Faculty of Mathematical Sciences t University of Twente The Netherlands P.O. Box 17 7500 AE Enschede The Netherlands Phone: +31-53-4893400 Fax: +31-53-4893114 Email: memo@math.utwente.nl www.math.utwente.nl/publications Memorandum No. 1617 Deformation and recursion for the N =α =1 supersymmetric KdV hierarchy A.S. Sorin 1 and P.H.M. Kersten March 00 ISSN 0169-690 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research (JINR), 141980 Dubna, Moscow Region, Russia
Deformation and Recursion for the N =α =1 Supersymmetric KdV hierarchy Alexander S. Sorin and Paul H. M. Kersten Abstract. A detailed description is given for the construction of the deformation of the N = supersymmetric α = 1 KdV-equation, leading to the recursion operator for symmetries and the zero-th Hamiltonian structure; the solution to a longstanding problem. 1. Introduction The N = supersymmetric α = 1 KdV-equation was originally introduced in [1] as a Hamiltonian equation with the N = superconformal algebra as a second Hamiltonian structure, and its integrability was conjectured there due to the existence of a few additional nontrivial bosonic Hamiltonians. Then its Lax pair representation has indeed been constructed in [], and it allowed an algoritmic reconstruction of the whole tower of highest commutative bosonic flows and their Hamiltonians belonging to the N = supersymmetric α =1KdV-hierarchy. Actually, besides the N = α = 1 KdV-equation there are another two inequivalent N = supersymmetric Hamiltonian equations with the N = super- conformal algebra as a second Hamiltonian structure (the N = α = and α =4KdV-equations[3, 1]), but the N =α = 1 KdV-equation is rather exceptional [4]. Despite knowledge of its Lax pair description, there remains a lot of longstanding, unsolved problems which resolution would be quite important for a deeper understanding and more detailed description of the N =α =1KdV hierarchy. Thus, since the time when the N = α = 1KdV-equationwaspro- posed, much efforts were made to construct a tower of its noncommutative bosonic and fermionic, local and nonlocal symmetries and Hamiltonians, bi-hamiltonian structure as well as recursion operator (see, e.g. discussions in [5, 6] and references therein). Though these rather complicated problems, solved for the case of the N = α = andα = 4 KdV-hierarchies, still wait their complete resolution for the N =α = 1 KdV- hierarchy, a considerable progress towards their solution arose quite recently. Thus, the puzzle [5, 6], related to the nonexistence of higher fermionic flows of the N =α = 1 KdV-hierarchy, was partly resolved in [7, 8] by explicit constructing a few bosonic and fermionic nonlocal and nonpolynomial flows and Hamiltonains, then their N = superfield structure and origin 1991 Mathematics Subject Classification. 58F07, 58G37, 58H15, 58F37. Key words and phrases. Complete Integrability, Deformations, Bi-Hamiltonian Structure, Recursion Operators, Symmetries, Conservation Laws, Coverings. 1
ALEXANDER S. SORIN AND PAUL H. M. KERSTEN were uncovered in [9]. A new property, crucial for the existence of these flows and Hamiltonians, making them distinguished compared to all flows and Hamiltonians of other supersymmetric hierarchies constructed before, is their nonpolynomiality. A new approach to a recursion operator treating it as a form valued vector field which satisfies a generalized symmetry equation related to a given equation was developed in [10, 11]. Using this approach the recursion operator of the bosonic limit of the N= α = 1 KdV-hierarchy was derived in [1], and its structure, underlining relevance of these Hamiltonians in the bosonic limit, gives a hint towards its supersymmetric generalization. The organisation of this paper is as follows. First the general notions from the mathematical theory of symmetries, nonlocalities, deformations and form-valued vector fields are exposed to some detail in Section and its subsections for the classical KdV-equation. For full details the reader is referred to e.g. [10, 7]. In Section 3 we expose all results obtained for the N =α = 1 supersymmetric KdV-equation in great detail. In Section 4 we present conclusions, while in Section 5, arranged as an Appendix, results of the Poisson bracket structure are given.. Nonlocal Setting for Differential Equations, the KdV Equation.1. Nonlocalities. As standard example, to illustrate the notions we are going to discuss for the N =α = 1 supersymmetric KdV-equation in Section 3, we take the KdV-equation u t = uu x + u xxx. (1) For a short theoretical introduction we refer to [1], while for more detailed expositions we refer to [10, 7, 11]. We consider Y J (x, t; u) the infinite prolongation of (1), c.f.[13, 14], where coordinates in the infinite jet bundle J (x, t; u) aregivenby(x, t, u, u x,u t, )and Y is formally described as the submanifold of J (x, t; u) defined by u t = uu x + u xxx, u xt = uu xx + u x + u xxxx, ().. As internal coordinates in Y one chooses (x, t, u, u x,u xx, ) while u t,u xt, are obtained from (). The Cartan distribution on Y is given by the total partial derivative vector fields D x = x + u n+1 un, n 0 D t = t + (3) u nt un n 0 where u = u 0,u 1 = u x, u = u xx, ; u 1t = u xt ; u t = u xxt.
