Multiform Description of the AKNS Hierarchy and Classical R-Matrix

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Regarding the practical aspect, the limitation appeared in connection with integrable classical field theories in the presence of a defect [10], specifically when one tried to under- stand Liouville integrability of that context, see [11, 12] and references therein. This was explained in [13, 14] where the idea of a “dual” Hamiltonian formulation of a given classical field theory was introduced to resolve the issue. An important observation, which was further developed in [15], is that in this dual formulation, both Lax matrices of the pair describing the model at hand possess the same classical r-matrix structure, pointing to a space-time duality of this structure. This was unlikely to be a coincidence and the question of the origin of this common structure led to the work [16] where, in the case of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [3], it was traced back to a Lie-Poisson bracket used by Flashka-Newell-Ratiu (FNR) in [17].

Regarding the conceptual limitation, this “dual” description was still unsatisfactory in the sense that one had to choose one of the independent variables or the other as a starting point, but it did not seem possible to construct a classical r-matrix formalism capable of including both independent variables simultaneously. This flaw of the (traditional) Hamiltonian formulation of a field theory has been known for a long time. This is what motivated the early work of De Donder and Weyl [18, 19] aimed at establishing a covariant Hamiltonian field theory. A wealth of subsequent developments ensued, driven in particular by the desire to construct a covariant canonical quantization scheme. This area is too vast to review faithfully here and we only refer, somewhat arbitrarily and with apologies to missed authors, to [20] and the references in [21]. As we showed in [21], some of the ideas in that area provide a solution to our conceptual problem: can we construct a covariant Poisson bracket such that the Lax form (sometimes also called Lax connection) possesses a classical r-matrix structure and such that the zero curvature equation can be written as a covariant Hamilton equation? This success in giving the r-matrix a covariant interpretation was illustrated explicitly on three models: the nonlinear Schrödinger (NLS) equation, the modified Korteweg-de Vries (mKdV) equation and the sine-Gordon model in laboratory coordinates. Each model was considered in its own right as an integrable classical field theory. However, as mentioned above, it is known that they come in hierarchies. Of those three models, two (NLS and mKdV) naturally fit into the AKNS hierarchy. We can now pose the problem that is addressed in the present work: how can we extend our results in [21] to the whole AKNS hierarchy? In other words, denoting by Q(k)(λ), k 0 the set of Lax matrices associated to the flows with respect to the variables xk, k 0 in the AKNS≥ hierarchy, is it possible to construct a Poisson bracket which can take the Lax form≥ ∞ W (λ)= Q(k)(λ) dxk kX=0 as an argument and which exhibits the classical r-matrix structure known to exist for each individual Q(k)(λ) [16]? Also, can we write the entire set of zero curvature equations of the hierarchy, i.e.

∂ Q(j)(λ) ∂ Q(k)(λ) + [Q(j)(λ),Q(k)(λ)]=0, j, k 0 k − j ≥ in Hamiltonian form? The appropriate generalisation of the covariant Poisson bracket and the covariant Hamiltonian used in [21] to tackle this problem for individual classical field theories was introduced recently by the authors in [22]. We call them multi-time Poisson bracket (for the obvious reason that it can be used to generate the flows with respect to all the “times” xk of the hierarchy) and Hamiltonian multiform respectively. The latter terminology comes from the fact that this object is derived from what is called a Lagrangian multiform, a notion introduced by Lobb and Nijhoff in [23] and which encodes integrability in a variational way. At first, Lagrangian multiforms appeared in the realm of fully discrete integrable systems, in order to encapsulate multidimensional consistency, which captures the analogue of the commutativity of Hamiltonian flows in continuous integrable systems. This original work stimulated further developments in discrete integrable systems [24, 25, 26, 27, 28, 29], then progressively in continuous finite dimensional systems, see e.g. [30, 31], 1 + 1-dimensional field theories, see e.g. [32, 33, 34, 35, 36, 37], and the first example in 2 + 1-dimensions [38]. The proposed generalised variational principle produces the standard Euler-Lagrange equations for the various equations forming an integrable hierarchy as well as additional equations, originally called corner equations, which can be interpreted as determining the allowed integrable Lagrangians themselves. The set of all these equations is called multiform Euler-Lagrange equations. In [22], we realised that one could apply successfully the strategy we used in [21] to obtain a covariant Poisson bracket and

2 covariant Hamiltonian, which is based on the data of a single Lagrangian, to a Lagrangian multiform instead. The outcome is the desired multi-time Poisson bracket and Hamiltonian multiform. Our main results are: 1. We introduce for the first time a Lagrangian multiform for the complete AKNS hierarchy, see equations (3.13)-(3.14). This is achieved using a generating Lagrangian L (λ, µ) which is a double formal series whose coefficients are the coefficients of the desired Lagrangian multiform. It is remarkable that we can produce a closed form expression in generating form while previous attempts involved complicated iterative procedures. 2. Given this Lagrangian multiform L , we obtain the multi-time Poisson bracket , and the Hamiltonian multiform associated with it. {| |} H ∞ 3. We prove that the Lax form W (λ) = Q(k)(λ) dxk possesses a classical r-matrix structure k=0 with respect to the multi-time PoissonX bracket above, i.e.

W (λ), W (µ) = [r (λ µ), W (λ)+ W (µ)], {| 1 2 |} 12 − 1 2

where r12(λ) is the so-called rational r-matrix. This shows, together with [21] and [22], the potential of covariant Hamiltonian ﬁeld theory to reproduce and generalise one of the crucial identities in integrable systems.

(j) 4. We show the Hamiltonian multiform nature of the set of zero curvature equations ∂iQ (λ) (i) (i) (j) − ∂j Q (λ) + [Q (λ),Q (λ)] = 0, i, j 0. We do this by proving that it is equivalent to the multiform Hamilton equations for the∀ Lax≥ form, i.e. 1

∞ dW (λ)= H , W (λ) dxi dxj , {| ij |} ∧ iphase space coordinates e(λ) and f(λ) that we use to our advantage for many of our proofs. For a better ﬂow, some of those long, and not necessarily illuminating, proofs are gathered in Appendix B.

2 Algebraic construction of the AKNS hierarchy

In the 1983 paper [17], Flashka, Newell and Ratiu introduced an algebraic formalism to cast the soliton equations associated with the AKNS hierarchy into what is known as the Adler-Kostant-Symes scheme [39, 40, 41]. At the same period, the russian school unraveled the structures underlying this type of construction which culminated in the classical r-matrix theory [7], and the introduction of the notion of Poisson-Lie group [8]. Here, we review some aspects of this topic, freely adapting and merging notations and notions coming from both sources. It had been known before [17], since the

∞ ∞ 1 Here and in the rest of the paper the notation X means X . i

3 work of [3], that the so-called AKNS hierarchy can be constructed by considering an auxiliary spectral problem of the form iλ q(x) ∂xψ = − ψ P ψ (λP0 + P1)ψ , (2.1) r(x) iλ ≡ ≡ where 0 q P = , P = iσ , (2.2) 1 r 0 0 − 3 as well as another equation of the form2 ∂ ∂ ψ = Q(n)(λ)ψ , ∂ := , (2.3) n n ∂xn with Q(n)(λ) = λnQ + λn−1Q + + Q where each Q is a 2 2 traceless matrix. Then the 0 1 ··· n i × compatibility condition ∂x∂nψ = ∂n∂xψ translates into the well-known zero-curvature equation ∂ P (λ) ∂ Q(n)(λ) + [P (λ),Q(n)(λ)]=0 . (2.4) n − x By setting to zero every coefficient of powers of λ one obtains a series of equations that allow to find Q0,..., Qn recursively. This produces Q0 = P0, Q1 = P1 (up to some normalisation constants) and the entries of Qj with j 2 are found to be polynomials in q, r and their derivatives with respect to x. The last of these equations≥ is ∂ P ∂ Q + [P ,Q ]=0 , (2.5) n 1 − x n 1 n and produces a partial differential equation for q and r viewed as functions of x and xn which is integrable. Different values of n gives the successive equations of the AKNS hierarchy. We list them for n =0, 1, 2, 3, giving the name of the corresponding famous example (which is usually obtained by a further reduction, e.g. r = q∗ for n = 2 gives the (de)focusing nonlinear Schrödinger equation). ± q0 = 2iq, r0 =2ir Scaling , − 1 q1 = qx , r1 = rx Translation x x + x , i 2 i 2 7→ (2.6) q2 = 2 qxx iq r , r2 = 2 rxx + iqr Non-linear Schrödinger equation , q = 1 q −+ 3 qrq , r = −1 r + 3 qrr Modified Korteweg-de Vries equation 3 − 4 xxx 2 x 3 − 4 xxx 2 x It turns out that all these equations can be interpreted as Hamiltonian flows which commute with each other and can therefore be imposed simultaneously on the variable q and r. This is ensured by that fact that the following zero curvature equations hold for any k,n 0 (by setting x = x1 and Q(1) = P ), ≥ ∂ Q(k)(λ) ∂ Q(n)(λ) + [Q(k)(λ),Q(n)(λ)]=0 . (2.7) n − k In [17], these facts and several others were cast into the algebraic setup of the Adler-Kostant-Symes scheme whereby one can introduce integrable Hamiltonian systems based on the decomposition of a Lie algebra into two Lie subalgebras which are isotropic with respect to an ad-invariant nondegenerate symmetric bilinear form on the Lie algebra. For the AKNS hierarchy, [17] use the the Lie algebra of formal Laurent series in a variable λ with coefficients in the Lie algebra sl(2, C), i.e. the Lie algebraL of elements of the form N X(λ)= X λj , X sl(2, C); for some integer N, (2.8) j j ∈ j=−∞ X with the bracket given by k [X, Y ](λ)= [Xi, Yj ]λ . (2.9) Xk i+Xj=k There is a decomposition of into Lie subalgebras = − where L L L ⊕ L+ −1 ∞ j j − = X λ , = X λ ; X =0 j>N for some integer N 0 . (2.10) L { j } L+ { j j ∀ ≥ } j=−∞ j=0 X X 2Traditionally, the flows thus defined are associated to “time” variable tn but one of the main points of [17] is that they all play the same role as x which could be viewed as t1 in this hierarchy. We simply denote them all by xn since whether they play the role of a space or time variable is really up to interpretation.

4 This yields two projectors P+ and P−. The following ad-invariant nondegenerate symmetric bilinear form is used, for all X, Y , ∈ L (X, Y )= Tr(X , Y ) Res Tr(X(λ)Y (λ)) , (2.11) i j ≡ λ i+j=−1 X Without entering the details of the construction, we present the summarised results of interest for us. The entire AKNS hierarchy can be obtained by considering Q(λ) as the following formal series ∞ Q Q Q Q(λ)= Q λ−i = Q + 1 + 2 + 3 + ..., (2.12) i 0 λ λ2 λ3 i=0 X ai bi −i −i −i Qi = a(λ)= aiλ , b(λ)= biλ , c(λ)= ciλ , (2.13) ci ai i i i − X X X and introducing the vector ﬁelds ∂n by

n n n ∂ Q(λ) = [P (λ Q(λ)),Q(λ)] = [P−(λ Q(λ)),Q(λ)] = [R(λ Q(λ)),Q(λ)] , (2.14) n + − 1 where R = 2 (P+ P−) is the endomorphism form of the classical r-matrix. It is well-known that this operator satisfies− the modified classical Yang-Baxter equation and allows one to define a second Lie bracket [ , ] on (see e.g. [42]) R L [X, Y ]R = [RX,Y ] + [X,RY ] . (2.15) The significance of this reformulation is that the authors achieved several important results: 1. The equations (2.14) are commuting Hamiltonian flows associated to the Hamiltonian functions 1 h (X)= (Sk(X),X) , k Z , (SkX)(λ)= λkX(λ) . (2.16) k −2 ∈ which are Casimir functions with respect to the Lie-Poisson bracket associated to the Lie bracket (2.9). As a consequence, these functions are in involution with respect to the Lie-Poisson bracket associated to the second Lie bracket (2.15) on and their Hamilton equations take the form of the Lax equation (2.14); L 2. In this construction, one can get rid of the special role of the x variable, which is now the variable x1, no different from any of the other xn. They then propose to define a hierarchy of integrable PDEs as follows: use (2.14) for a fixed n as a starting point to determined all the Qj . This yields that bj , cj for j>n and aj , j > 1 are polynomials in bj ,cj , j =1,...,n, which are now viewed as functions of xn, and in their derivatives with respect to xn. Then, one can use any one of the other variables xk to induce a Hamiltonian flow on the infinite dimensional phase n n space bj(x ),cj (x ), j = 1,...,n. The Hamilton equations takes the form of a zero curvature equation ∂ Q(n)(xn, λ) ∂ Q(k)(xn, λ) + [Q(n)(xn, λ),Q(k)(xn, λ)] = 0 (2.17) k − n (n) n n where Q (x , λ) denotes P+(λ Q(λ)) where the above substitution for aj,bj ,cj in terms of the n n n finite number of fields bj (x ),cj (x ), j =1,...,n and their x derivatives has been performed. See [16] for more details about this.

