We present here an overview for the Encyclopaedia of Mathematics of the various forms and properties of this system of equations together with its geometric and Lie algebraic background

Helminck, G.F.

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Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1534The AKNS-hierarchyG.F. HelminckJuly 2000ISSN 0169-2690The AKNS-hierarchyG.F. HelminckJuly 6, 2000AbstractWe present here an overview for the Encyclopaedia of Mathematics of thevarious forms and properties of this system of equations together with itsgeometric and Lie algebraic background.1991 MSC : 22E65, 22E70, 35Q53, 35Q58, 58B25Keywords : AKNS hierarchy, Zero curvature form, Lax equations,Kac-Moody Lie algebra, Grassmann manifold,Stationary AKNS-equationsThe AKNS-hierarchy is an infinite tower of non linear evolution equations thatderives its name from the simplest non trivial system of equations contained in it,the AKNS-equationsi∂∂tq(x, t) := iqt = −12qxx + q2ri∂∂tr(x, t) := irt =12rxx − qr2.(1)It were Ablowitz, Kaup, Newell and Segur who showed that the initial valueproblem of this system of equations could be solved with the Inverse ScatteringTransform. To get a natural embedding of the AKNS-equations in a larger systemwe rewrite (1) in the zero curvature form as∂∂t(P1)−∂∂xQ2 + [P1, Q2] = 0, where (2)P1 =(0 qr 0)and Q2 =− i2qri2qx− i2rxi2qr . (3)Consider now the polynomial expressions in the parameter zP = P0z + P1 :=(−i 00 i)z +(0 qr 0)andQ(n) := Q0zn +Q1zn−1 · · ·Qn,(4)where the Qi are sl2-valued functions depending of the variables x, t = (tn) andQ0 = P0 and Q1 = P1. For these data we have the zero curvature equations∂∂tn(P )− ∂∂xQ(n) + [P,Q(n)] = 0⇔[∂∂x− P, ∂∂tn−Q(n)]= 0, (5)12which is an infinite tower of equations extending the system (1). The system (5)generalizes from Sl2(C) to a general simple complex Lie algebra g, a regularelement P0 in a Cartan subalgebra h of g and an element Q0 in h, see [DS] and[W1]. Solutions of the equations (5) can be obtained by the dressing method ofZacharov and Shabat. Consider namely the functionφ(x, t, z) = exp(xP0z +∞∑r=1Q0zr)g(z)exp(−xP0z −∞∑r=1Q0zr), (6)with g(z) belonging to the loop group C∞(S1, Sl2(C)). If this φ factorizes as φ =φ−φ+, withφ−(x, t, z) = exp(∞∑i=1χi(x, t)z−i) and φ+ = exp(∞∑j=1φj(x, t)zj), (7)then conjugating the trivial connections∂∂x−P0z and∂∂tn−Q0zn with φ− givesyou the connections of the required formφ−1− (∂∂x− P0z)φ− =∂∂x− P and φ−1−∂∂tn−Q0znφ− =∂∂tn−Q(n). (8)Since flatness is preserved by this procedure, this leads to solutions of (5). If Q0 =P0, as in the AKNS-case, one can take just as well∂∂x=∂∂t1.It was observed in ([FNR]) that the equations (5) for the Sl2(C)-case could becatched in the following system∂∂tn(Q) = [Q(n), Q], n ≥ 1, (9)for the single seriesQ =∞∑j=0Qjz−j, Qj =(hj ejfj −hj)and Q(n) =n∑j=0Qjzn−j. (10)They showed that these equations are commuting Hamiltonian flows on the Liealgebra∑∞i=0Xiz−i, Xi ∈ sl2(C), w.r.t. a natural Poisson bracket. Further theyintroduced the flux tensor Fjk byFjk =12Trace(j∑r=0(j − r)QrQk+j−r +12j∑r=0(r − k)QrQk+j−r)(11)and proved the local conservation laws of the system, namely∂∂tk(Fij) =∂∂ti(Fjk). (12)The left hand side of (12) is in fact even symmetric under permutations of theindices i, j, k and this property made them introduce a potential τ byFjk =∂∂tj∂∂tklog(τ). (13)The equations (10) are called the Lax equations of the AKNS-hierarchy. As suchthe AKNS-hierarchy is a natural reduction of the two-component KP -hierarchy, see([HP]), a fact that enables a description in the Grassmanian of that hierarchy.3It was shown by Bergvelt and ten Kroode, see ([BtK1]), that it is natural toconsider a system of zero curvature relations (5) labeled by the root lattice of theLie algebra, where the operators at different sites of the lattice are linked by Todatype differential-difference equations. For example, for nearest neighbour sites thereholdsq(l+1) = −(q(l))2r(l) + q(l)log(q(l)) and r(l+1) = 1q(l), (14)where P (l) =(−i 00 i)z +(0 q(l)r(l) 0). (15)This phenomenon is due to the fact that there is a natural lattice group that com-mutes with the commuting flows corresponding to the parameters tn.In the representation theoretic approach to soliton equations, see ([DKJM]) and([K]) the soliton equations occur as the equations describing the group orbit of thehighest weight vector. A similar description holds for these combined differential-difference equations. Let L(Λ0) be the basic representation