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY3 Classically the notion of a generalized or higher symmetry Y of a differential equation {F =0} is defined as a vertical vector field V V = З f = f u + D x (f) u1 + D x(f) u +... (4) where f C (Y )aresuchthat l F (f) =0. (5) Here, l F is the universal linearisation operator [15, 13], also denoted as Fréchet derivative of F, which reads in the case of the KdV-equation () D t (f) u D x (f) u 1 f ( D x ) 3 (f) =0. (6) Let now W R m with coordinates (w 1, w m ). The Cartan distribution on Y W is given by D x = D x + m X j wj, j=1 D t = D t + m (7) T j wj where X j,t j C (Y W ) such that [D x, D t ]=0 (8) which yields the so called covering condition D x (T ) D t (X)+[X, T] =0 whereas in (8) [*,*] is the Lie bracket for vector fields X = m j=1 Xj wj, T = m j=1 T j wj defined on W. A nonlocal symmetry is a vertical vector field on Y W, i.e. of the form (4), which satisfies (f C (Y W )) l F (f) =0 (9) which for the KdV-equation results in D t (f) ud x (f) u 1 f (D x ) 3 (f) =0. (10) Formally, this is just what is called the shadow of the symmetry, i.e., not bothering about the w j (j =1...m)components. In effect the full symmetry should also satisfy the invariance of the equations governing the nonlocal variables w j (j =1,...,m); i.e., (w j ) x = X j, (w j ) t = T j. The construction of the associated w j (j =1,...,m) components is called the reconstruction problem [16]. For reasons of simplicity, we omit this reconstruction problem, i.e. reconstructing the complete vector field or full symmetry from its shadow. The classical Lenard recursion operator R for the KdV-equation, j=1 which is just such, that R = D x + 3 u + 1 3 u 1D 1 x (11)
4 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN f 0 = u 1, Rf 0 = f 1 = uu 1 + u 3, (1) Rf 1 = f = u 5 + 5 3 u 3u + 10 3 u u 1 + 5 6 u 1u, i.e., creating the (x, t) independent hierarchy of higher symmetries, has an action on the vertical symmetry З f 1 (Gallilei-boost) f 1 =(1+tu 1 )/3, Rf 1 = f 0 =u + xu 1 +3t(u 3 + uu 1 ), (13) f 1 = Rf 0 =3t(f )+x(f 0 )+4u + 4 3 u + 1 3 u 1Dx 1 (u). If we introduce the variable p (= w 1 ) through p x = u, p t = u + 1 u, (14) i.e. D t (u) =D x (u + 1 u ), then З f 1 is the shadow of a nonlocal symmetry in the one-dimensional covering of the KdV-equation by p = w 1,X 1 = u, T 1 = u + 1 u. So, by its action the Lenard recursion operator creates nonlocal symmetries in a natural way. More applications of nonlocal symmetries can be found in e.g.[7]... A Special Type of Covering: the Cartan-covering. We discuss a special type of the nonlocal setting indicated in the previous section, the so called Cartan-covering. As mentioned before we shall illustrate this by the KdV-equation. Let Y J (x, t; u) be the infinite prolongation of the KdV-equation (). Contact one forms on TJ (x, t; u) aregivenby α 0 = du u 1 dx u t dt, α 1 = du 1 u dx u 1t dt, (15) α = du u 3 dx u t dt. From the total partial derivative operators of the previous section we have D x (α 0 )=α 1, Dx (α 1 )=α,..., D t (α 0 ) = α 0 u x + α 1 u + α 3 = α t, D t (α i ) = ( D x ) i (α t ). We now define the Cartan-covering of Y by Y R D C x = D x + i D C t = D t + i (α i+1 ) α i, (16) ( D x ) i α t α i (17)
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY5 where local coordinates are given (x, t, u, u 1,...,α 0,α 1,...). It is a straightforward check, and obvious that [Dx C,DC t ]=0, (18) i.e. they form a Cartan distribution on Y R. Note 1: Since at first α i (i =0,...) are contact forms, they constitute a Grassmann algebra (graded commutative algebra) Λ(α), where α i α j = α j α i, i.e., xy =( 1) x y yx where x, y are contact ( )-forms of degree x and y, respectively. So in effect we are dealing with a graded covering. Note : Once we have introduced the Cartan-covering by (17) we