∂Fjk ∂Fℓk 3. There exist generalised conservation laws ∂xℓ = ∂xj for all j,k,ℓ 0 where Fkj can be obtained efficiently from a generating function. For j = 1, they reproduce≥ the usual AKNS conservation laws with F1k being the conserved densities and Fℓk the corresponding fluxes. Those results are reviewed in detail in [16] where the observation that one can start from an arbitrary flow xn is used to prove a general result on the r-matrix structure of dual Lax pairs which was first observed in [13] and [14]. More explicitly, recall that the first r-matrix structure appeared in [6] with the objective of quantizing the nonlinear Schrödinger equation. The fundamental observation is that one can encode the Poisson bracket q(x), r(y) = iδ(x y) allowing to describe NLS as a Hamiltonian field theory into the (linear ultralocal){ Sklyanin} bracket− [6]

Q(1)(x, λ) ,Q(1)(y,µ) = δ(x y) r (λ µ),Q(1)(x, λ)+ Q(1)(y,µ) , (2.18) { 1 2 } − 12 − 1 2 h i 5 where the indices 1 and 2 denote in which copy of the tensor product a matrix acts non trivially, e.g. M = M I, M = I M, and r (λ) is the classical r-matrix related to the operator R by 1 ⊗ 2 ⊗ 12 X , (RX)(λ) = res Tr (r (λ µ)X (µ)) , (2.19) ∀ ∈ L µ 2 12 − 2 where Tr2 means that the trace is only taken over the second space in the tensor product. The main results of [16] is that starting from a given xn and inducing the xk ﬂow by considering the zero curvature equation

∂ Q(n)(xn, λ) ∂ Q(k)(xn, λ) + [Q(n)(xn, λ),Q(k)(xn, λ)] = 0 (2.20) k − n or starting from xk and inducing the xn ﬂow by considering the zero curvature equation

∂ Q(k)(xk, λ) ∂ Q(n)(xk, λ) + [Q(k)(xk, λ),Q(n)(xk, λ)] = 0 (2.21) n − k yields the same set of integrable PDEs which possesses two Hamiltonian formulations with respect to two distinct Poisson brackets , and , for which one has { }n { }k Q(n)(xn, λ) ,Q(n)(yn,µ) = δ(xn yn) r (λ µ),Q(n)(xn, λ)+ Q(n)(yn,µ) , (2.22) { 1 2 }n − 12 − 1 2 h i and

Q(k)(xk, λ) ,Q(k)(yk,µ) = δ(xk yk) r (λ µ),Q(k)(xk, λ)+ Q(k)(yk,µ) . (2.23) { 1 2 }k − 12 − 1 2 h i We want to stress that despite the deep observation that all independent variables xj play the same role, both in [17] and [16], the authors still implement the step of using (2.14) first for a fixed (but arbitrary) xn in order to produce a phase space for a field theory consisting of a finite number of fields n n bj (x ), cj (x ) j =1,...,n. This leads to a rather complicated construction of the Poisson brackets , n and , k in [16] whose common r-matrix structure is traced back to the original Lie-Poisson {bracket} associated{ } to the second Lie bracket (2.15). This also points to the need of a truly covariant Poisson bracket capable of accommodating any pair of independent variables xn and xk simultaneously and producing an r-matrix structure for the associated Lax form W (λ)= Q(n)(λ) dxn + Q(k)(λ) dxk . This was achieved by us in [21]. Another essential question was still pending: how to go beyond only a pair of times xn and xk, corresponding to a single zero curvature equation, in order to include the entire hierarchy of flows? How to construct a Poisson structure capable of dealing with the ∞ corresponding Lax form W (λ)= Q(j)(λ) dxj ? j=0 In this paper, we answer theseX questions by avoiding altogether the first step of fixing a given time xn and by working with all the equations (2.14) at once. They are interpreted as commuting Hamiltonian flows on a phase space with a countable number of coordinates bj , cj, j 1. This interpretation is less known than the standard field theory viewpoint but it provides a deeper≥ insight into the structure of the hierarchy. It is possible thanks to the very ideas behind a covariant approach to field theories which promote a finite dimensional phase space over a space-time manifold of dimension greater than one. Classical mechanics is viewed as a particular case where the space time manifold is reduced to R for a single time only. In our opinion, our interpretation is also a true implementation of the original observation that all independent variables x0, x1, x2 ... play a symmetric role. This is captured by our use of a Lagrangian and Hamiltonian multiform which do not distinguish any particular independent variable as being special. Our main objective is to construct a Poisson bracket , , called multi-time Poisson bracket, and ∞ {| |} a Hamiltonian multiform H = H dxi dxj such that: ij ∧ i

6 (k) 3 2. The collection of all the equations ∂kQ(λ) = [Q (λ),Q(λ)], k 0 or, equivalently , of all the zero curvature equations ≥ ∂ Q(j)(λ) ∂ Q(i)(λ) + [Q(j)(λ),Q(i)(λ)]=0 , i,j 0 , (2.24) i − j ≥ ∞ can be written in Hamiltonian form as dW (λ)= H , W (λ) dxi dxj . {| ij |} ∧ i

DµQ(λ)= DλQ(µ) , (2.29) which in component is ∂ Q = ∂ Q , j, k 0 . (2.30) k j+1 j k+1 ≥ Moreover, by means of the Jacobi identity we have

DλDµQ(ν)= DµDλQ(ν) , (2.31)

4 which means that the flows ∂j and ∂k commute . Finally, noting that the generating function of the Hamiltonian functions (2.16) is given by ∞ 1 1 1 g(λ) Tr Q2(λ)= Tr Q2 + g , (2.32) ≡−2 −2 0 λk+1 k Xk=0 we find Dµg(λ)=0 . (2.33) This shows that the flows take place on the level surface g(λ) = C(λ) where C(λ) is a series in λ−1 with constant coefficients. Therefore, in line with [17], we fix Tr Q2(λ)= 2 , (2.34) − in the rest of this paper.

3This equivalence does not seem to be well-known but we use it all along and deal interchangeably with the FNR equations (2.14) and the zero curvature equations (2.24). The implication (2.14)⇒(2.24) is shown for instance in [16, Lemma 3.13]. The converse is discussed in [43, Chapter 5]. 4Of course, this had to be the case in the first place so as to allow us to consider those flows simultaneously and to define Dµ, but this is a good check of the generating function formalism and an argument in favour of its efficiency.

7 3 Lagrangian and symplectic multiforms for the AKNS hierarchy

The framework of this work is the variational bi-complex, whose algebraic description is giving N in [44]. Very roughly, let M = R be the multi-time manifold with coordinates (x0, x1, x2,... ), and let it be the base manifold of a fibred manifold, in which the section of the fibres are the fields and k their derivatives u(j), (j) = (j0, j1, j2,... ) where only a finite number of ji’s are non zero. The base coordinates will be called horizontal, and the fibred coordinates will be called vertical. We will model the fibres using a differential algebra , with derivations ∂ acting in the usual way, ∂ uk = uk . i i (j) (j)+ei We will use two different differentials,A a vertical one δ, which resembles the variational derivative, and a horizontal one d, which is the usual total differential, such that (δ + d)2 = 0. With the symbol (p,q) we mean the set of forms with vertical degree p and horizontal degree q of the form A (i) kp (i) ω = f δuk1 δu dxj1 dxjq , f . (3.1) (k),(j) (i1) ∧···∧ (ip) ∧ ∧···∧ (k),(j) ∈ A (i),X(k),(j) The reader can find more details on how to use these tools in the context of multiforms in [22].

Lagrangian multiforms were first introduced by Lobb and Nijhoff in 2019 [23] to describe integrable hierarchies using a variational principle, in a way that preserves multidimensional consistency. For 1 + 1-dimensional field theories, the starting point is to consider a two-form ∞ ij L [u]= Lij [u] dx . (3.2) i

In [22] we introduced and developed the Hamiltonian counterpart called Hamiltonian multiforms, which for 1 + 1-ﬁeld theories are two-forms ∞ [u]= H [u] dxij . (3.4) H ij i

8 3.1 Lagrangian multiform We now introduce a Lagrangian multiform which allows us to implement the strategy [22] recalled brieﬂy above and obtain and Ω for the AKNS hierarchy. Recall that the collection of ﬂows in the AKNS hierarchy is writtenH in generating form as

∞ [Q(µ),Q(λ)] Q D Q(λ)= ,Q(λ)= i , (3.7a) µ µ λ λi i=0 − X a(λ) b(λ) a b Q(λ)= ,Q = i i , (3.7b) c(λ) a(λ) i c a − i − i 1 Tr Q(λ)2 = a2(λ)+ b(λ)c(λ)= 1 . (3.7c) 2 − where λ and µ are formal parameters. In order to ﬁnd an appropriate Lagrangian multiform, it is convenient to note that we can write Q(λ) as

−1 Q(λ)= ϕ(λ)Q0ϕ(λ) (3.8) with Q0 = iσ3 being constant and − ∞ ϕ ϕ(λ)= I + j . (3.9) λj j=1 X This has been established independently from various angles, in relation to the factorization theorem, see e.g. [42] or in relation to vertex operators, see e.g. [43, Chapter 5]. Contrary to the parametriza- tion used in the latter book, we ﬁnd it useful to use the following remarkable set of coordinates found in [17]

∞ ∞ b(λ) e c(λ) f e(λ)= = i , f(λ)= = i , (note: e = f = 0), (3.10) λi λi 0 0 i a(λ) i=1 i a(λ) i=1 − X − X and set p p 1 2i e(λ)f(λ) e(λ) ϕ(λ)= − . (3.11) √2i f(λ) 2i e(λ)f(λ) p − − 2 −1 A direct calculation using a (λ)+ b(λ)c(λ) = 1 shows thatp det ϕ(λ) = 1 and ϕ(λ)( iσ3)ϕ(λ) = Q(λ) as required. The reader can find more about− the coordinates e(λ), f(λ) in Appendix− A. Their main property is that they provide Darboux coordinates for all the single-time Poisson brackets , k constructed from the single-time symplectic forms ωk, see Corollary 3.3 and Proposition 4.3 below.{ } We can now formulate the first main result of this section. We obtain the desired Lagrangian ∞ ij multiform L = Lij dx using the generating function formalism and collecting the coefficients i

∞ L L (λ, µ)= ij . (3.12) λi+1µj+1 i,j=0 X By a slight abuse of language, we will also call L (λ, µ) a Lagrangian multiform. Theorem 3.1 (Lagrangian multiform and multiform Euler-Lagrange equations). Deﬁne

L (λ, µ)= K(λ, µ) V (λ, µ) , (3.13) − where 1 (Q(λ) Q(µ))2 K(λ, µ) = Tr ϕ(µ)−1D ϕ(µ)Q ϕ(λ)−1D ϕ(λ)Q , V (λ, µ)= Tr − . λ 0 − µ 0 −2 λ µ − (3.14) Then L (λ, µ) is a Lagrangian multiform for the AKNS hierarchy equations (3.7a).

