Integrability of the KdV hierarchies via group theoretical approach

This paper explores the integrability of the Korteweg-de Vries (KdV) hierarchies via a renowned group theoretical approach. Briefly this work employs the group representation method in identifying a (physically significant) nonlinear dynamical system as an integrable Hamiltonian system. | Integrability | KdV hierarchy | Group theoretical approach |

tersely explores the integrability of the KdV hierarchies (or simply the system of generalised KdV equation) by reworking on two important propositions.Finally we end this paper with some comments on this approach.

The Gelfand-Dickey Programme
Gelfand and Dickey [11] developed the algebra and variational calculus in the ring of polynomial functions of Η ∈ ϕ and their derivatives with respect to x, where T S , Η is the function space with integers S ≥ 0, T∈ R.

χ χ
By using the symbolic technique and expressions involving the resolvent diagonal and fractional powers to L, Gelfand and Dickey [10] were able to show the Lax representation in the 'symbol space'; i. ; r = 0,1, … , n-2 (1) where K a polynomial function.Essentially equation (1) embodies the KdV hierarchies or the system of generalised KdV equation.In addition, it can be shown (refer Gelfand and Dickey [10]) that the KdV hierarchies are equivalent to the Hamiltonian equations and interestingly Gelfand and Dickey [10] had shown that all the first integrals j H of the system (2) would form the involutive system, i.e.

Concept of integrability in this formulation
Dynamical formulation in this framework is based on the following facts: A. Infinite dimensional representation theory for each Lie G is well connected to a finite dimensional representation of G.The representation acts in the dual space g * to g for G and is called the coadjoint representation.
B. Orbit of a Lie group in a space of coadjoint representation represents a symplectic manifold and can be interpreted as phase space of a system of Hamiltonian mechanics.The Lie group is a symmetry group.
The splitting of a Lie algebra into a direct sum of vector spaces for several Lie algebras is found (Konstant [13], Symes [22]) to be responsible for the integrability of a Hamiltonian system with Lax representation.The Konstant-Symes theorem discusses elaborately this key aspect.

Konstant-Symes Theorem (refer Adler & van Moerbeke [2])
Let g be a Lie algebra having a direct sum decomposition of vector spaces as follows , k h g ⊕ = h and k are Lie subalgebras.Assume that g can be identified with its dual vector space, g * via a nondegenerate bilinear form and ad-invariant ( ).
,⋅ ⋅ Subsequently it is induced on g * a direct sum decomposition The nondegeneracy of ( ) , can be written as ), (g This means that the system of Hamiltonian equations take the form of g and ad * =ad, then (*) and (**) can be represented by the Lax equation

Remark 1
We discuss here several important remarks that can be concluded from the above theorem: a.The coadjoint orbit ∝ k of the dual space algebra Lie g represents the 'phase space' of a system of Hamiltonian mechanics and certainly the Lax pairs naturally live in g.

b. The coadjoint orbit
∝ k inherits the symplectic structure, and thus is defined the Poisson bracket over it.

For two functions
This equation can also be expressed as , where the vector fields The result below shows the relevancy of the algebra g of the formal pseudo differential operators in Gelfand-Dickey programme and within this group theoretical formulation.
, where the 'plane' ) , associated with a localized function H at 2 p , i.e.Θ , H X , generated by H at 2 p is given by Interestingly, by using the above results and the standard binomial identities, Adler [1] was able to show that

Integrability of the KdV hierarchies
We have ℘ be the Lie algebra of the formal pseudo differential operator, ; , where From Konstant-Symes theorem we can observe the following pertinent propositions for the KdV hierarchies.
) (   ], q is of the form (3) and also    { } ( ) By following the Konstant-Symes theorem and the above-mentioned theorem, this bracket is based from the symplectic structure at any coadjoint orbit in ∝ k ; i.e. we can write  From the results above and together with the flow generated by H X (coadjoint action of G on * k ), definitively we can conclude that the dynamical system related to KdV is a Hamiltonian mechanical system over the coadjoint orbits in ∝ k .