can forget about the specifics of α i (i =0,...) and just treat them as (odd) ordinary variables, associated with their differentiation rules. One can discuss nonlocal symmetries in this type of covering just as in the previous section, the only difference being: f C (Y ) (α). In the next section we shall combine constructions of the previous subsection and this one, in order to construct the recursion operator for symmetries..3. The Recursion Operator as Symmetry in the Cartan-covering. We shall discuss the recursion operator for symmetries of the KdV-equation as a geometrical object, i.e., a symmetry in the Cartan-covering. Our starting point is the four dimensional covering of the KdV-equation in Y R 4 where D x = D x + u w1 + 1 u w +(u 3 3u 1) w3 + w 1 w4, D t = D t +( 1 u + u ) w1 +( 1 3 u3 1 u 1 + uu ) w (19) +( 3 4 u4 6u 1 u 3 +3u u 6uu 1 +3u ) w 3 +(u 1 + w ) w4. D x, D t satisfy the covering condition (8), and note, that due to the fact that the coefficients of wi (i =1,, 3) in (19) are independent of w j (j =1,, 3), these coefficients constitute local conservation laws for the KdV-equation. The coefficients w 1 and u 1 + w of w4 constitute the first nonlocal conservation law of KdV-equation.. We have the following formal variables:
6 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN w 1 = udx, 1 w = u dx, w 3 = (u 3 3u 1 )dx, (0) w 4 = w 1 dx. where w 4 is of a higher or deeper nonlocality. We now build the Cartan-covering of the previous section on the covering given by (19) by introduction of the contact forms α 0,α 1,α,... (15) and α 1 = dw 1 udx ( 1 u + u )dt, α = dw 1 u dx ( 1 3 u3 1 u 1 + uu )dt (1) as well as similarly for α 3,α 4. It is straightforward to prove the following relations D x (α 1 )=α 0, D t (α 1 )=uα 0 + α 0, D x (α )=uα 0, D t (α )=u α 0 u 1 α 1 + uα + u α 0, () D x (α 3 )=3u α 0 6u 1 α 1,.... We are now constructing symmetries in this Cartan-covering of the KdV-equation which are linear w.r.t. α i (i = 4,...,0, 1,...). The symmetry condition for f C (Y R 4 ) Λ 1 (α) isjustgivenby(6) l C F (f) = 0 (3) which for the KdV-equation results in D C t (f) udc x (f) u xf (D C x )3 f =0. As solutions of these equations we obtained f 0 = α 0, f 1 =( 3 u)α 0 + α +( 1 3 u 1)α 1, (4) f =( 4 9 u + 4 3 u )α 0 +(u 1 )α 1 +( 4 3 u)α + α 4 + 1 3 (uu 1 + u 3 )α 1 + 1 9 (u 1)α. As we mentioned above we are working in effect with form-valued vector fields З f 0, З f 1З f. For these objects one can define Frölicher-Nijenhuis and (by contraction) Richardson-Nijenhuis brackets [10],[7] Without going into details, for which the reader is referred to [10], we can construct the contraction of a (generalized) symmetry and a form-valued symmetry e.g. R =( 3 uα 0 + α + 1 3 u 1α 1 ) +.... (5) u
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY7 The contraction formally defined by V R u = α V α R α u where α runs over all local and nonlocal variables, is given by (V R) =(V R u ) u + D C x (V R u) u1 +.... (6) Start now with V 1 = u 1 u + u +... (7) u 1 whose prolongation in the setting Y R 4 is V 1 = u 1 u + u +...+ u + 1 u 1 w 1 u +(u 3 3u 1 w ) w 3 + w 1 (8) w 4 then (V 1 R) =[( 3 u)u 1 +1 u 3 + 1 3 u 1 u] u +... =(u 3 + uu 1 ) u +...= V 3 (9) and similarly (V 3 R) =(u 5 + 5 3 u 3u + 10 3 u u 1 + 5 6 u u 1 ) u +...= V 5. (30) The result given above means that the well known Lenard recursion operator for symmetries of the KdV-equation is represented as a symmetry, З f1,inthecartancovering of this equation and in effect is a geometrical object.