9 Indeed, the multiform Euler-Lagrange equations δdL = 0 are given by

[Q(µ),Q(λ)] D Q(λ)= , (3.15) µ µ λ − and the closure relation dL = 0 is satisfied on those equations. In generating form, the latter is equivalent to Dν L (λ, µ)+ DλL (µ,ν)+ DµL (ν, λ)=0. (3.16) The proof is given in Appendix B.1. Remark. Although we discovered it differently, the Lagrangian multiform (3.13)-(3.14) bears some striking resemblance with the Zakharov-Mikhailov (ZM) Lagrangian appearing in [46], despite the fact that the latter is a standard Lagrangian and not a multiform. The ZM Lagrangian was introduced to provide a variational description of the system of compatibility conditions (zero curvature equations) corresponding to a Lax pair of matrices which are rational functions of the spectral parameter with distinct simple poles. Interestingly, one can formally relate our Lagrangian multiform to the ZM Lagrangian but we do not elaborate on this here as we have not gained any insight on either our results or the ZM Lagrangian by doing so. A Lagrangian multiform constructed on the ZM Lagrangian was obtained in [36] and used to obtain a variational derivation of Lax pair equations themselves. In that same paper, the authors presented the first few coefficients of the Lagrangian multiform for the AKNS hierarchy but it was not clear how these derive directly from the ZM Lagrangian multiform. Our Lagrangian multiform and Theorem 3.1 fill in this gap and provide the complete set of coefficients Lij of the Lagrangian multiform for the AKNS hierarchy. We note that Lagrangians producing the zero curvature equations (2.24) in potential form were obtained in [47]. They involved a potential function denoted by H in that paper which produces the Lax matrices Q(k) we use here via the (k) relation Q = ∂k−1H. However, assembling all those Lagrangians into a two-form does not seem to provide a Lagrangian multiform for the set of AKNS equations. The closure relation does not hold for instance. To help the reader recognize the most familiar models, we write some of the coefficients of the La- ∞ i+1 j+1 grangian multiform explicitly using our formula. Using the expansion L (λ, µ)= Lij /λ µ i

i Vij = (2akai+j+1−k + bkci+j+1−k + ckbi+j+1−k) . (3.18) Xk=0

Recall that the elements aj, bj and cj can all be expressed in terms of the coordinates ej and fj (see Appendix A). At this stage, no particular choice of time has been made to write these Lagrangians as field theory Lagrangian, in the spirit of [15] for instance. Hence, as an example, we simply have 1 1 L = (f ∂ e e ∂ f + f ∂ e e ∂ f ) (f ∂ e e ∂ f ) V , (3.19) 12 2 1 1 2 − 1 1 2 2 1 1 − 2 1 1 − 2 1 2 1 − 1 2 1 − 12 and 1 1 L = (f ∂ e e ∂ f + f ∂ e e ∂ f + f ∂ e e ∂ f ) (f ∂ e e ∂ f ) V , (3.20) 13 2 1 1 3 − 1 1 3 2 1 2 − 2 1 2 3 1 1 − 3 1 1 − 2 1 3 1 − 1 3 1 − 13 which produce partial differential equations for the phase space coordinates ej , fj , j = 1, 2, 3. Now to make contact with the more familiar form of these Lagrangians and the corresponding equations 1 5 of motion, we express the phase space coordinates in terms of b1 = q, c1 = r and their x derivatives . 5 1 The reader can find the relations between the ei’s and fi’s and q and r and their derivative with respect to x in Appendix A.

10 1 Note that this amounts to choosing the x equation in (2.14) and use it to solve for Qj (standard field theory point of view). Doing so yields, i 1 1 L = (q r qr )+ (rq + qr ) q2r2 , (3.21) 12 4 2 − 2 8 11 11 − 4 and i i 3i L = (rq qr )+ (q r qr )+ qr(qr rq ) , (3.22) 13 4 3 − 3 16 111 − 111 16 1 − 1 which are known Lagrangians whose Euler-Lagrange equations are 1 1 iq + q q2r =0 , ir r + qr2 , (3.23a) 2 2 11 − 2 − 2 11 1 3 1 3 q + q qrq =0 , r + r qrr =0 . (3.23b) 3 4 111 − 2 1 3 4 111 − 2 1 These are the (unreduced) NLS and mKdV systems respectively. We can just as easily produce the Lagrangian L23, first in the e and f coordinates and then, if desired, in the q and r coordinates as before. It reads i i i L = (rq qr )+ (q r q r ) (q r q r ) 23 16 112 − 112 16 1 12 − 12 1 − 16 11 2 − 2 11 3i 1 1 qr(rq qr ) (q r + qr )+ (r q + q r ) (3.24) − 16 2 − 2 − 8 13 13 8 1 3 1 3 1 qr 1 1 + q r (qr + q r)+ (qr q r)2 + q3r3 , 16 11 11 − 8 11 11 16 1 − 1 4 and its Euler-Lagrange equations are just consequence of (3.23a)-(3.23b). 12 Remark. The partial Lagrangian multiform thus derived here for the first three times L12 dx + 23 13 i L23 dx + L13 dx is equivalent to the one first obtained in [36], up to an overall coefficient 2 and the (total) differential of 1 (q r + r q) dx2 i (q r qr ) dx3. − 8 1 1 − 16 11 − 11 3.2 Symplectic multiform Equipped with a Lagrangian multiform for the AKNS hierarchy, we now construct the associated symplectic multiform Ω. As always, it is very convenient to work with generating functions so we introduce ∞ (1) ∞ ω ω Ω(1)(λ)= j , Ω(λ)= j , (3.25) λj+1 λj+1 j=0 j=0 X X to represent respectively ∞ ∞ Ω(1) = ω(1) dxj , Ω= ω dxj . (3.26) j ∧ j ∧ j=0 j=0 X X As before, by a slight abuse of language, we also call Ω(λ) symplectic multiform. Proposition 3.2. The symplectic multiform associated to L (λ, µ) is given by

Ω(λ)= Tr Q ϕ(λ)−1δϕ(λ) ϕ(λ)−1δϕ(λ) . (3.27) − 0 ∧ The proof is in Appendix B.2. Remark. The expression for Ω(λ) is reminiscent of the well-known expression for the (pull-back to the group of the) Kostant-Kirillov symplectic form on a coadjoint orbit of the loop algebra through L the element Q0. To make this more precise, let us use for instance the formulas in [48, Section 3.3] giving the expression of the pull-back to the group of the Kostant-Kirillov form for the orbit through a diagonal matrix polynomial A(λ),

ω = Res Tr A(λ)g−1(λ)δg(λ) g−1(λ)δg(λ) . (3.28) λ ∧

11 k Here, choosing A(λ)= iλ σ3, k 0, and g(λ)= ϕ(λ), we get the connection between our symplectic multiform and the Kostant-Kirillov− ≥ form

k ω = Resλλ Ω(λ)= ωk . (3.29)

k In particular, each single-time symplectic form ωk corresponds to ω on the orbit of the element iλ σ3. Therefore, our symplectic multiform contains in a single object all those symplectic forms. This− is the ﬁrst time such an object is derived and, to our knowledge, it is the ﬁrst time that a Kostant-Kirillov symplectic form is derived from a Lagrangian perspective. As a consequence of the explicit formula for Ω, we get the following remarkable result that the e,f coordinates provide Darboux coordinates. Corollary 3.3. Ω(λ)= δf(λ) δe(λ) , (3.30) ∧ and hence, ω0 =0 and, k ω = δf δe − , k 1 . (3.31) k i ∧ k+1 i ∀ ≥ i=1 X Proof. Direct calculation by inserting (3.11) into (3.27).

4 Classical r-matrix structure

4.1 Hamiltonian forms and multi-time Poisson bracket As explained in [22], only for a specific class of horizontal forms called Hamiltonian forms, i.e. a form F such that there exist a (multi)vector field ξF (called Hamiltonian vector field) that satisfies the relation ξF yΩ = δF . Having the symplectic multiform Ω at our disposal, we can investigate in detail under which conditions a horizontal form is Hamiltonian and then compute the multi-time Poisson bracket for two such forms. Recall (see [22]) that in our case, only 0- and 1-forms can be non-trivial Hamiltonian forms. We have the following two propositions, the proofs of which are given in Appendix B.3 and B.4. ∞ k Proposition 4.1. A 1-form F = Fk dx is Hamiltonian with respect to Ω if and only if F0 is k=0 constant and, for all k 1, F dependsX only on the coordinates (e ,...,e ,f ,...,f ) and ≥ k 1 k 1 k ∂F ∂F ∂F ∂F k = k+1 , k = k+1 , j =1, . . . , k . (4.1) ∂ej ∂ej+1 ∂fj ∂fj+1 Its Hamiltonian vector field is given by

∞ ∂Fk ∂Fk ξF = ∂ek + ∂fk . (4.2) − ∂f1 ∂e1 kX=1 Proposition 4.2. Every 0-form H(e1,...,f1,... ) is Hamiltonian with respect to Ω, with Hamiltonian vector ﬁeld given by ∞ ∂H ∂H ξ = ∂ ∂ + ∂ ∂ . (4.3) H − ∂f e1 ∧ i ∂e f1 ∧ i i=1 i i X Note that in practice, we will deal with 0-forms that depend only on a ﬁnite number of coordinates ej , fj in which case the sum in (4.3) truncates accordingly.

We can now construct the multi-time Poisson bracket with respect to Ω between two Hamiltonian forms F and G as F, G = ( 1)rξ yδG (4.4) {| |} − F where r is the horizontal degree of F . Theorem 2.18 in [22] gives the decomposition of the multi-time Poisson brackets in terms of the single-time Poisson brackets , constructed from the symplectic { }k

12 forms ω : for every F , G functions of e ,...,e ,f ,...,f we have F, G = ξ yδG where ξ yω = k 1 k 1 k { }k − F F k δF . Given that we know the explicit form of the single-time symplectic forms ωk, see (3.31), we obtain the following specialisation as a consequence. Proposition 4.3 (Decomposition of the multi-time Poisson brackets). The multi-time Poisson brack- ∞ ∞ k k ets with respect to Ω of two Hamiltonian 1-forms F = Fk dx and G = Gk dx satisﬁes the k=0 k=0 following decomposition: X X

∞ k k ∂Fk ∂Gk ∂Fk ∂Gk F, G = Fk, Gk k dx , Fk, Gk k = , k 1 (4.5) {| |} { } { } ∂f ∂e − − ∂e ∂f − ≥ j=1 j k j+1 j k j+1 Xk=0 X and F , G =0. { 0 0}0 Thanks to the the propositions above, we can prove by direct but long calculations that the multi-time Poisson bracket , satisﬁes the Jacobi identity. {| |} Proposition 4.4 (Jacobi identity). If F,G,K (0,1) and H are Hamiltonian forms, we have that ∈ A ∈ A 1. F, G and F,H are respectively a Hamiltonian 1-form and a Hamiltonian 0-form, {| |} {| |} 2. F, G ,K + K, F , G + G, K , F =0, {|{| |} |} {|{| |} |} {|{| |} |} 3. F, G ,H + H, F , G + G, H , F =0. {|{| |} |} {|{| |} |} {|{| |} |} Remark. It is known (see e.g. [49]) that the Jacobi identity is not necessarily satisﬁed by a covariant Poisson bracket. This problem could therefore be present in general for a multi-time Poisson bracket (which can be viewed as a generalisation of a covariant Poisson bracket). This is why the Jacobi identity was not discussed in [22] and why we checked it here directly.

