftbulk ( λ ) = Z x ∞ λ ˜ q e Γ dx , (4.7) I rightbulk ( λ ) = Z ∞ x λq Γ dx , (4.8) I defect ( λ ) = − ln( M + M Γ) | x = x , (4.9) and M ij ’s are the entries of the B¨acklund matrix M . For this scheme, Γ and e Γ have adifferent expansion as λ → ∞ Γ = ∞ X n =0 Γ n (2 iλ ) n +1 , (4.10) with Γ = − r , Γ n +1 = 2 i Γ nx + q n X p =0 Γ p Γ n − p , (4.11) and similarly for e Γ . Proof: The proof is the same as that of Proposition 1.1, the only difference being that theconservation equations now read( q Γ) t = 1 λ ( B Γ + A ) x , ∀ x > x , (4.12) (cid:16) ˜ q e Γ (cid:17) t = 1 λ (cid:16) e B e Γ + e A (cid:17) x , ∀ x < x , (4.13)due to the different λ dependence of U KN . The latter is also responsible for the differentseries expansions of Γ and e Γ coming from the following Ricatti equationΓ x = λr + 2 iλ Γ − λq Γ , (4.14)and similarly for e Γ.To apply this to specific models, we would need some sort of classification of thematrices M along the lines of what is available for L . However, the author is not awareof such results. It is a problem for future investigation.19 Discussion of integrability
So far, we have shown how the infinite set of conservation laws is modified by the presenceof a defect described by a matrix for any evolution equation falling into the AKNS or KNschemes. The role of an infinite set of conserved quantities is well-known in the construc-tion of action-angle variables in the inverse scattering method for evolution equations onthe line. In turn, this construction ensure the integrability of the system in the sense thatin terms of the new variables, the evolution in time is very easy to solve. Then, usingthe inverse part of the method (the Gelfan’d-Levitan-Marchenko equations [22, 23]) onecan deduce the time evolution of the original fields (see e.g. [16]). So, in this sense, thesystems with defect we have considered are integrable .A traditional complementary view is to reformulate the evolution equation as Hamil-tonian systems with a Poisson structure. In this formalism, the idea is to show that theconserved quantities previously constructed form a commutative Poisson algebra contain-ing the Hamiltonian which generates the time evolution of the fields. Then, one talksabout integrability in the sense of Liouville. One of the advantages of this reformulationis the possibility of quantization and this was the object of the quantum inverse scatteringmethod, see e.g. [24].The method of the classical r -matrix [25] has proved very useful and fundamentalin the discussion of these issues for classical integrable systems on the whole line. Thesituation on the half-line is also well understood [26]. Our aim in this section is to discussthe situation with a defect.Let us first recall some facts about the method of the classical r -matrix for ultralocalmodels. The basic ingredient is the 2 × T ( x, y, λ ), x < y , definedas the fundamental solution of the x -part of the auxiliary problem at a given time (notexplicitely displayed in T) ∂ y T ( x, y, λ ) = U ( y, t, λ ) T ( x, y, λ ) , T ( x, x, λ ) = 1I , (5.1)where U is given as in (1.3). The important result reads (see e.g. [27]) { T ( x, y, λ ) , T ( x, y, µ ) } = [ r ( λ − µ ) , T ( x, y, λ ) T ( x, y, µ )] , (5.2)where r ( λ ), the classical r -matrix, is a 4 × T rT ( x, y, λ ) in λ are in involution and the Hamiltonian is oneof the them. For systems on the line, the principle is the same but one has to take theinfinite volume limit of (5.2), usually with special care.For the problem with defect, we would like to mimic this procedure. The main dif-ference here is that there is something nontrivial going on at a single point x = x and For completeness, one should perform the inverse scattering method and identify the action-anglevariables in this context. This is left for future work. L (or M ). The standard procedure becomes ill-defined in the continu-ous case as it involves singular Poisson brackets of the type ” δ (0)” where δ ( x ) is the Diracdistribution.Indeed, the analog of the transition matrix for x < x < y is T x ( x, y, λ ) = T ( x , y, λ ) L − ( x , t, λ ) e T ( x, x , λ ) , (5.3)and the task of computing {T x ( x, y, λ ) , T x ( x, y, µ ) } , (5.4)involves computing { L − ( x , t, λ ) , L − ( x , t, λ ) } . There does not seem to be a direct ap-proach starting from the explicit form of L as classified in this paper. However, in thecontext of finite-dimensional integrable systems, important results were obtained by E.Sklyanin in [29] concerning the canonicity of B¨acklund transformations in the formalismof the r -matrix approach. Transposing the results in the present context and assuming wehave the same theories on both sides of the defect, i.e. T ( x, y, λ ) and e T ( x ′ , y ′ , λ ) satisfy(5.2), for x, y > x and x ′ , y ′ < x respectively, it seems reasonable to postulate that L − is just another representation of the Poisson algebra (5.2) so that { L − ( x , t, λ ) , L − ( x , t, λ ) } = [ r ( λ − µ ) , L − ( x , t, λ ) L − ( x , t, λ )] . (5.5)From this it immediately follows that {T