8 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN 3. The N =α =1Supersymmetric KdV Equation In this section we shall discuss all computations leading at the end to the complete integrability of the N = α = 1 supersymmetric KdV-equation. The N = α = 1 supersymmetric KdV-equation is described by J t = {J zz +3J[D, D]J + J 3 } z (31) in the N = superspace with a coordinate Z = (z,θ,θ), z denotes derivative with respect to z and D, D are the fermionic covariant derivatives of the N = supersymmetry, governed by definitions D = θ 1 θ z, D = θ 1 θ z, D = D { } =0, D, D = z. Formally, the nonlocal setting for differential equations of the previous section was done for classical equations, but applies too for supersymmetric equations([7]). In order to discuss conservation laws, symmetries and deformations we choose local coordinates in the infinite jet bundle Y ((z,θ,θ); J), where we choose as local even coordinates z,t,j,ddj, J z,ddj z,j zz,ddj zz,j zzz,... and odd coordinates θ, θ, DJ, DJ, DJ z, DJ z,dj zz, DJ zz,.... In order to have a complete setting we first describe the construction of conservation laws and the associated introducton of nonlocal variables. In effect we construct an abelian covering of the equation structure. In the first subsection we shall discuss conservation laws and the associated nonlocalities. In the next subsection we obtain nonlocal symmetries associated to these nonlocalities, which turn out to be (θ, θ) - dependent and arise in so called quadruplets, similar to the nonlocalities. In the subsection 3.3 we derive an explicit deformation of the equation structure leading to the recursion operator for symmetries. Finally in the last subsection we obtain the factorization of the recursion operator as product of the second Hamiltonian operator and the inverse of the zero-th Hamiltonian operator. 3.1. Conservation laws and nonlocal variables. Here we shall construct conservation laws for (31) in order to arrive at an abelian covering. So we construct X = X(z,t,J,...), T = T (z,t,j,...) such that D z (T )=D t (X) (3) and in a similar way we construct nonlocal conservation laws by the requirement D z ( T )= D t ( X) (33) where D z, D t are defined by D z = z + J z J + DDJ z DDJ +... + DJ z DJ + DJ z DJ +... D t = t + J t J + DDJ t DDJ +... + DJ t DJ + DJ t DJ +.... (34)
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY9 Moreover X, T are dependent on local variables z, t, J,..., as well as the already determined nonlocal variables, denoted here by p, which are associated to the conservation laws (X, T) by the formal definition D z (p )=(p ) z = X, D t (p )=(p ) t = T. Proceeding in this way, we obtained a number of conservation laws, which arise as multiplets of four conservation laws each. The corresponding nonlocal variables are P 0,DP 0, DP 0,DDP 0, Q 1,DQ 1, DQ 1,DDQ 1, Q 1,DQ 1, DQ 1,DDQ1, (35) P 1,DP 1, DP 1,DDP 1 where their defining equations are given by (P 0 ) z = J, (Q 1 ) z = e (+) DJ, (Q 1 ) z = e ( ) DJ, (36) (P 1 ) z = e (+) (DJ)(DP 0 )+e ( ) (DJ)(Q 1 ). In (36) e (+) and e ( ) refer to e (+P0) and e ( P0) respectively. It should be noted that the quadru-plet P 0 satisfies differentiation rules as follows: D(P 0 )=DP 0, D(DP 0 )=0, D(DP 0 )=DDP 0, D(DDP 0 )=0, (37) D(P 0 )=DP 0, D(DP 0 )= (P 0 ) z DDP 0, D(DP 0 )=0, D(DDP 0 )= D(P 0 ) z, and similarly for other quadru-plets Q 1, Q 1,P 1. It should be noted that for the two other N =KdV-hierarchies(α=4 and -), despite their original N = supersymmetric structure, their conservation laws do not form supersymmetric multiplets. So in effect we have at this moment sixteen local and nonlocal conservation laws leading to a similar number of new nonlocal variables. They perfectly match with those ones obtained previously ([7])(page 340,(7.78)). If we arrange them according to their respectively degrees we arrive at:
10 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN 0:P 0 ; 1 : DP 0, DP 0,Q1, Q 1 ; 1:DDP 0,DQ1,DQ1, DQ 1, DQ 1,P 1; 3 : DDQ 1,DDQ1,DP 1, DP 1 ; :DDP 1. In Subsection 3. we shall discuss local and nonlocal symmetries of the (31) while in Subsection 3.3 we construct the recursion operator or deformation of the equation structure. 3.. Local and nonlocal symmetries. In this section we shall present results for the construction of local and nonlocal symmetries of (31). In order to construct these symmetries, we consider the system of partial differential equations obtained by the infinite prolongation. First we present the local symmetries as they are required for explicit formulae for the coefficients which arise in the form-valued vector field of the next subsection leading to the recursion operator for symmetries. Y 1 := J z, Y 3 := {J zz +3J[D, D]J + J 3 } z, Y 5 := 10 DJ DJ zz J 0 DJ DJ z J 10 DJ DJ z J z 10 DJ DJ zz J +0 DJ DJ z J 10 DJ DJ z J z +40 DJ DJ J J z +5 J 4 J z +0 J 3 DDJ z +10 J 3 J zz + 60 J DDJ J z +30 J J z +10 J J zzz +80 J DDJ DDJ z +40 J DDJ J zz +5 J J zzzz +40 J J z DDJ z +50 J J z J zz +10 J DDJ zzz +40 DDJ J z +40 DDJ J z + 10 DDJ J zzz +J zzzzz +15 J 3 z +0 J z DDJ zz +15 J z J zzz +0 DDJ z J zz +10 J zz, Y 7 := ( 4 DJ z DJ zzz J 16 DJ z DJ zz J 8 DJ z DJ zz J z 4 DJ z DJ zzz J +16 DJ z DJ zz J 8 DJ z DJ zz J z +364 DJ z DJ z J J z 8 DJ DJ zzzz J 70 DJ DJ zzz J 4 DJ DJ zzz J z 70 DJ DJ zz J 3 168 DJ DJ zz J DDJ 350 DJ DJ zz J J z 8 DJ DJ zz J zz 84 DJ DJ z J 4 336 DJ DJ z J DDJ 378 DJ DJ z J J z 168 DJ DJ z J DDJ z 94 DJ DJ z J J zz 168 DJ DJ z DDJ J z 38 DJ DJ z J z 14 DJ DJ z J zzz 8 DJ, 1) J +70 DJ DJ zzz J 4 DJ DJ zzz J z 70 DJ DJ zz J 3 168 DJ DJ zz J DDJ+18 DJ DJ zz J J z 8 DJ DJ zz J zz +84 DJ DJ z J 4 +336 DJ DJ z J DDJ 4 DJ DJ z J J z 168 DJ DJ z J DDJ z +16 DJ DJ z J J zz 168 DJ DJ z DDJ J z + 70 DJ DJ z J z 14 DJ DJ z J zzz +336 DJ DJ J 3 J z +336 DJ DJ J DDJ z + 168 DJ DJ J J zz +67 DJ DJ J DDJ J z +336 DJ DJ J J z +84 DJ DJ J J zzz +140 DJ DJ J z J zz +7 J 6 J z +4 J 5 DDJ z +1 J 5 J zz +10 J 4 DDJ J z + 105 J 4 J z +35 J 4 J zzz +504 J 3 DDJ DDJ z +5 J 3 DDJ J zz +35 J 3 J zzzz + 5 J 3 J z DDJ z +350 J 3 J z J zz +70 J 3 DDJ zzz +756 J DDJ J z +756 J DDJ J z +4 J DDJ J zzz +1 J J zzzzz +315 J J 3 z +406 J J z DDJ zz +315 J J z J zzz +40 J DDJ z J zz +10 J J zz+840 J DDJ DDJ z +40 J DDJ
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY 11 J zz +98 J DDJ J zzzz +840 J DDJ J z DDJ z +1148 J DDJ J z J zz +196 J DDJ DDJ zzz +147 J J zzzz J z +14 J DDJ zzzzz +7 J J zzzzzz +74 J J z DDJ z + 735 J J z J zz+98 J J z DDJ zzz +40 J DDJ z DDJ zz +10 J DDJ z J zzz +10 J J zz DDJ zz +45 J J zz J zzz +80 DDJ 3 J z +40 DDJ J z +84 DDJ J zzz + 14 DDJ J zzzzz +350 DDJ J 3 z +364 DDJ J z DDJ zz +66 DDJ J z J zzz +336 DDJ DDJ z J zz +168 DDJ J zz +4 J zzzz DDJ z +56 J zzzz J zz +4 DDJ zzzz J z +8 J zzzzz J z + J zzzzzzz + 105 J 4 z + 18 J z DDJ zz + 10 J z J zzz + 80 J z DDJ z +448 J z DDJ z J zz +80 J z J zz+70 J zz DDJ zzz +70 DDJ zz J zzz +35 J zzz). It should be noted that the symmetries Y 5 and Y 7 are rather massive, containing 8 and 104 terms respectively. We made our choice for a presentation as shown above, in order to have expressions, which are quite huge, as readable as possible. Now, we present the four multiplets of nonlocal (θ, θ) dependent symmetries as they were constructed in the above described nonlocal setting. The nonlocal (θ, θ) dependent symmetries at level 0, 1/, 1, 3/, are given as follows: This represents the first quadru-plet associated to P 0 Y P0,θθ := θθ J z + θ DJ θ DJ, Y P0,θ := θ J z + DJ, Y P0,θ := θ J z + DJ, Y P0 := J z. The second quadru-plet associated to Q 1 is represented by Y Q 1,θθ := ( θθ e(+) ( DP 0 DDJ DP 0 J z DJ z + DJ DP 0 DP 0 + DJ DDP 0 )+ θθ (Q 1 J z + DJ DQ 1 + DJ DQ 1 ) θ e(+) DJ DP 0 + θ DJ Q 1 θ e(+) (DJ DP 0 +DDJ +J z ) θ DJ Q 1 e(+) DJ)/, Y Q 1,θ := θ e(+) ( DP 0 DDJ DP 0 J z DJ z + DJ DP 0 DP 0 + DJ DDP 0 )+θ (Q 1 J z + DJ DQ 1 + DJ DQ 1 ) e(+) DJ DP 0 + DJ Q 1, Y Q 1,θ := θ e (+) ( DP 0 DDJ DP 0 J z DJ z + DJ DP 0 DP 0 +DJ DDP 0 )+θ (Q 1 J z +DJ DQ 1 +DJ DQ 1 )+e(+) (DJ DP 0 +DDJ +J z )+DJ Q 1, Y Q 1 := e (+) ( DP 0 DDJ DP 0 J z DJ z + DJ DP 0 DP 0 + DJ DDP 0 )+Q1 J z + DJ DQ 1 + DJ DQ 1. The third quadru-plet associated to Q 1 is represented by Y Q 1,θθ := ( θθ e(+) (Q 1 J z + DJ DQ 1 + DJ DQ 1 )+ θθ ( DP 0 DDJ + DJ z + DJ DP 0 DP 0 DJ J DJ DDP 0 )+ θ e (+) DJ Q 1 +