4.2 Classical r-matrix structure of the multi-time Poisson bracket Definition 4.5. We call Lax form the following horizontal 1-form with matrix coefficient ∞ W (λ)= Q(i)(λ) dxi (4.6) i=0 X where, for i 0, ≥ (i) i Q (λ) := P+(λ Q(λ)) . (4.7) Note that the definition of a Hamiltonian form extends naturally to the case of matrix coefficients by requiring that each entry be a Hamiltonian form. We also extend the tensor notation used in the Sklyanin bracket, as reviewed in Section 2, to the present situation as follows ∞ ∞ W (λ) Q(i)(λ) I dxi = Q(i)(λ) σ I dxi W k(λ)σ I (4.8) 1 ≡ ⊗ k k ⊗ ≡ k ⊗ i=0 i=0 k=+,−,3 k=+,−,3 X∞ X∞ X X W (λ) I Q(i)(λ) dxi = Q(i)(λ) I σ dxi W k(λ)I σ , (4.9) 2 ≡ ⊗ k ⊗ k ≡ ⊗ k i=0 i=0 − − X X k=+X, ,3 k=+X, ,3 where we have written the matrices in terms of the sl(2, C) basis σ+, σ−, σ3. We define the multi-time Poisson bracket between W1(λ) and W2(µ) by

k ℓ W1(λ), W2(µ) = W (λ), W (µ) σk σℓ . (4.10) {| |} − {| |} ⊗ k,ℓ=+X, ,3 Finally, we deﬁne the commutator of a matrix 0-form M and a matrix 1-form W by ∞ [M, W ] [M, W ] dxi . (4.11) ≡ i i=0 X We are now ready to formulate the main result of this section, the proof of which is long but straightforward and is given in Appendix B.5.

13 Theorem 4.6. The Lax form W (λ) is Hamiltonian, with Hamiltonian vector ﬁeld

∞ ∂Q(k)(λ) ∂Q(k)(λ) ξW (λ)= ∂ek + ∂fk . (4.12) − ∂f1 ∂e1 Xk=1 Its multi-time Poisson bracket possesses the linear Sklyanin bracket structure i.e.

W (λ), W (µ) = [r (λ µ), W (λ)+ W (µ)] , (4.13) {| 1 2 |} 12 − 1 2 where r12(λ, µ) is the so-called rational classical r-matrix given by P r (λ)= 12 . (4.14) 12 − λ Remark. We have already shown directly that our multi-time Poisson bracket , satisﬁes the Jacobi identity for 0- and 1-forms. In the case of 1-forms, this is also a corollary{| of|} Theorem 4.6 since W (λ) contains all the coordinates of our phase space and it is known that the rational r-matrix satisﬁes the classical Yang-Baxter equation which implies the Jacobi identity.

5 Hamiltonian multiform description of the AKNS hierarchy

5.1 Multiform Hamilton equations for the AKNS hierarchy According to formula [22, Deﬁnition 2.3], the coeﬃcients of the Hamiltonian multiform = ∞ H ij (1) Hij dx associated to L and Ω are given by i

1 (Q(λ) Q(µ))2 (λ, µ)= V (λ, µ)= Tr − , (5.3) H −2 λ µ − or, in components, i Hij = Tr QkQi+j−k+1 . (5.4) kX=0 Hence, (λ, µ) satisﬁes the closure relation. H (1) (1) Proof. A direct calculation shows that DλyΩ (µ) DµyΩ (λ)= K(λ, µ) hence (λ, µ)= V (λ, µ). Performing a series expansion of the explicit formula− for V (λ, µ), one obtains (5.4).H Finally, the closure relation of is a general resulte [22, Corollarye 2.6] but here, we get a direct conﬁrmation from the structureH of the proof of Theorem 3.1 which established that V is closed on the equations of motion, separately from K. For completeness, we now check the validity of the general result of Equation (3.5) in our case. Proposition 5.2. The multiform Hamilton equations associated to and Ω are equivalent to H [Q(λ),Q(µ)] D Q(µ)= . (5.5) λ λ µ −

14 Proof. The multiform Hamilton equations read δ = dxj ∂ yΩ , (5.6) H ∧ j j X or, in components, e δH = ∂ yω ∂ yω . (5.7) ij j i − i j This is reformulated in generating form as, e e δ (λ, µ)= D yΩ(λ) D yΩ(µ) . (5.8) H µ − λ We have already computed δ (λ, µ)= δV (λ, µ) as H e e 1 1 δ (λ, µ) = Tr ϕ(λ)−1[Q(µ),Q(λ)]δϕ(λ) ϕ(µ)−1[Q(λ),Q(µ)]δϕ(µ) . (5.9) H µ λ − λ µ − − Now the right hand-side is

D yΩ(λ) D yΩ(µ) = Tr Q ϕ(λ)−1D ϕ(λ)ϕ(λ)−1δϕ(λ)+ Q ϕ(λ)−1δϕ(λ)ϕ(λ)−1D ϕ(λ) µ − λ − 0 µ 0 µ e e + Q ϕ(µ)−1D ϕ(µ)ϕ(µ)−1δϕ(µ) Q ϕ(µ)−1δϕ(µ)ϕ(µ)−1D ϕ(µ) 0 λ − 0 λ = Tr ϕ−1(λ)D Q(λ)δϕ(λ) ϕ−1(µ)D Q(µ)δϕ(µ) . µ − λ (5.10) The result follows by reading the coeﬃcient of δϕ(µ) or equivalently δϕ(λ).

5.2 Zero curvature equations as multiform Hamilton equations It is one of the most important results of the theory of integrable classical ﬁeld theories that their zero curvature representation admits a Hamiltonian formulation. This is established for all famous models and goes as follows, for the example of the NLS equation, see e.g. [9]. We choose Q(1)(x, λ)= iλσ3 + Q1(x) satisfying the Sklyanin linear Poisson algebra (2.18), and H being the (appropriate) − (1) Hamiltonian extracted from the monodromy matrix of the spectral problem (∂x Q )Ψ = 0. One can then show that the time ﬂow induced on Q(1)(x, λ) by H i.e. − ∂ Q(1)(x, λ)= H,Q(1)(x, λ) (5.11) t { } takes the form of a zero curvature equation ∂ Q(1)(x, λ)= H,Q(1)(x, λ) = ∂ Q(2)(x, λ) + [Q(2)(x, λ),Q(1)(x, λ)] . (5.12) t { } x Q(2)(x, λ) is derived from the classical r-matrix and the monodromy matrix. In [21] the authors cast this result into a covariant framework, for the NLS and mKdV equations separately: the covariant Hamilton equations for the Lax form associated to each equation (thus containing only the two rele- vant Q(j)(x, λ)) produce the respective zero curvature condition.

Here, we are in a position to prove the analogous result for the whole AKNS hierarchy at once, thanks to our Hamiltonian multiform and multi-time Poisson bracket. The following is the main result of this section ∞ Theorem 5.3. The multiform Hamilton equations for the Lax form W (λ)= Q(k)(λ) dxk , i.e. kX=0 ∞ dW (λ)= H , W (λ) dxij , (5.13) {| ij |} i

15 5.3 Conservation laws An important aspect of the theory of integrable PDEs is the presence of an infinite number of conservation laws. When formulated in the traditional Hamiltonian framework, this means that the Hamiltonian h is one element in a countable family of independent Hamiltonians functions hk, k 1 say, which are in involution with respect to the Poisson bracket , used to write the PDE of interest≥ as a Hamiltonian system { } h ,h =0 , n,m 1 . (5.15) { n m} ≥ When using the classical r-matrix formalism to tackle this problem, the strategy is to compute the Poisson bracket of the entries of the monodromy matrix, starting from the linear Sklyanin bracket and deduce that (5.15) holds, where the hk’s are extracted in an appropriate way from the monodromy matrix. The details depend on the model and on the boundary conditions imposed on the fields, see e.g. [9] for a full account of this procedure. It is natural to ask what happens to this approach in our context and where to find the standard Hamiltonians hn, which are rather different from the coefficients Hij of our Hamiltonian multiform. In our opinion, it is rather remarkable to we do not need to resort to any notion of monodromy matrix to answer this question. Instead, all the required information is encoded in our Hamiltonian multiform and the notion of conservation law which we naturally define as follows. H ∞ i Definition 5.4. A conservation law is a Hamiltonian 1-form A = Ai dx such that dA = 0 on i=0 the equations of motion. X In components, this yields of course the familiar form of an infinite family of conservation laws

∂ A ∂ A =0 . (5.16) i j − j i We have proved in [22] that A is a conservation law if it Poisson-commutes with the Hamiltonian multiform, i.e. ∞ H , A dxij =0 . (5.17) {| ij |} i

∞ n ∂Ak ∂Hmn ∂Ak ∂Hmn mn ∂Hmn ∂Hmn mn dA = ξAyδ = + dx = ek + fk dx H − ∂f1 ∂ek ∂e1 ∂fk − ∂ek ∂fk mn (without ∂e1 ∂f1 ∂ek ∂fk loss of generality, we consider m

(i) (j) Hmn = h(i)(j)(e) (f) , (5.20) ∈Nn (i),(Xj) where the sum is finite (only a finite number of coefficients h(i)(j) C are non zero) and we have ∈ n (i) i1 i2 im (j) j1 j2 jn used the notations (e) = e1 e2 ...em , (f) = f1 f2 ...fn , and has the property that ik = Xk=1

16 n n n ∂ ∂ jk. The result then follows since ek and fk are Euler operators with respect to ∂ek ∂fk Xk=1 Xk=1 Xk=1 the coordinates ek and fk respectively.

1 1 This result provides a reinterpretation of the known fact the quantities hk = k ak+1 dx , viewed as the traditional hierarchy of standard, single-time, Hamiltonians are indeed constant of the motion R and in involution with respect to the traditional (single-time) Poisson bracket , 1 (see e.g. [44, Section 9.3]). { }

6 Recovering previous results and the ﬁrst three times

It is straightforward to recover our previous results [21] by “freezing” all times except a given pair. This singles out a single 1 + 1-dimensional ﬁeld theory within the hierarchy and our Lagrangian multiform, symplectic multiform, Hamiltonian multiform and multi-time Poisson bracket reduce respectively to a Lagrangian, multisymplectic form, covariant Hamiltonian and covariant Poisson bracket. As the simplest example, let use freeze all times except x1 = x and x2 = t: we specialise to NLS and recover all the results of Section 4.2 in [21] by direct calculation. The Lax form is simply W (λ)= Q(1)(λ) dx + Q(2)(λ) dt , (6.1) which can be computed using again the coordinates q, r and derivatives with respect to x for instance to reproduce the well known NLS Lax pair. The Lagrangian multiform reduces to = L dx dt L 12 ∧ where L12 is given in (3.21) while the Hamiltonian multiform only involves H12. Using our general formula, i Hij = (2akai+j+1−k + bkci+j+1−k + ckbi+j+1−k) , (6.2) kX=0 we ﬁnd