x ( x, y, λ ) , T x ( x, y, µ ) } = [ r ( λ, µ ) , T x ( x, y, λ ) T x ( x, y, µ )] . (5.6)Note that the generalization to N defects can be described in this formalism as well.Using the same notations as Section 1.3, one construct the monodromy matrix T N ( λ ) = T N +1 ( x N , x N +1 , λ )( L N ) − ( x N , t, λ ) . . . ( L ) − ( x , t, λ ) T ( x , x , λ ) × ( L ) − ( x , t, λ ) T ( x , x , λ ) , (5.7)with all the L j ’s satisfying (5.5).This discussion brings us to the connection between the lagrangian approach of [11, 12]to integrable defects and a quite standard procedure to implement inhomogeneities orimpurities in discrete integrable systems. Indeed, through the approach of this paper,one can reformulate the lagrangian approach in terms of a transition matrix made oftwo bulk parts and a localised defect part realising a different representation of the samePoisson algebra. But this is exactly what is usually done to implement so-called impuritiesor inhomogeneities in discrete integrable systems. Let us consider a discrete ultralocalsystem of length ℓ with N sites. The transition matrix T ( λ ) is then a product of localmatrices t j ( λ ), j = 1 , . . . , N satisfying { t j ( λ ) , t k ′ } = δ jk [ r ′ ( λ − µ ) , t j ( λ ) t k ′ ( µ )] , (5.8) For convenience, in the rest of the paper, we stick to a single notation L for the defect matrix.
21n the same representation (the index j represents the space of dynamical variables while0, 0 ′ are auxiliary spaces, C here). One can check then that T ( λ ) = t N ( λ ) . . . t ( λ )satisfies { T ( λ ) , T ′ ( µ ) } = [ r ′ ( λ − µ ) , T ( λ ) T ′ ( µ )] , (5.9) i.e. exactly (5.2). To introduce an inhomogeneity at site j say, one then chooses adifferent representation ˆ t j ( λ ) of the same algebra. This does not change the properties of T ( λ ) = t N ( λ ) . . . ˆ t j ( λ ) . . . t ( λ ) but influences the physical quantities (e.g. the integrals ofmotion) that can be computed since the latter depend on the representation at each site.Again, the introduction of several inhomogeneities can be done straightforwardly. Thisactually provides a solution to the above problem of computing Poisson brackets of defectmatrices by considering lattice regularizations of integrable field theories and changingrepresentations appropriately at local sites to generate defects.From this point of view, one can anticipate the outcome of the quantization of thisapproach. It is known that (5.9) is the classical ( ~ →
0) limit of the quantum Yang-Baxteralgebra R ′ ( λ − µ ) τ j ( λ ) τ j ′ ( µ ) = τ j ′ ( µ ) τ j ( λ ) R ′ ( λ − µ ) , (5.10)where R ( λ ) is the quantum R matrix associated to rR ( λ ) = 1I + i ~ r ( λ ) + O ( ~ ) , (5.11)and τ j ( λ ) → t j ( λ ) in the ~ → L ( λ ) encoding thedefect conditions will satisfy the quantum Yang-Baxter algebra. This gives some supportto the ad hoc quantization procedure adopted in [30] for the sine-Gordon with integrabledefect. We would like to stress that the above programme of discretization has beencompleted for the defect sine-Gordon model in the important paper [31], both at classicaland quantum level. The approach is based on the notion of ancestor algebra [32]. Conclusions and outlook
In this paper, we have reformulated the lagrangian approach to the question of integrabledefects in the language of the inverse scattering method, taking advantage of the commonfeatures that had been observed on a case by case study: frozen B¨acklund transformationsas defect conditions ensure integrability. The reformulation allows a systematic proof ofthis as well as an efficient computation of the modified conserved quantities to all ordersin terms of the defect matrix. The latter, and the associated defect conditions, can beclassified and we performed these computations for a certain class of matrices. Takingparticular examples, we recovered and even generalized all the previous results obtained bythe lagrangian method. It should be emphasized that this procedure provides a sufficient22pproach to the question of integrable defects in classical field theories and, by no means,represents a complete picture of the story.Rather, it is a first step for future developments among which further study of theclassical r matrix approach and quantization of the method are important. Let us mentionalso the contruction of other integrable defects allowing if possible reflection as well. Ifapplicable, the quantization should then be related to existing quantum algebraic frame-works like the Reflection-Transmission algebras [6]. Finally, the complete setup of thedirect and inverse part of the method for the actual construction of the solutions, espe-cially of soliton type, should shed new light on the results already obtained by the moredirect approach of [11, 12]. Acknowledgements
It is a pleasure to thank E. Sklyanin and E. Corrigan for discussions and encouragementsin the course of this paper. We also warmly thank E. Ragoucy for useful comments inthe final stage of this work.
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