THE n × n KDV FLOWS

We introduce a new integrable system hierarchy which is a restriction of the AKNS n×n hierarchy coming from an unusual splitting of the loop algebra. This splitting comes from an automorphism of the loop algebra instead of an automorphism of SL(n,C). It is known that the 2 × 2 KdV is the standard KdV hierarchy. We construct a gauge equivalence of this n× n KdV hierarchy with the Gelfand-Dickey hierarchy. We show that the tau function of Wilson has many of the same properties of the usual tau function. Introduction While an exact definition of an integrable system is illusive, mathematicians generally agree that the hallmarks of the the theory are an infinite number of commuting flows, formal direct and inverse scattering theory, symplectic structures and Bäcklund transformations. More recently, because of the connections with quantum cohomology, one might add tau functions and actions of Virasoro algebra. From our viewpoint we take the basic model formulated by Ablowitz, Kaul, Newell and Segur (AKNS), the n× n model generalizing the 2 × 2 non-linear Schröedinger hierarchy, as the basic construction. The various ingredients in this model are fairly well-known, and a definition of a tau function we use is due to Wilson. There are a number of possible restrictions of this theory in which the general hierarchies described by AKNS restrict to more specialized hierarchies.These include the Kuperschmidt-Wilson flows and the modified KdV hierarchy defined by Drinfeld and Sokolov. There are also several models in which matrices are replaced by more specialized Lie Algebras, and the formalism carries over. However, this same formalism does not carry over directly to the Gelfand-Dickey (GD) hierarchies. In this paper we construct a Lie subalgebra of the algebra of power series with values in SL(n,C) and a splitting to construct an apparently new hierarchy, which we call the n × n KdV hierarchy. The KuperschmidtWilson and mKdV flows turn out to be equivalent; likewise under a gauge transformation these n× n flows are equivalent in the sense of DrinfeldSokolov to the GD hierarchy. The advantage is that the same recipe for scattering, inverse scattering, Bäcklund transformations, symplectic structures and tau Research supported in part by NSF Grant DMS-0529756. Research supported in part by the Sid Richardson Regents’ Chair Funds, University of Texas system and NSF grant DMS-0305505 . 1 2 CHUU-LIAN TERNG AND KAREN UHLENBECK functions holds for AKNS and n× n KdV. Wilson’s tau function generates the n 1 variables. In a later paper we show that the Virasoro algebra which arises naturally as a symmetry of the formal scattering data of AKNS, is the Virasoro symmetry of GD flows. The plan of the paper is as follows: Section 1 reviews a general construction of a hierarchy of commuting flows from a Lie algebra splitting and factorization in the big cell. We also describe the basic examples of the theory, which include the AKNS hierarchy, the Kuperschmidt-Wilson hierarchy, and the n×n mKdV hierarchy. In section 2 we give the restriction which is used to obtain the n×n KdV and prove that the splitting is consistent with the restriction. In section 3 we show the equivalence with the GD flows. In section 4 we review Wilson’s constructions of a μ function on the big cell of the loop group via the central extension and apply it to construct tau functions for n × n KdV flows. This construction is similar in spirit to the construction of Segal and Wilson using determinant bundles (which give a central extension for groups given by operators in a Hilbert space). We also prove in this section that the n-1 dependent variables in the n×n KdV flows are computable from the second derivatives of ln(τ). 1. A general method of constructing soliton hierarchy We review the method that generates a hierarchy of commuting flows from a splitting of a Lie algebra. Definition 1.1. Let L be a formal Lie group, L its Lie algebra, and L± subgroups of L with Lie subalgebra L±. The pair (L+,L−) is called a splitting of L if (1) L+ ∩ L− = 0, (2) L = L+ + L− as linear subspaces, (3) the multiplication maps L+×L− → L and L−×L+ → L defined by (f+, f−) 7→ f+f− and (f−, f+) 7→ f−f+ are one to one and the images are open and dense. We call the open set O = (L+L−) ∩ (L−L+) the big cell of L. Definition 1.2. A sequence J = {Ji | i ≥ 1, integer} in L+ is called a vacuum sequence of the splitting (L+,L−) if (1) [Ji, Jj ] = 0 for all i, j and J generates a maximal abelian subalgebra of L+, (2) the Ji’s are linearly independent in L+, (3) Jj lies in the subalgebra generated by J1 in the universal enveloping algebra. Let π± denote the projection of L to L± with respect to the sum L = L+ + L−. Set Y to be the subspace Y = π+([J1,L−]). THE n× n KDV FLOWS 3 In the examples we give L is a subalgebra of the algebra of loops in gl(n), and J1 = aλ + b and π+([J1,L−]) = [a,L−1], where L−1 = {ξ ∈ gl(n) | ξλ −1 ∈ L−}. Assume further that given u ∈ C (R, Y ), there is a unique M ∈ L− such that M(∂x − J1)M −1 = ∂x − (J1 + u). SuchM is usually called a Baker function or bare wave function of ∂x−(J1+ u), and f = M(0) ∈ L− the scattering data for the operator ∂x − (J1 + u). Then there is a standard method of generating a hierarchy of commuting flows on C(R, Y ). The j-th flow associated to (L+,L−) and the vacuum sequence J is defined by the Lax pair, [∂x − (J1 + u), ∂tj − π+(MJjM )] = 0, i.e., ∂u ∂tj = [∂x − (J1 + u), π+(MJjM )]. So if u is a solution of the j-th flow, then there is a unique solution E(x, tj) such that