H12 =2a0a4 + b0c4 + c0b4 +2a1a3 + b1c3 + c1b3

i 2 i 2 = 2i(e1f3 + e2f2 + e3f1)+2ie1(f3 + e1f1 )+2if1(e3 + e1f1) − 4 4 (6.3) = 2ie f e2f 2 − 2 2 − 1 1 1 = (q r q2r2) . − 4 1 1 − This is the covariant Hamiltonian for NLS found in [21] (up to an irrelevant factor). The symplectic multiform boils down to the following multisymplectic form Ω= ω dx + ω dt , (6.4a) 1 ∧ 2 ∧ i 1 1 ω = δq δr, ω = δr δq + δq δr , (6.4b) 1 2 ∧ 2 4 ∧ 1 4 ∧ 1 also found first in [21] (up to irrelevant factors). It gives rise to a covariant Poisson bracket which is simply the reduction of our multi-time Poisson bracket to only two times and our main results, Theorems 4.6 and 5.3 restrict accordingly to the results of [21]. We stress however that we can instead choose any pair of times xn and xk and apply the same reasoning. Doing so provides a way to unify the results in [16] which established the r-matrix structure of dual Lax pairs for an arbitrary pair of times and the results in [21] which provided a covariant formulation of this structure but only for the pair of times (x1, x2) and (x1, x3). The salient features of the multiform theory appear when at least three times are combined together. In general, the coefficients L1n (resp. H1n) are not too difficult to construct but all the other ones are, and indeed up to now, it was not known how to obtain them. For instance, freezing 1 2 3 all times except x , x , x , the coefficient L23 was first obtained in [36] by complicated calculations. Here, we obtain it rather easily, see (3.24), as well as the associated coefficient H23 in the Hamiltonian multiform which reads 1 i H = 2ie f + e f (f e + e f ) (e f + f e )2 + e3f 3 (6.5) 23 − 3 3 2 1 1 1 3 1 3 − 1 2 1 2 8 1 1 1 qr 1 1 = q r + (rq + qr ) (rq qr )2 q3r3 (6.6) −16 11 11 8 11 11 − 16 1 − 1 − 4

17 For completeness, let us also give 3 H = 2i(e f + e f ) e f (f e + e f ) , (6.7) 13 − 2 3 3 2 − 2 1 1 1 2 1 2 i = (q r r q ) . (6.8) 8 1 11 − 1 11

i We remark that these coefficients differ from those in [22] by a factor 2 . In the rest of this section, we illustrate in every detail the calculations involved in our general results when restricted to the first three times. This has only pedagogical value. We hope that this will help the reader familiarise themselves with some of the new formalism while dealing with the most familiar and easiest levels of 1 2 3 the AKNS hierarchy. We now turn to the symplectic multiform Ω = ω1 dx + ω2 dx + ω3 dx , where ∧ ∧ ∧

ω = δf δe , ω = δf δe + δf δe , ω = δf δe + δf δe + δf δe . (6.9) 1 1 ∧ 1 2 1 ∧ 2 2 ∧ 1 3 1 ∧ 3 2 ∧ 2 3 ∧ 1

As done above for ω1 and ω2, it is interesting to write ω3 using b1 = q, c1 = r and their derivatives 1 with respect to x , denoted by q1, r1, q11, r11. We ﬁnd i i i 3iqr ω = δr δq + δr δq + δq δr + δq δr , (6.10a) 3 8 ∧ 11 8 11 ∧ 8 1 ∧ 1 4 ∧

i and we remark that they also diﬀer from the ones in [22] by the same factor 2 , so that the multiform j y Hamilton equations δ = j dx ∂j Ω are the same. Let us compute them, in the new e and f coordinates. In componentsH we have∧ P e • δH = ∂ yω ∂ yω : 12 2 1 − 1 2 ∂ f =2if , ∂ e = 2ie , (6.11a) 1 1 2 1 1 − 2 ∂ f ∂ f =2e f 2 , ∂ e ∂ e =2e2f . (6.11b) 1 2 − 2 1 1 1 2 1 − 1 2 1 1 i i i i The top equations give the relations b2 = 2 ∂1b1 = 2 q1 and c2 = 2 ∂1c1 = 2 r1, and the bottom ones give the NLS equations. − − • δH = ∂ yω ∂ yω : 13 3 1 − 1 3 ∂ f =2if , ∂ e = 2ie , (6.12a) 1 1 2 1 1 − 2 3 3 ∂ f =2if + e f 2 , ∂ e = 2ie e2f , (6.12b) 1 2 3 2 1 1 1 2 − 3 − 2 1 1 3 3 ∂ f ∂ f = e f 2 +3e f f , ∂ e ∂ e = e2f +3e f e , (6.12c) 1 3 − 3 1 2 2 1 1 1 2 3 1 − 1 3 2 1 2 1 1 2

i i 1 1 2 where the top four equations give the relations b2 = 2 q1 and c2 = 2 r1, and b3 = 4 q11 + 2 q r and c = 1 r + 1 qr2, and the bottom ones are the mKdV equations.− − 3 − 4 11 2 • δH = ∂ yω ∂ yω : 23 3 2 − 2 3 1 1 ∂ f =2if e f 2 , ∂ e = 2ie + e2f , (6.13a) 2 1 3 − 2 1 1 2 1 − 3 2 1 1 ∂ f ∂ f =2f 2e +2e f f , ∂ e ∂ e =2e2f +2e f e , (6.13b) 2 2 − 3 1 1 2 1 1 2 3 1 − 2 2 1 2 1 1 2 1 3i ∂ f ∂ f = f 2e + e f f + e2f 3 2e f 2 2f e f , (6.13c) 3 2 − 2 3 2 1 3 1 1 3 8 1 1 − 1 2 − 1 2 2 1 3i ∂ e ∂ e = e2f + e f e + e3f 2 2f e2 2e e f , (6.13d) 2 3 − 3 2 2 1 3 1 1 3 8 1 1 − 1 2 − 1 2 2 which reduce to diﬀerential consequences of the previous equations. The single-time Poisson brackets , for k =1, 2, 3 { }k k ∂ ∂ ∂ ∂ , k = , (6.14) { } ∂f ∂e − − ∂e − ∂f i=1 i k+1 i k+1 i i X

18 can be re-expressed in the q and r coordinates as ∂ ∂ ∂ ∂ , =2i , (6.15) { }1 ∂r ∂q − ∂q ∂r ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , =4 + , (6.16) { }2 ∂r ∂q ∂q ∂r − ∂q ∂r − ∂r ∂q 1 1 1 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , = 8i + + +6qr { }3 − ∂r ∂q ∂r ∂q ∂q ∂r ∂q ∂r 11 11 1 1 11 11 (6.17) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 6qr . − ∂q11 ∂r − ∂q ∂r11 − ∂r1 ∂q1 − ∂r11 ∂q11 These differ from the ones obtained in [22] by a factor 2i. − We will now show how to obtain the classical r-matrix structure within the multi-time Poisson brackets for the first three times. We will use the first three Lax matrices repackaged into the Lax form W (λ)= Q(1)(λ) dx1 + Q(2)(λ) dx2 + Q(3)(λ) dx3 + 1 2 2 3 W (λ)= b1 dx + (λb1 + b2) dx + (λ b1 + λb2 + b3) dx , (6.18a) − 1 2 2 3 W (λ)= c1 dx + (λc1 + c2) dx + (λ c1 + λc2 + c3) dx , (6.18b) i iλ i W 3(λ)= iλdx1 + ( iλ2 b c ) dx1 + ( iλ3 b c (b c + c b )) dx3 , (6.18c) − − − 2 1 1 − − 2 1 1 − 2 1 2 1 2 which we can also write in terms of the coordinates e and f as in i W +(λ)= √2ie dx1 + √2i(λe + e ) dx2 + √2i(λ2e + λe + e + e2f ) dx3 , (6.19a) 1 1 2 1 2 3 4 1 1 i W −(λ)= √2if dx1 + √2i(λf + f ) dx2 + √2i(λ2f + λf + f + e f 2) dx3 , (6.19b) 1 1 2 1 2 3 4 1 1 W 3(λ)= iλdx1 + ( iλ2 + e f ) dx1 + ( iλ3 + λe f + e f + e f ) dx3 . (6.19c) − − 1 1 − 1 1 1 2 2 1 We can then compute the Hamiltonian vector field associated to each component of the Lax form:

2 i i 2 ξ + (λ)= √2i ∂ + λ∂ + (λ + e f )∂ e ∂ , (6.20a) W f1 f2 2 1 1 f3 − 4 1 e3 2 i i 2 ξ − (λ)= √2i ∂ λ∂ + ( λ e f )∂ + f ∂ , (6.20b) W − e1 − e2 − − 2 1 1 e3 4 1 f3 ξ 3 (λ)= e ∂ + f ∂ + ( λe e )∂ + (λf + f )∂ . (6.20c) W − 1 e2 1 f2 − 1 − 2 e3 1 2 f3 Let us now compute the multi-time Poisson bracket. W (λ), W (µ) = W i(λ), W j (µ) σ σ {| 1 2 |} {| |} i ⊗ j i,j=+,−,3 X + + + − = W (λ), W (µ) σ σ + W (λ), W (µ) σ σ− {| |} + ⊗ + {| |} + ⊗ + 3 − + + W (λ), W (µ) σ+ σ3 + W (λ), W (µ) σ− σ+ (6.21) {| |} ⊗ {| |} ⊗ − − − 3 + W (λ), W (µ) σ− σ− + W (λ), W (µ) σ− σ {| |} ⊗ {| |} ⊗ 3 3 + 3 − + W (λ), W (µ) σ σ + W (λ), W (µ) σ σ− {| |} 3 ⊗ + {| |} 3 ⊗ + W 3(λ), W 3(µ) σ σ . {| |} 3 ⊗ 3 The reader can check that W +(λ), W +(µ) = W −(λ), W −(µ) = W 3(λ), W 3(µ) = 0, while the other non-zero Poisson brackets{| are |} {| |} {| |} W +(λ), W −(µ) = 2i dx1 2i(λ + µ) dx2 2i (λ2 + λµ + µ2 + ie f ) dx3 (6.22a) {| |} − − − 1 1 W −(λ), W +(µ) =2i dx1 +2i(λ + µ) dx2 +2i (λ2 + λµ + µ2 + ie f ) dx3 (6.22b) {| |} 1 1 W +(λ), W 3(µ) = √2ie dx2 √2i((λ + µ)e + e ) dx3 , (6.22c) {| |} − 1 − 1 2 W 3(λ), W +(µ) = √2ie dx2 + √2i((λ + µ)e + e ) dx3 , (6.22d) {| |} 1 1 2 W −(λ), W 3(µ) = √2if dx2 + √2i((λ + µ)f + f ) dx3 , (6.22e) {| |} 1 1 2 W 3(λ), W −(µ) = √2if dx2 √2i((λ + µ)f + f ) dx3 . (6.22f) {| |} − 1 − 1 2

19 Adding everything together one realises that W (λ), W (µ) = [ P12 , W (λ)+ W (µ)], as desired. {| 1 2 |} µ−λ 1 2 Let us verify that for the ﬁrst three times

H , W (λ) = W (λ) W (λ)= [Q(i)(λ),Q(j)(λ)] dxij (6.23) {| ij |} ∧ i

ξ =2e2f ∂ ∂ 2e f 2∂ ∂ +2ie ∂ ∂ 2if ∂ ∂ , (6.25) 12 1 1 e1 ∧ 1 − 1 1 f1 ∧ 1 2 e1 ∧ 2 − 2 f1 ∧ 2 so that the left hand-side reads H , W (λ) =ξ yδW (λ) {| 12 |} 12 =ξ y(e δf dx2 + f δe dx2)σ 12 1 1 ∧ 1 1 ∧ 3 + ξ y(√2iδe dx1 + √2iδe dx2 + √2iλδe dx2)σ 12 1 ∧ 2 ∧ 1 ∧ + 1 2 2 (6.26) + ξ y(√2iδf dx + √2iδf dx + √2iλδf dx )σ− 12 1 ∧ 2 ∧ 1 ∧ 2 2 =2i(e f f e )σ + √2i( 2e f 2iλe )σ + √2i(2e f +2iλf )σ− 1 2 − 1 2 3 − 1 1 − 2 + 1 1 2 =[Q(1)(λ),Q(2)(λ)] .