1 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN θ (DJ DP 0 + DDJ) θ e (+) DJ Q 1 + θ DJ DP 0 DJ)/( e (+) ), Y Q 1,θ := (θ e(+) (Q 1 J z + DJ DQ 1 + DJ DQ 1 )+θ ( DP 0 DDJ + DJ z + DJ DP 0 DP 0 DJ J DJ DDP 0 )+e (+) DJ Q 1 +DJ DP 0+DDJ)/e (+), Y Q 1,θ := (θ e(+) (Q 1 J z + DJ DQ 1 + DJ DQ 1 )+θ ( DP 0 DDJ + DJ z + DJ DP 0 DP 0 DJ J DJ DDP 0 )+e (+) DJ Q 1 DJ DP 0)/e (+), Y Q 1 := (e (+) (Q 1 J z + DJ DQ 1 + DJ DQ 1 )+ DP 0 DDJ + DJ z + DJ DP 0 DP 0 DJ J DJ DDP 0 )/e (+). The fourth quadru-plet associated to P 1 is represented by Y P1,θθ := (θθ e(+) (4 DP 0 Q 1 DDJ +4 DP 0 Q 1 J z + DJ z Q 1 DJ DDQ 1 DJ Q 1 DDP 0 + DJ DP 0 DQ 1 4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 + DDJ DQ 1 + J z DQ 1 )+θθ e(+) ( Q1 Q1 J z DJ Q1 DQ 1 + DJ Q 1 DQ 1 DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ DJ + J J z + DDJ z + J zz )+θθ (4 DP 0 Q 1 DDJ + DJ z Q 1 + DJ DDQ 1 DJ Q 1 J DJ Q 1 DDP 0 + DJ DP 0 DQ 1 +4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 DDJ DQ 1 )+θ e(+) ( DJ DP 0 Q 1 +DJ DQ 1 )+θ e(+) (DP 0 J z + DJ z DJ Q1 Q 1 +DJ J DJ DDP 0)+θ ( Q1 DDJ + DJ DP 0 Q1 DJ DQ 1 over )+θ e (+) ( Q 1 DDJ + Q1 J z + DJ DP 0 Q 1 +DJ DQ 1 )+ θ e (+) ( DP 0 J z + DJ z + DJ Q 1 Q 1 DJ J DJ DDP 0)+θ ( DJ DP 0 Q 1 DJ DQ 1 )+e(+) DJ Q 1 +e(+) (DJ DP 0 DJ DP 0 ) DJ Q 1 )/e(+), Y P1,θ := (θ e(+) (4 DP 0 Q 1 DDJ +4 DP 0 Q 1 J z + DJ z Q 1 DJ DDQ 1 DJ Q 1 DDP 0 + DJ DP 0 DQ 1 4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 + DDJ DQ 1 + J z DQ 1 )+θ e(+) ( Q 1 Q 1 J z DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ DJ + J J z + DDJ z +J zz )+θ (4 DP 0 Q 1 DDJ + DJ z Q 1 +DJ DDQ 1 DJ Q 1 J DJ Q 1 DDP 0 + DJ DP 0 DQ 1 +4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 DDJ DQ 1 )+e(+) ( DJ DP 0 Q 1 + DJ DQ 1 )+e(+) (DP 0 J z + DJ z DJ Q 1 Q 1 +DJ J DJ DDP 0)+ Q 1 DDJ + DJ DP 0 Q 1 DJ DQ 1 )/e(+), Y P1,θ := (θ e (+) (4 DP 0 Q 1 DDJ+4 DP 0 Q 1 J z+ DJ z Q 1 DJ DDQ 1 DJ Q 1 DDP 0+ DJ DP 0 DQ 1 4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 + DDJ DQ 1 + J z DQ 1 )+θ e(+) ( Q 1 Q 1 J z DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ DJ + J J z + DDJ z +J zz )+θ (4 DP 0 Q 1 DDJ + DJ z Q 1 +DJ DDQ 1 DJ Q 1 J DJ Q 1 DDP 0 + DJ DP 0 DQ 1 +4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 DDJ DQ 1 )+ e (+) ( Q1 DDJ Q1 J z DJ DP 0 Q1 DJ DQ 1 )+e(+) (DP 0 J z
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY 13 DJ z DJ Q 1 Q1 + DJ J +DJ DDP 0) DJ DP 0 Q 1 +DJ DQ 1 )/e(+), Y P1 := (e (+) (4 DP 0 Q 1 DDJ +4 DP 0 Q 1 J z + DJ z Q 1 DJ DDQ 1 DJ Q 1 DDP 0 + DJ DP 0 DQ 1 4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 + DDJ DQ 1 + J z DQ 1 )+e(+) ( Q 1 Q 1 J z DJ Q 1 DQ 1 + DJ Q 1 DQ 1 DJ Q1 DQ 1 + DJ Q1 DQ 1 DJ DJ + J J z + DDJ z +J zz )+ 4 DP 0 Q1 DDJ + DJ z Q1 +DJ DDQ 1 DJ Q1 J DJ Q1 DDP 0 + DJ DP 0 DQ 1 +4 DJ DP 0 DP 0 Q 1 DJ DP 0 DQ 1 DDJ DQ 1 )/e(+). Note that in previous formulas e (+) refers to e 4P0. 3.3. Recursion operator. HerewepresenttherecursionoperatorR for symmetries for this case obtained as a higher symmetry in the Cartan covering of equation (31) augmented by equations governing the nonlocal variables (36 37). As explained in the previous section, the recursion operator is in effect a deformation of the equation structure. As demonstrated there, this deformation is a form-valued vector field and has to satisfy l C F (R) =0. (38) In order to arrive at a nontrivial result as was explained for the KdV equation, we have to introduce associated to the nonlocal variables P 0,DP 0, DP 0, DDP 