Concluding Remarks
We have shown briefly that the above group theoretical formulation exhibits a coherent connection between Lax representation and mechanics on the coadjoint orbit space of a Lie group action.Clearly this framework is largely based on sterling works done by Adler [1], Lebedev & Manin [15] and Berezin & Perelomov [4], which asserts firstly that the relevant symplectic structure is the symplectic structure of the relevant orbit and secondly the integrability of the KdV hierarchy's dynamical system and its Lax representation are closely related to the Lie algebra splitting.This manifestation naturally explains the integrability of the KdV hierarchies via this group theoretical approach.In fact, we think that the approach is also applicable to other general class of nonlinear partial differential equations (eg.Drinfeld [7], Semenov-Tian-Shansky [20], Terng & Uhlenbeck [23]).We believe (as Palais [17]) that the secret sources of soliton symmetries and as well as the many remarkable properties of soliton equations are closely related to the existence of large and non-obvious groups of symplectic automorphisms.These groups would act on the phase spaces of these Hamiltonian systems (the coadjoint orbit space of a Lie group action) and leave the Hamiltonian function invariant.Furthermore, this framework simply brings forth further understanding with respect to the link between (affine) Lie algebra representations and integrable systems.
The concept of integrability, which is being extended to infinite dimensions or to partial differential equation, is just a portion of a rich structure found in the class of 'completely integrable' equations or the 'soliton' equations residing within the integrable systems (as Batlle [3]).The 'unreasonable effectiveness' of these full-blown structures can be seen in the remarkable recent emergence of a most exotic range of actively researched disciplines: two dimensional quantum field theory, intersection theory on the moduli space of Riemann surfaces, integrable hierarchies, matrix integrals, random surfaces, Gromov-Witten invariants, and many more (eg.Chen et al [5], Dijkgraaf [6], Marshankov [16]).

,
the variational derivative of H (representing the gradient of j H ) and J the skew symmetric matrix consisting of the differential operator of the form d.The integrability condition via Konstant-Symes theorem actually states that, if Ad * -invariant over g * , then i.
respect to x; k = 0,1, … , n and υ = 1,2, …, then the induced Hamiltonian vector field via ω is given formal derivative with respect to j ϕ .In addition, the formal Poisson bracket, basically from ω, {⋅ , ⋅}, is given by the Poisson structure is the Poisson bracket when 0 extracted from the above theorem related to the form of 1 p Θ , tangent space 1 2 p p T Θ , and the orbit's symplectic structure, 2 p ω , are clearly derived from this group theoretical formulation (particularly from Konstant-Symes theorem) formal derivative of P with respect to j ϕ .In Frechet sense, this becomes vector field of the orbit structure 1 p

2 p 1 pΘ
is equivalent to the Gelfand-Dickey formula, i.e. the form of a KdV hierarchy, when . in the ∝ k background.This indirectly validates the integration of the Gelfand-Dickey programme and the applied group theoretical approach.c) Both equations (3) and (4) are localised with values ) x and are dependent on the values at x of k ϕ and derivatives ν ϕx k , , and various variational derivatives H.This fact implies that the right hand sides of equations (3) and (4) are admissible without any assumptions on the compactness of H and , the KdV hierarchy acts as an Hamiltonian system over the coadjoint orbit , the dual space * k of Lie algebra g of formal pseudo differential operator.
of negative type formal pseudo differential).(• , •) is defined via the trace functional and becomes the symmetrical inner product, then

;
m = 0,1, … , then the Hamiltonian equation for the KdV hierarchy is of the form

[
e. the Lax isospectral equation.From the Hamiltonian formalism, we have Poisson bracket

2 p
Konstant-Symes theorem we have the Lax equation (6).As a conclusion, following this perspective naturally shows the complete integrability of the KdV hierarchies.b) From the results above, we examine the Lie geometric structure of the well-known KdV equation: to Gelfand-Dickey programme, which requires that the coadjoint orbit over ∝ k is given with conditions that the integral orbit are invariants, i.e. is related to the time independent Schroedinger operator with potentialϕ , then via equation (3) we have ,

2 p
can be written as (via (5)) are ignored since these are of negative degree.If we assume that there exists an algebraic isomorphism between the 'formal pseudo differential operator space' and 'symbol space', then the KdV(7).
H m is in involution w.r.t.Poisson bracket.