Similarly one obtains H , W (λ) = [Q(1)(λ),Q(3)(λ)] and H , W (λ) = [Q(2)(λ),Q(3)(λ)]. {| 13 |} {| 23 |} 1 2 3 We can also verify that A = a2dx + a3dx + a4dx is indeed a conservation law in the usual coordinates q and r. In fact, we have that i a = e f = qr , (6.27a) 2 1 1 −2 1 a = e f + e f = (q r qr ) , (6.27b) 3 1 2 2 1 4 1 − 1 i i 3i i a = e f + e f + e f = qr + q r q2r2 q r . (6.27c) 4 1 3 2 2 3 1 8 11 8 11 − 8 − 8 1 1 Imposing dA = 0 is equivalent to the equations

∂1a3 = ∂2a2 , (6.28a)

∂1a4 = ∂3a2 , (6.28b)

∂2a4 = ∂3a3 , (6.28c) which hold on the equations of motion.

7 Conclusions and perspectives

This work constitutes progress towards the understanding of the role played by multi-time consistency and the application of covariant Hamiltonian field theory to integrable systems, which is a line of research that started with [21] and continued with [22] and [50]. In this paper, we have presented a new Lagrangian and Hamiltonian multiforms to describe the complete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, and recognised in this description the main features of integrability, including the presence of an infinite number of conservation laws and the classical r-matrix structure. Unlike the usual approach to an integrable hierarchy, our formalism preserves equal footing with respect to all the times of the integrable hierarchy, and avoids altogether the infinite-dimensional Hamiltonian formalism in favour of a more “finite-dimensional” approach to a field theory. The Lagrangian multiform is written as a double generating series in the negative loop algebra, and is proved to satisfy the closure relation. In turn, via a “multi-time Legendre transform”, we obtain a Hamiltonian and a

20 symplectic multiform, which encapsulate respectively all the covariant Hamiltonians and the symplectic structures of the hierarchy. The classical r-matrix structure is found within a multi-time Poisson bracket (that is derived from the symplectic multiform), which contains all the single-time Poisson brackets of the hierarchy. The set of zero curvature equations are obtained as multiform Hamilton equations for the complete Lax connection. Conservation laws have a vanishing multi-time Poisson bracket with the Hamiltonian multiform, emulating the familiar condition in ﬁnite dimensional Hamil- tonian mechanics. The well-known ﬂuxes and conserved quantities are re-obtained naturally from this requirement.

These results point to some interesting open questions. In [50] we have identiﬁed a covariant equivalent of the famous relation “H = Tr L2” between the Hamiltonian multiform and the Lax connection. An open question is if there is such a formula for the AKNS hierarchy as well, and what are its implications. The conservation laws are obtained in this paper without resorting to the monodromy matrix [51], and without involving the r-matrix structure at the group level, which is the starting point of the traditional and well-known (quantum) Inverse Scattering Method. This is in our opinion certainly remarkable, and it needs to be understood whether the monodromy matrix is really out of the picture and if so, what is the reason. Moreover, we were not able to relate the closure of the Lagrangian multiform with the “mutual involution” of the single-time Hamiltonians, which would connect our work to the one of [52]. Finally, we believe that a similar approach will be useful in describing other integrable hierarchies, such as the ones containing the (potential) Korteweg-de Vries equation or the sine-Gordon equation, which were ﬁrst analysed with a Hamiltonian multiform formalism in [22]. This is left for future investigation.

As mentioned above, the classical r-matrix structure of a single-time Poisson bracket is the starting point of a well-established procedure of canonical quantisation [6]. We wish to remark that this work (together with [21], [22] and partly [50]) belongs to a programme whose overarching goal is a new approach to canonical covariant quantisation of an integrable system, and builds an important step towards this objective. We believe that the classical r-matrix structure within the covariant (and multi-time) Poisson bracket can provide a new outlook on how to perform this canonical quantisation in a covariant fashion.

A The e and f coordinates

In this section we discuss some of the properties of the coordinates e and f, which are deﬁned as

b(λ) c(λ) e(λ)= , f(λ)= . (A.1) i a(λ) i a(λ) − − We remember that we are restrictingp to the subset where a2p(λ)+ b(λ)c(λ) = 1, which means that a(λ)= e(λ)f(λ) i, in fact − − b(λ)c(λ) 1 a2(λ) e(λ)f(λ)= = − − = i + a(λ) . (A.2) i a(λ) i a(λ) − − We list the ﬁrst few coeﬃcients of e and f:

e0 =0 , f0 =0 , (A.3a) 1 1 e1 = b1 , f1 = c1 , (A.3b) √2i √2i 1 1 e2 = b2 , f2 = c2 , (A.3c) √2i √2i 1 1 2 1 1 2 e3 = (b3 b c1) , f3 = (c3 b1c ) , (A.3d) √2i − 8 1 √2i − 8 1 1 1 1 2 1 1 1 2 e4 = (b4 b1c1b2 b c2) , f4 = (c4 b1c1c2 c b2) . (A.3e) √2i − 4 − 8 1 √2i − 4 − 8 1

21 Conversely, we have that

b1 = √2ie1 , c1 = √2if1 , (A.4a)

b2 = √2ie2 , c2 = √2if2 , (A.4b) i i b = √2i(e + e2f ) , c = √2i(f + e f 2) , (A.4c) 3 3 4 1 1 3 3 4 1 1 i i i i b = √2i(e + e f e + e2f ) , c = √2i(f + e f f + f 2e ) . (A.4d) 4 4 2 1 1 2 4 1 2 4 4 2 1 1 2 4 1 2

k−1 Also a = ef i, so a = e f − : − k i=1 i k i a = i, a =0 ,P a = e f , a = e f + e f , a = e f + e f + f e . (A.5) 0 − 1 2 1 1 3 1 2 2 1 4 1 3 2 2 1 3 It is also useful to express these relations in terms of the usual q and r coordinates (and their derivatives with respect to x1 = x) we have the following identities

b1 = q, c1 = r , (A.6a) i i b = q , c = r , (A.6b) 2 2 1 2 −2 1 1 1 1 1 b = q + q2r, c = r + qr2 , (A.6c) 3 −4 11 2 3 −4 11 2 i 3i i 3i b = q + qrq , c = r qrr . (A.6d) 4 −8 111 4 1 4 8 111 − 4 1

1 1 e1 = q, f1 = r , (A.7a) √2i √2i 1 i 1 i e2 = q1 , f2 = r1 , (A.7b) √2i 2 −√2i 2 1 1 3 1 1 3 e = q + q2r , f = r + qr2 , (A.7c) 3 √ −4 11 8 3 √ −4 11 8 2i 2i 1 i 5i i 1 i 5i i e = q + qrq + q2r , f = r qrr q r2 . (A.7d) 4 √ −8 111 8 1 16 1 4 √ 8 111 − 8 1 − 16 1 2i 2i Conversely:

q = √2ie1 , r = √2if1 , (A.8a) q = √2i2ie , r = √2i2if , (A.8b) 1 − 2 1 2 q = √2i 4e +3ie2f , r = √2i 4f +3ie f 2 , (A.8c) 11 − 3 1 1 11 − 3 1 1 2 2 q111 = √2i 8ie4 + 20e1f1e2 2e1f2 , r111 = √2i 8if 4 20e1f1f2 +2 f1 e2 . (A.8d) − − − We can also write the expressions for the derivatives of Q with respect to the coordinates e and f:

−k i−3a(λ) −k b2(λ) ∂Q(λ) λ c(λ) 2 ∂Q(λ) λ b(λ) − 2 2(i a(λ)) = c (λ) , = i−3a(λ) − . (A.9) ∂ek i a(λ) 2(i−a(λ)) c(λ) ! ∂fk i a(λ) b(λ) ! − − − − 2 − Therefore wep have p

∂ai = fi−j , (A.10a) ∂ej ∂b i 3a(λ) i = − , (A.10b) ∂ej 2 i a(λ) ! − i−j 2 2 ∂ci f (λ) p c (λ) = − = − 3/2 , (A.10c) ∂ej 2 i a(λ) ! 2(i a(λ)) i−j − i−j − p 22 ∂ai = ei−j , (A.11a) ∂fj 2 2 ∂bi e (λ) b (λ) = − = − 3/2 , (A.11b) ∂fj 2 i a(λ) ! 2(i a(λ)) i−j − i−j − p∂c i 3a(λ) i = − . (A.11c) ∂fj 2 i a(λ) ! − i−j p B Proofs

B.1 Proof of Theorem 3.1 Proof. We need to calculate δdK and then δdV . We do so with the help of the generating functions as follows. Note that ijk dK = (∂iKjk + ∂kKij + ∂j Kki) dx , (B.1) i

−1 −1 Tr Q(λ)δ(Dν Q(µ)) = Tr ϕ(µ) [Dν Q(µ),Q(λ)] + Dν ϕ(µ)ϕ(µ) [Q(λ),Q(µ)] δϕ(µ) −1 Tr ϕ(µ) [Q(λ),Q(µ)]δ(Dν ϕ(µ)) , (B.2) − −1 Tr Dν Q(λ)δQ(µ) = Tr ϕ(µ) [Q(µ),Dν Q(λ)]δϕ(µ) . (B.3)

We have that −1 −1 −1 Dν K(λ, µ) = Tr( ϕ(µ) Dν ϕ(µ)ϕ(µ) Dλϕ(µ)Q0 + ϕ (µ)Dν Dλϕ(µ)Q0 − (B.4) + ϕ(λ)−1D ϕ(λ)ϕ(λ)−1D ϕ(λ)Q ϕ−1(λ)D D ϕ(λ)Q ) . ν µ 0 − ν µ 0 We now apply the δ-diﬀerential.

−1 −1 −1 δDν K(λ, µ) = Tr(ϕ(µ) Dν ϕ(µ)ϕ(µ) Dλϕ(µ)ϕ(µ) Q(µ) + D ϕ(µ)ϕ(µ)−1Q(µ)D ϕ(µ)ϕ(µ)−1 D D ϕ(µ)ϕ(µ)−1Q(µ) δϕ(µ) λ ν − ν λ ϕ(µ)−1D ϕ(µ)ϕ(µ)−1Q(µ)δ(D ϕ(µ)) (B.5) − λ ν ϕ(µ)−1Q(µ)D ϕ(µ)ϕ(µ)−1δ(D ϕ(µ)) − ν λ + ϕ(µ)−1Q(µ)δ(D D ϕ(µ)) (λ µ)) . ν λ − ↔

We add the cyclic sum and we select the coeﬃcients of δϕ(µ), δDν ϕ(µ), etc. adding the corresponding terms from δDλK(µ,ν).

−1 −1 −1 δdK = Tr(ϕ(µ) Dµϕ(µ)ϕ(µ) DλQ(µ) Dλϕ(µ)ϕ(µ) DµQ(µ))δϕ(µ) − (B.6) ϕ(µ)−1D Q(µ)δ(D ϕ(µ)) + ϕ(µ)−1D Q(µ)δ(D ϕ(µ))+ ) . − λ µ µ λ We do the same for V (λ, µ)= 1 Tr(Q(λ) Q(µ))2. 2(µ−λ) − 1 D V (λ, µ)= Tr(D Q(λ) D Q(µ))(Q(λ) Q(µ)) ν µ λ ν − ν − − (B.7) 1 = Tr(D Q(λ)Q(µ)+ D Q(µ)Q(λ)) , λ µ ν ν − where we used Tr Q(λ)Dν Q(λ) = Tr Q(µ)Dν Q(µ) = 0. We now apply the δ-diﬀerential. 1 δD V (λ, µ)= Tr (D Q(λ)δQ(µ)+ Q(λ)δ(D Q(µ))) (λ µ) (B.8) ν λ µ ν ν − ↔ − 7With we mean the cyclic permutations of (ν, λ, µ).