0, Q1,... their Cartan forms ω P0,ω DP0, ω DP0, ω DDP0, ω Q 1, ω DQ 1, ω DQ 1, ω DDQ 1, ω Q 1,ω DQ 1, ω DQ 1, ω DDQ 1, ω P1, ω DP1, ω DP1,ω DDP1. Motivated by the results of the previous subsections and the grading of the equation our search is for a one-form-valued vector field whose defining function is of degree 3. So besides the Cartan forms associated to the nonlocal variables P 0,... which will account for the pseudo differential part of the recursion operator, we also have to introduce the local Cartan forms ω J, ω DJ,ω DJ, ω DDJ, ω Jz, ω DJz, ω DJz, ω DDJz, ω Jzz,ω DJzz, ω DJzz,ω DDJzz which will represent the pure differential part of the recursion operator. We have to remark that the coefficients of the one-form-valued vector field are just functions dependent on all local and nonlocal variables. Moreover it should be noted that the representation of the vector field with respect to its form part is to be understood as to be equipped with a right-module structure. This will account for the correct action of contraction. Since the coordinate P 0 is of degree zero, this procedure requires the introduction of approximately 400, yet free, functions dependent on this variable. Now this quite extensive one-form-valued vector field has to satisfy eq. (38). Although this is just the deformation condition which should be satisfied, we decided, in order to reduce the already extremely extensive computations, first to require that the resulting recursion operator performs its duties, sending symmetries to
14 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN symmetries. In order to achieve these goals we required the operator to satifies the following requirements R(Y 1 )=Y 3, R(Y 3 )=Y 5, R(Y 5 )=Y 7. (39) Due to these conditions it has been possible to fix all 400 free functions in the defining function R of the one-form-valued vector field R. In effect the condition R(Y 5 )=Y 7 was itself sufficient to fix all coefficients. After this, the result has been substituted into (38), satisfying it completely. So, the final result is the following: Starting form the defining function R for the form-valued vector field R, givenby R = α ω α Φ α (40) where α runs over DDJ zz,..., DJ, J, P 0,DP 0,...,D DP 1, the coefficients are given by Φ DDP1 := 0, Φ P1 := J z, Φ DQ 1 := e (+) DJ DP 0 DJ Q 1, Φ DQ 1 := (DJ DP 0 + DDJ) e ( ), Φ DQ 1 := (e (+) DJ DP 0 + e (+) DDJ + e (+) J z + DJ Q 1 ), Φ DQ 1 := e ( ) DJ DP 0, Φ DDP0 := DJ DP 0 +3 J z, Φ P0 := DJ DJ + J J z + DDJ z + J zz, Φ DP1 := DJ, := DJ, Φ DP1 Φ DDQ 1 Φ DDQ 1 := e (+) DJ/, := e ( ) DJ/, Φ Q 1 := e (+) DP 0 DDJ + e (+) DP 0 J z + e (+) DJ z e (+) DJ DP 0 DP 0 e (+) DJ DDP 0 Q 1 J z DJ DQ 1 DJ DQ 1, Φ Q 1 := e ( ) ( DP 0 DDJ DJ z DJ DP 0 DP 0 + DJ J + DJ DDP 0 ), Φ DP0 := DP 0 J z DJ z DJ J + DJ DDP 0, Φ DP0 := DJ z DJ J, Φ Jzz := 1, Φ DDJz := 0, Φ Jz := J, Φ DDJ := 4 J, Φ J := J +3 DDJ +3 J z,
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY 15 Φ DJz := 0, Φ DJz := 0, Φ DJ := 0, Φ DJ := ( DJ)/. This now finishes the longstanding problem of the existence of the recursion operator for the N = supersymmetric α =1KdV-equation. Transformation of the formvaluedness to the Fréchet derivatives leads to the presentation of the recursion operator in classical form ([17]). We have as mentioned before checked that the form-valued vector field satisfies (38) and so indeed gives the proper recursion operator for symmetries of the N =α =1KdV-hierarchy. 