of the Kac-MoodyLie algebra A(1)1 . In A(1)1 one takes the homogeneous Heisenberg algebras =∑i>0Cλi(−1 00 1)⊕∑i>0Cλ−i(−1 00 1)⊕ Cc,where c is the central element of A(1)1 that is in the kernel of the projection ofA(1)1 onto the loop algebra of Sl2(C). The A(1)1 -module L(Λ0) decomposes withrespect to the action of the homogeneous Heisenberg algebra s as a direct sumof irreducible s-modules C[t] = C[t1, t2, . . . ] labeled by the root lattice. Thuswe can write each element of L(Λ0) as τ(t) = (τl(t))l∈Z. The group orbit of thehighest weight vector can then be characterized by a set of so-called Hirota bilinearrelations for the components (τl). By using the representation theory one constructsa series of elements (gl) in a suitable completion of the Kac-Moody group associatedwith A(1)1 such that the vacuum expectation value of gl is exactly τl. The Birkhoffdecomposition of the (gl) in that group enables you then to construct solutions ofthe lattice zero curvature equations, see ([BtK2]). In particular, the operators P (l)from (14) one obtains in this way, can be expressed in the components (τl) byq(l) = 2iτl+1τland r(l) = −2i τl−1τl. (16)By using the adjoint action of the Kac-Moody group Bergvelt and ten Kroode alsoshowed in ([BtK2]) that theF(l)jk :=∂∂tj∂∂tklog(τl) (17)give you exactly the flux tensor from (11), thus furnishing a representation theoreticbasis for the results in ([FNR]).A geometric way to look at τ -functions, see e.g.([SW]), is to consider a homoge-neous space over the relevant loop group L, a holomorphic line bundle L over thisspace and its pull-back over the corresponding central extension L̂ of L. If this lastline bundle has a global holomorphic section, then τ -functions measure the failureof equivariance of partial liftings from L to L̂ w.r.t. this section. With this pointof view, one can also arrive at the formulae in (16) by lifting the discrete group oftransformations that commute with the flows from (8) appropriately, see ([W2]).An important class of equations associated with the AKNS-hierarchy are theso-called stationary AKNS-equations. These are the differential equations for theREFERENCES 4functions q and r from the first order differential operatorL := P0ddx+ P1 =(−i 00 i)ddx+(0 qr 0), (18)resulting from the existence of a 2× 2-matrix valued differential operatorPn+1 =n+1∑i=0ui(ddx)ithat commutes with L. Such a pair is naturally associated with a hyperellipticcurve of genus n and that is why one calls P1 an algebro-geometric AKNS-potential.The elliptic algebro-geometric AKNS-potentials have been characterized in ([GW]).They correspond exactly to the potentials for which the equation L(ψ) = zψ has ameromorphic fundamental system of solutions with respect to x for all valuesof the spectral parameter z ∈ C.References[BtK1] M.J.Bergvelt and A.P.E. ten Kroode, Differential-difference AKNS equa-tions and homogeneous Heisenberg algebras, J.Math.Phys. 28 (1987), 302-306.[BtK2] M.J.Bergvelt and A.P.E. ten Kroode, τ-functions and zero curvature equa-tions of Toda-AKNS type, J.Math.Phys. 29 (1988), 1308-1320.[DKJM] E. Date, M. Jimbo, M.Kashiwara, T. Miwa, Transformation groups forsoliton equations III, Operator approach to the Kadomtsev-Petviashviliequation, J. Phys. Soc. Japan 50, no.11, (1981), 3806–3812.[DS] V.G.Drinfel’d and V.V.Sokolov, Lie algebras and equations of Korteweg deVries type, Itogi Nauki i Tekhniki, ser. Sovremennye Problemy Matematiki,24 (1984), 81-180.[FNR] H.Flaschka, A.C.Newell and T.Ratiu, Kac-Moody Lie algebras and solitonequations II. Lax equations associated with A(1)1 , Physica 9D(1983), 300-323.[GW] F.Gesztesy, R.Weikard, A characterization of all elliptic algebro-geometricsolutions of the AKNS hierarchy, Acta Math. 181, (1998), 63-108.[HP] G.F. Helminck, G.F. Post, A convergent framework for the multicomponentKP -hierarchy, Trans. Amer. Math. Soc. 324, no.1, (1991), 271–292.[K] V.G.Kac, Infinite dimensional Lie algebras, Cambridge University Press,third edition 1989.[SW] G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ.Math. IHES 63 (1985), 1–64.[W1] G.Wilson, The modified Lax and two-dimensional Toda lattice equationsassociated with simple Lie algebras, Ergod.Th.and Dynam. Sys.1 (1981),361-380.[W2] G.Wilson, The τ-functions of the AKNS Equations, in Integrable Systemsthe Verdier memorial conference: actes du colloque international de Lu-miny, Progress in Mathematics Volume 115, (1993), 131-145.

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