23 and using the identities above we get 1 δD V (λ, µ)= Tr ϕ(µ)−1[Q(λ),Q(µ)]δ(D ϕ(µ)) ν λ µ − ν − −1 −1 + ϕ(µ) [Dν Q(µ),Q(λ)] + [Q(µ),Dν Q(λ)] + Dν ϕ(µ)ϕ(µ) [Q(λ),Q(µ)] δϕ(µ) (λ µ). − ↔ (B.9)

We add the cyclic sum and we select the coeﬃcients of δϕ(µ), δDν ϕ(µ), etc. adding the corresponding terms from δDλV (µ,ν). 1 δdV = Tr ϕ(µ)−1[Q(λ),Q(µ)]δ(D ϕ(µ)) λ µ − ν − −1 −1 + ϕ(µ) [Dν Q(µ),Q(λ)] + [Q(µ),Dν Q(λ)] + Dν ϕ(µ)ϕ(µ) [Q(λ),Q(µ)] δϕ(µ) 1 + Tr ϕ(µ)−1[Q(ν),Q(µ)]δ(D ϕ(µ)) ν µ λ − ϕ(µ)−1 [D Q(µ),Q(ν)] + [Q(µ),D Q(ν)] + D ϕ(µ)ϕ(µ)−1[Q(ν),Q(µ)] δϕ(µ) + . − λ λ λ (B.10)

By comparing the coefficients of δDν ϕ(µ) and of δDλϕ(µ) we get the desired equations (3.7a). The equations coming from the coefficients of δϕ(µ) are differential consequences of them. We turn to the closure relation. We are going to use the following identities: 1 1 λ µ = − , (B.11) µ ν − λ ν (µ ν)(λ ν) − − − − 1 1 1 + + =0 , (B.12) (µ ν)(λ ν) (ν λ)(µ λ) (ν µ)(λ µ) − − − − − − Tr[Q(λ),Q(µ)]Q(λ)=0 , (B.13) Tr[Q(λ),Q(ν)]Q(µ) = Tr[Q(µ),Q(λ)]Q(ν) . (B.14) A direct computation shows that the kinetic term brings −1 Dν K(λ, µ)+ DλK(µ,ν)+ DµK(ν, λ) = Tr(Dν ϕ(λ)ϕ(λ) DµQ(λ)+ ) 1 = Tr( D Q(λ)Q(µ)+ ) λ µ ν − 1 (B.15) = Tr( [Q(λ),Q(ν)]Q(µ)+ ) (λ µ)(λ µ) − − 1 = Tr + [Q(λ),Q(ν)]Q(µ) =0 (λ µ)(λ µ) − − We have that the potential term brings: 1 D V (λ, µ)= 2 Tr(D Q(λ) D Q(µ))(Q(λ) Q(µ)) ν −2(λ µ) ν − ν − − 1 [Q(λ),Q(ν)] 1 [Q(µ),Q(ν)] = Tr (Q(λ) Q(µ)) Tr (Q(λ) Q(µ)) µ λ λ ν − − µ λ µ ν − − − − − 1 1 = Tr[Q(λ),Q(ν)]Q(µ)+ Tr[Q(µ),Q(ν)]Q(λ) (B.16) (λ µ)(λ ν) (λ µ)(µ ν) − − − − 1 1 1 = Tr[Q(λ),Q(ν)]Q(µ) λ µ λ ν − µ ν − − − 1 = Tr[Q(λ),Q(ν)]Q(µ) (λ ν)(ν µ) − − The cyclic sum then is 1 1 1 + + Tr[Q(µ),Q(λ)]Q(ν) = 0 (B.17) (λ ν)(ν µ) (ν µ)(µ λ) (µ λ)(λ ν) − − − − − − so that we have that the Lagrangian multiform satisfies the closure relation dL = 0.

24 B.2 Proof of Proposition 3.2

Proof. First, we claim that Ω(1) is given by the generating function

(1) −1 Ω (λ) = Tr Q0ϕ(λ) δϕ(λ) . (B.18)

(1) We need to show that δL + dΩ = 0 on the multiform Euler-Lagrange equations DµQ(λ) = [Q(µ),Q(λ)] −1 µ−λ . For convenience, let us denote ψ(λ) := ϕ (λ). A direct computation shows that

δK(λ, µ) =Tr Dλϕ(µ)Q0δψ(µ)+ Q0ψ(µ)δ(Dλϕ(µ)) (B.19) D ϕ(λ)Q δψ(λ) Q ψ(λ)δ(D ϕ(λ)) , − µ 0 − 0 µ and 1 1 δV (λ, µ) = Tr ψ(λ)[Q(λ),Q(µ)]δϕ(λ) ψ(µ)[Q(λ),Q(µ)]δϕ(µ) . (B.20) λ µ − λ µ − − ∞ (1) (1) k+1 (1) The coeﬃcient of the generating function Ω (λ)= ωk /λ are obtained as (note that ω0 = 0) Xk=0 k (1) ωk = Tr Q0ψiδϕk+1−i . (B.21) i=1 X Hence, for the corresponding form, we have using the variational bicomplex calculus,

∞ (1) (1) k dΩ =d ωk dx ∧ ! kX=1 ∞ k jk = Tr (∂ ψ δϕ − + ψ δ(∂ ϕ − ) ∂ ψ δϕ − ψ δ(∂ ϕ − )) dx . k i j+1 i i k j+1 i − j i k+1 i − i j k+1 i ∧ i=1 j

The associated generating function is given by

(1) dΩ (λ, µ) = Tr Q0Dµψ(λ)δϕ(λ)+ Q0ψ(λ)δ(Dµϕ(λ)) (B.23) Q D ψ(µ)δϕ(µ) Q ψ(µ)δ(D ϕ(µ)) . − 0 λ − 0 λ So the sum δK(λ, µ) δV (λ, µ)+ dΩ(1)(λ, µ) reads − 1 Tr ψ(λ)D Q(λ)δϕ(λ) ψ(λ)[Q(λ),Q(µ)]δϕ(λ) µ − λ µ − (B.24) 1 ψ(µ)D Q(µ)+ ψ(λ)[Q(λ),Q(µ)]δϕ(µ) . − λ λ µ − This vanishes on the multiform Euler-Lagrange equations

[Q(µ),Q(λ)] [Q(λ),Q(µ)] D Q(λ)= ,D Q(µ)= , (B.25) µ µ λ λ λ µ − − thus completing the argument. As a consequence,

Ω(λ)= δΩ(1)(λ)= Tr Q ϕ(λ)−1δϕ(λ) ϕ(λ)−1δϕ(λ) , (B.26) − 0 ∧ as required.

25 B.3 Proof of Proposition 4.1 Proof. We start with the general expression of the vertical vector ﬁeld

∞

ξF = Aj ∂fj + Bj ∂ej , (B.27) j=1 X and determine Aj , Bj such that ξF yΩ= δF holds, or equivalently,

ξ yω = δF , k 0 . (B.28) F k k ∀ ≥

Since ω0 = 0 we instantly get that F0 has to be constant. The left-hand side reads

k ∞ ξ yω = (A δ δe − B δ − δf ) F k j i,j k i+1 − j j,k i+1 i i=1 j=1 X X (B.29) k = (A − δe B − δf ) , k i+1 i − k i+1 i i=1 X whilst the right hand-side is ∞ ∂F ∂F k δe + k δf . (B.30) ∂e i ∂f i i=1 i i X Comparing the two we get ∂F ∂F k = k =0 , i > k , (B.31a) ∂ei ∂fi ∀ ∂Fk ∂Fk = Ak−i+1 , = Bk−i+1 , i k . (B.31b) ∂ei ∂fi − ∀ ≤ The latter brings that ∂Fk ∂Fk+1 = Ak−i+1 = A(k+1)−(i+1)+1 = , (B.32) ∂ei ∂ei+1 and similarly for fi. These conditions are necessary and suﬃcient.

B.4 Proof of Proposition 4.2 Proof. We need to show that

∞ k k ξ yΩ= δH where Ω= δf δe − dx . (B.33) H m ∧ k+1 m ∧ m=1 kX=1 X We start with the left hand-side

∞ ∞ k ∂H ∂H k ξ yΩ= ∂ ∂ + ∂ ∂ y δf δe − δx H − ∂f e1 ∧ i ∂e f1 ∧ i m ∧ k+1 m ∧ i=1 m=1 i i X Xk=1 X ∞ ∞ k ∞ ∞ k ∂H ∂H = δikδk+1−m,1δfm + δikδm,1δek+1−m (B.34) ∂fi ∂ei i=1 k=1 m=1 i=1 k=1 m=1 X∞ X X ∞ X X X ∂H ∂H = δf + δe = δH . ∂f i ∂e i i=1 i i=1 i X X

26 B.5 Proof of Theorem 4.6

Lemma B.1. For each k 0, the only non-zero single-time Poisson of ai, bi and ci, 0 i k, are given by ≥ ≤ ≤

a ,b = b − − , (B.35) { i j}k i+j k 1 a ,c = c − − , (B.36) { i j }k − i+j k 1 b ,c =2a − − . (B.37) { i j}k i+j k 1 For convenience, we use the convention that a coeﬃcient in a series vanishes when its index is negative. Hence, it is understood that a ,b = a ,c = b ,c =0 whenever i + j < k +1. { i j}k { i j}k { i j }k Proof. We start with the fact that for any power series α and β we have

k αi−ℓβj+ℓ−k−1 = (αβ)i+j−k−1 . (B.38) Xℓ=1 In fact, by limiting the sum only to the non-zero terms:

k i i+j−k−1 αi−l βj+ℓ−k−1 = αi−ℓ βj+ℓ−k−1 = αi+j−k−1−m βm = (αβ)i+j−k−1 . (B.39) − m=0 Xℓ=1 ℓ=kX+1 j X We study the case where k +1 j j, namely i + j k +1. If i + j < k + 1 then the sum is empty, and the result is zero. We− are≤ now ready to compute≥ the following Poisson brackets using the formulas in Appendix A.

k ∂ai ∂bj ∂bj ∂ai ai,bj k = { } ∂fℓ ∂ek+1−ℓ − ∂fℓ ∂ek+1−ℓ Xℓ=1 k i 3a e2 = e − − − f − − i ℓ i+ℓ k 1 (B.40) 2√i a j+ℓ−k−1 − 2√i a j−ℓ ! Xℓ=1 − − ie 3ae + e2f ie 3ae + (i + a)e = − = − 2√i a − − 2√i a − − − i+j k 1 − i+j k 1 =(e√i a) − − = b − − . − i+j k 1 i+j k 1 k ∂ai ∂cj ∂cj ∂ai ai,cj k = { } ∂fℓ ∂ek+1−ℓ − ∂fℓ ∂ek+1−ℓ Xℓ=1 k f 2 i 3a = e − − − f − − i ℓ i+ℓ k 1 (B.41) 2√i a j+ℓ−k−1 − 2√i a j−ℓ ! Xℓ=1 − − ef 2 f(i 3a) f(i + a) if +3af = − − − = − − 2√i a − − 2√i a − − − i+j k 1 − i+j k 1 = (f√i a) − − = c − − . − − i+j k 1 − i+j k 1 k ∂ai ∂aj ∂aj ∂ai ai,aj k = { } ∂fℓ ∂ek+1−ℓ − ∂fℓ ∂ek+1−ℓ Xℓ=1 k e2 f 2 i 3a i 3a = − − − − (B.42) 2√i a i−ℓ 2√i a j+ℓ−k−1 − 2√i a j−ℓ 2√i a i+ℓ−k−1! Xℓ=1 − − − − e2f 2 (i 3a)2 (i + a)2 (i 3a)2 = − − = − − 4(i a) 4(i a) − i+j−k−1 − i+j−k−1 =2ai+j−k−1 .