3.4. Factorization of the Recursion Operator and the Bi-Hamiltonian structure. Here we shall present the factorization of the recursion operator obtained in last section. Factorization in this respect means R = J J0 1 (41) where J is the second Hamiltonian structure J 1 [D, D ] + DJD + DJD + J + J (4) and J 0 will be the zero Hamiltonian structure. We assume J0 1 to be a one-form-valued function, i.e., in effect a pseudo differential operator, the pseudo part of which is realized through the Frechet derivatives of the nonlocal variables. So we describe J0 1 in a similar way as the defining function of the deformation structure J0 1 = ω α Φ 0 α (43) α where α runs over local and nonlocal variables and Φ 0 α being function of appropriate degree. A rather straightforward computation does lead to the following result: α and the associated Φ 0 α are given by Φ 0 P 1 := 1, Φ 0 DQ 1 := e ( ) )/, Φ 0 DQ 1 := e (+) /, Φ 0 DDP 0 := 3, Φ 0 P 0 := J, Φ 0 Q 1 := (e (+) DP 0 + Q 1 ), Φ 0 Q 1 := e ( ) DP 0, Φ 0 DP 0 := DP 0,
16 ALEXANDER S. SORIN AND PAUL H. M. KERSTEN Φ 0 J := 3/. If we use now Frechet derivatives associated to the occurring nonlocal variables in the presentation, we arrive at J0 1 =[D, D ] 1 + 1 J 1 + 1 f T 1 1 f 1 1 f T 1 1 f 1 (44) where f 1, f 1 are the Fréchet derivatives of Q 1, Q 1 respectively. The factorization of the recursion operator for symmetries has wider applicability in the construction of Hamiltonian operators and will be discussed more deeply elsewhere. 4. Conclusion We gave an outline of the theory of deformations of the equation structure of differential equations, leading to the construction of recursion operators for symmetries of such equations. The extension of this theory to the nonlocal setting of differential equations is essential for getting nontrivial results. The theory has been applied to the construction of the recursion operator for symmetries for a coupled KdV mkdv system, leading to a highly nonlocal result for this system. Moreover the appearance of nonpolynomial nonlocal terms in all results, e.g., conservation laws, symmetries and recursion operator is striking and reveals some unknown and intriguing underlying structure of the equations. Acknowledgments. A.S. is grateful to University of Twente for the hospitality extended to him during this research. This work was partially supported by the grants NWO NB 61-491, FOM MF 00/39, RFBR 99-0-18417, RFBR-CNRS 98-0-034, PICS Project No. 593, Nato Grant No. PST.CLG 974874,RFBR-DFG 0-0-0400 and the Heisenberg-Landau program.
DEFORMATION AND RECURSION FOR THE N = α = 1 SUPERSYMMETRIC KDV HIERARCHY 17 5. Appendix: The Second layer of Nonlocalities Here we present the second set of sixteen nonlocal variables, conservation laws and Hamiltonians. We shall present here the results for P, DP, DP,DDP ; Q 5,DQ5, DQ 5,DDQ 5 ; Q 5,DQ5, DQ 5,DDQ5; P 3, DP 3, DP 3,DDP 3. The explicit formulae for (P ) z, (Q 5 ) z, (Q 5 ) z are (P ) z := (( ) ( e (+) DJ Q 1 J e(+) DJ Q 1 DDP 0 e (+) DJ DP 0 DQ 1 e(+) DJ DP 0 DQ 1 + e(+) ḑotdj Q 1 DQ 1 + e(+) DJ DP 0 J + e (+) DJ DP 0 DDP 0 + e (+) DJ Q 1 DQ 1 e(+) DJ DP 0 DDP 0 e (+) DJ DJ + DJ Q 1 DDP 0 DJ DP 0 DQ 1 ))/e(+), (Q 5 ) z := ( ( e (+) DP 1 DDJ e (+) DP 1 J z +e (+) DP 0 DDJ DDP 0 + e (+) DP 0 J z DDP 0 e (+) DJ DP 0 DP 1 e (+) DJ J e (+) DJ J DDP 0 e (+) DJ DDJ e (+) DJ J z + e (+) Q 1 J DDJ + e(+) Q 1 DDJ DDP 0 e (+) DP 0 DDJ DQ 1 + e(+) DJ DP 0 Q 1 e (+) DJ DP 0 Q 1 DDP 0 e (+) DJ J DQ 1 e(+) DJ DDP 0 DQ 1 e (+) DJ DP 0 Q 1 DDP 0 e (+) DJ DJ Q 1 DJ DP 0 Q 1 DQ 1 ))/e(+), (Q 5 ) z := ( ( e (+) Q 1 DDJ DDP 0 e (+) Q 1 J z DDP 0 + e (+) DP 0 DP 0 Q 1 J z + e (+) DJ DP 0 Q 1 DDP 0 + e (+) DJ DP 0 Q 1 J e(+) DJ DP 0 Q 1 DDP 0 e (+) DJ DJ Q 1 e(+) DJ DDP 0 DQ 1 DP 1 DDJ Q 1 DDJ DQ 1 + Q 1 DDJ DQ 1 DJ z DDP 0 DJ DP 0 Q 1 DQ 1 DJ DP 0 DP 1 DJ DP 0 Q 1 DQ 1 + DJ DP 0 Q 1 DQ 1 + DJ DP 0 DP 0 J +3 DJ J DDP 0 DJ DQ 1 DQ 1 ))/e(+). Other quantities can be obtained by action of D and D. The nonlocal variable P 3 results from the Poisson bracket of other Hamiltonians, i.e., {Q 1, Q 5 }. The general Poisson algebra structure of the Hamiltonians will be discussed elsewhere ([18]).
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