27 Remark. These Poisson bracket coincides with the , − in [16]. In this instance we don’t take the { } k Poisson brackets of ai,bi,ci for i > k because they do not belong to the k-th single-time phase space. Proof of Theorem 4.6. We start by proving that ∂Q ∂Q ∂Q ∂Q (λ)= λ (λ) , (λ)= λ (λ) . (B.43) ∂ek ∂ek+1 ∂fk ∂fk+1 This is done for each matrix element. In fact ∂Q ∂Q ∂e ∂Q ∂Q ∂Q ∂e ∂Q = = λ−k = λλ−k−1 = λ = λ . (B.44) ∂ek ∂e ∂ek ∂e ∂e ∂e ∂ek+1 ∂ek+1

∂Q ∂Q Similarly = λ . By virtue of this result, and since Q0 is constant, we have the following: ∂fk ∂fk+1 ∞ ∂Q ∂Q = j λ−j (B.45a) ∂e ∂e k j=0 k ∞ X∞ ∞ ∂Q ∂Q ∂Q ∂Q λ = λ i λ−i = i λ−i+1 = j+1 λ−j . (B.45b) ∂e ∂e ∂e ∂e k+1 i=0 k+1 i=0 k+1 j=0 k+1 X X X If we look at the coeﬃcients in λ we see that, for all j and k, ∂Qj = ∂Qj+1 . Similarly one can obtain ∂ek ∂ek+1 that ∂Qj = ∂Qj+1 . ∂fk ∂fk+1 Finally, we check that the Lax form is Hamiltonian, using Proposition 4.1, i.e. that

i i i ∂Q(i) ∂Q ∂Q ∂Q = λi−k k = λi−k k+1 = λ(i+1)−(k+1) k+1 ∂ej ∂ej ∂ej+1 ∂ej+1 kX=0 kX=0 kX=0 (B.46) i+1 ∂Q ∂Q(i+1) ∂Q ∂Q(i+1) = λi+1−k k = λi+1 0 = , ∂ej+1 ∂ej+1 − ∂ej+1 ∂ej+1 kX=1 (i) (i+1) where we used that Q is constant. Similarly ∂Q = ∂Q . 0 ∂fj ∂fj+1

We now turn to the proof of (4.13). Thanks to the decomposition of the multi-time Poisson bracket into single-time Poisson brackets, we have that W1(λ), W2(µ) = [r12(λ, µ), W1(λ)+ W2(µ)] if and only for all k 0, {| |} ≥ Q(k)(λ),Q(k)(µ) = [r (λ, µ),Q(k)(λ)+ Q(k)(µ)] . (B.47) { 1 2 }k 12 1 2 (k) (k) (k) (k) Writing Q (λ)= Q+ (λ)σ+ + Q− (λ)σ− + Q3 (λ)σ3, the right hand-side of (B.47) reads

(k) (k) 2 (k) (k) [r (λ, µ),Q (λ)+ Q (µ)] = (Q (µ) Q (λ))(σ σ− σ− σ ) 12 1 2 µ λ 3 − 3 + ⊗ − ⊗ + (k) (k) − (k) (k) (B.48) Q+ (µ) Q+ (λ) Q− (λ) Q− (µ) + − (σ σ σ σ )+ − (σ σ− σ− σ ) , µ λ 3 ⊗ + − + ⊗ 3 µ λ 3 ⊗ − ⊗ 3 − − while the left hand-side is given by

k k (k) (k) (λµ) Q (λ),Q (µ) = a ,a σ σ + b ,b σ σ + c ,c σ− σ− { 1 2 }k λiµj { i j }k 3 ⊗ 3 { i j }k + ⊗ + { i j }k ⊗ i,j=0 X (B.49) + b ,c σ σ− + c ,b σ− σ + a ,b σ σ { i j }k + ⊗ { i j}k ⊗ + { i j}k 3 ⊗ + + b ,a σ σ + a ,c σ σ− + c ,a σ− σ . { i j }k + ⊗ 3 { i j }k 3 ⊗ { i j }k ⊗ 3 We now invoke Lemma B.1 which gives the necessary single-time Poisson brackets and allows us to check directly that (B.48) is equal to (B.49). We show it for the σ+ σ− component, as the others are obtained similarly. In the left hand-side we have ⊗

k k k−j−1 2 2 (Q(k)(µ) Q(k)(λ)) = (µk−j λk−j )a =2 λiµk−i−j−1a , (B.50) µ λ 3 − 3 µ λ − j j j=0 j=0 i=1 − − X X X

28 while right hand-side is equal to

k k k i−1 k k−n−1 (λµ) am n k−n−m−1 2 a − − =2 =2 λ µ a . (B.51) λiµj i+j k 1 λi−kµm+1−i m i,j=0 i=0 m=0 n=0 m=0 X X X X X This concludes the proof.

B.6 Proof of Theorem 5.3 Proof. Note the set of zero curvature equations can be written as

dW (λ)= W (λ) W (λ) , (B.52) ∧ where the right-hand side is understood as

∞ ∞ W (λ) W (λ)= Q(i)(λ)dxi Q(j)(λ)dxj = [Q(i)(λ),Q(j)(λ)] dxij , (B.53) ∧ ∧   i=0 ! j=0 i

j ∂Q(k)(λ) ∂H ∂Q(k)(λ) ∂H [Q(i)(λ),Q(j)(λ)] = ij ij . (B.56) ∂e1 ∂fk − ∂f1 ∂ek kX=1 We prove the latter in generating form as follows. We multiply both sides by µ−i−1ν−j−1 and form the following sums over i and j

∞ j ∞ j j (k) (k) 1 (i) (j) 1 ∂Q (λ) ∂Hij ∂Q (λ) ∂Hij i+1 j+1 [Q (λ),Q (λ)] = i+1 j+1 . µ ν µ ν ∂e1 ∂fk − ∂f1 ∂ek j=0 i=0 j=0 i=0 k=1 X X X X X (B.57) We can rearrange the sums in the right-hand side to get

∞ ∞ j ∞ ∞ j 1 1 ( )= ( ) , (B.58) µi+1νj+1 ··· µi+1νj+1 ··· i=0 j=0 i=0 kX=1 Xj=k X kX=1 X X where we have used the fact that Hij depends only on e1,...,ej and f1,...,fj in the second step to extend the sum over j from 0 instead of k. We can similarly form the sums with µ ν and use the same trick to rearrange the sums in the right-hand side. Using the anti-symmetry↔ of both left and right-hand side of (B.56), we come to the following generating form of (B.56)

∞ 1 [Q(i)(λ),Q(j)(λ)] µi+1νj+1 i,j=0 X ∞ ∞ ∞ (B.59) ∂Q(k)(λ) ∂ H ∂Q(k)(λ) ∂ H = ij ij ,  ∂e ∂f µi+1νj+1 − ∂f ∂e µi+1νj+1  1 k i,j=0 1 k i,j=0 Xk=1 X X   29 i.e. , ∞ ∞ [Q(µ),Q(ν)] ∂Q(k)(λ) ∂ (µ,ν) ∂Q(k)(λ) ∂ (µ,ν) = H H , (B.60) (µ λ)(ν λ) ∂e1 ∂fk − ∂f1 ∂ek − − kX=1 Xk=1 where we have used ∞ Q(i)(λ) Q(µ) = . (B.61) µi+1 µ λ i=0 X − We now show that (B.60) holds by computing its right-hand side recalling that 1 (µ,ν)= Tr(Q(µ) Q(ν))2 . (B.62) H 2(ν µ) − − For convenience, denote a(µ),a(ν),a(λ) by a,a′,a′′ respectively and similarly for b and c. We have

∂ (µ,ν) 1 ∂Q(µ) ∂Q(ν) H = Tr (Q(µ) Q(ν)) (B.63) ∂f (ν µ) ∂f − ∂f − k − k k and ∞ 1 ∂Q(k)(λ) ∂Q(µ) Tr (Q(µ) Q(ν)) ν µ ∂e1 ∂fk − − k=1 X ∞ (k) b2 ′ ′ 1 ∂Q (λ) 1 b 2(i−a) a a b b = Tr i−3a − − ′ ′− ν µ ∂e1 µk√i a b c c a a − k=1 − 2 − ! − − X∞ 1 ∂Q(k)(λ) 1 b2(c c′) (i 3a)(b b′) = 2b(a a′) − + − − (B.64) k ν µ ∂e1 µ √i a − − 2(i a) 2 − k=1 − − X 2 ′ ′ µ ∂ Q(µ) 1 ′ b (c c ) (i 3a)(b b ) = 2b(a a ) − + − − (ν µ) ∂e1 µ λ √i a − − 2(i a) 2 − − − − 2 ′ ′ i−3a 1 1 ′ b (c c ) (i 3a)(b b ) c 2 = 2b(a a ) − + − − 2 , (ν µ)(µ λ) i a − − 2(i a) 2 c c − − − − − 2(i−a) − ! ∞ Q(k) Q(µ) Q where we have used that = µ( 0 ) and that Q is constant. Similarly, we have µk µ λ − µ 0 kX=1 − ∞ 1 ∂Q(k)(λ) ∂Q(ν) Tr (Q(µ) Q(ν)) = ν µ ∂e1 ∂fk − − k=1 X ′ (B.65) ′2 ′ ′ ′ ′ i−3a 1 1 ′ ′ b (c c ) (i 3a )(b b ) c ′ 2 ′ (2b (a a ) − ′ + − − ) c 2 ′ , (ν µ)(ν λ) i a − − 2(i a ) 2 ′ c − − − − − 2(i−a ) − ! ∞ 1 ∂Q(k)(λ) ∂Q(µ) Tr (Q(µ) Q(ν)) = ν µ ∂f1 ∂ek − k=1 − X (B.66) 2 ′ ′ b2 1 1 ′ c (b b ) (i 3a)(c c ) b (2c(a a ) − + − − ) − 2(i−a) , (ν µ)(µ λ) i a − − 2(i a) 2 i−3a b − − − − 2 − ! and ∞ 1 ∂Q(k)(λ) ∂Q(ν) Tr (Q(µ) Q(ν)) = ν µ ∂f1 ∂ek − − k=1 X ′ (B.67) ′2 ′ ′ ′ ′ b 2 1 1 ′ ′ c (b b ) (i 3a )(c c ) b 2(i−a′) ′ (2c (a a ) − ′ + − − ) − ′ − ′ . (ν µ)(ν λ) i a − − 2(i a ) 2 i 3a b ! − − − − 2 −

30 We collect all the contributions on the σ3 component for instance (the other two are obtained similarly). N1 The numerator of (ν−µ)(µ−λ)(i−a) is

cb2(c c′) (i 3a)(cb cb′) bc2(b b′) (i 3a)(bc bc′) N =2bc(a a′) − + − − 2cb(a a′)+ − − − 1 − − 2(i a) 2 − − 2(i a) − 2 − − 1 ′ ′ i 3a ′ ′ = ( c2b2 + cc b2 + b2c2 bb c2)+ − (cb cb bc + bc ) 2(i a) − − 2 − − − bc i 3a = (bc′ b′c)+ − (bc′ cb′) 2(i a) − 2 − − =(i a)(bc′ b′c) − − (B.68) where in the last equality, we have used that bc = 1 a2 = (i a)(i + a). Similarly, the numerator N2 ′ ′ ′ − − − of ′ is (i a )(bc b c), by simply swapping µ and ν. So, in total the σ component (ν−µ)(ν−λ)(i−a ) − − − 3 of the right-hand side of (B.60) is given by

bc′ b′c 1 1 bc′ b′c − = − . (B.69) ν µ µ λ − ν λ (µ λ)(ν λ) − − − − − [Q(µ),Q(ν)] This is exactly the coeﬃcient of the σ3 component of (µ−λ)(ν−λ) as is readily seen. The other components are dealt with in the same way, and are omitted for brevity.

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