Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW ₁ ^B ₊ andw ₁ ^B ₊ of theW-infinity algebrasW _{1 +} andw _{1 +} are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW ₁ ^B ₊ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW ₁ ^B ₊ symmetries are shown to be contracted intow ₁ ^B ₊ symmetries of the dispersionless BKP hierarchy.     

Symmetries of the super KP hierarchy

Symmetries of the super Kadomtsev-Petviashvili hierarchy are studied. A key role is played by a D-module structure, which connects the nonlinear system with the geometry of an infinite-dimensional super Grassmannian manifold. Infinitesimal action of a Lie superalgebra on the super Grassmannian manifold, via this connection, gives rise to symmetries of the nonlinear system.Supported in part by the Grant in Aid for Scientific Research, the Ministry of Education.     

SDiff(2) Toda equation — Hierarchy,Tau function,and symmetries

A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-Kähler version, however now based upon a symplectic structure on a cylinderS ¹×R. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.     

Remarks on Hamiltonian structures of the KP hierarchy and W KP( q ) algebra

It is known that second Hamiltonian structures of the KP hierarchy are parameterized by a continuous complex parameter q and correspond to the W-infinite algebra of W _infKP ^sup(q) . In this Letter, by constructing a Miura map, we first show a generalized decomposition theorem to the second Hamiltonian structures and then establish a relation between those structures which corresponds to values (q+1) and q of the parameter, respectively. This discussion also gives a better understanding to the structures of W _infKP ^sup(q) , its reduced algebras, and their free fields realizations.     

On the constrained KP hierarchy

An explanation for the so-called constrained hierarchies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows one to reduce the KP to the constrained KP.     

Generalization of the Toda chain system to the elliptic curve case

We propose a system of two equations which, when some of its parameters vanish, separates into two equations describing independent one-dimensional Toda chains. The system has its foundation in the discrete transformations of the Landau-Lifshitz equation which is closely connected with elliptic curves. Nontrivial solutions of the system are found in an explicit form.     

Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

Remarks on the integrable systems associated with rank 2 perturbations

In a paper by Moser, a class of completely integrable systems associated with the rank 2 perturbations of a symmetrical matrixA is given in the case that all eigenvalues ofA are distinct. This problem is also discussed by Alder and van Moerbeke in terms of the Kac-Moody algebra. In this Letter, we prove that these systems are also completely integrable in the case thatA has multiple eigenvalues by use of the moment map and the isospectral deformations.     

Solution of the Cauchy problem for a continuous limit of the Toda lattice and its superextension

A supersymmetric equation associated with a continuum limit of the classical superalgebra sl(n/n+1) is constructed. This equation can be considered as a superextension of a continuous limit of t the Toda lattice with fixed end-points or, in other words, as a supersymmetric version of the heavenly equation. A solution of the Cauchy problem for the continuous limit of the Toda lattice and for its superextension is given using some formal reasonings.     

Integrable hierarchy underlying topological Landau-Ginzburg models of D-type

A universal integrable hierarchy underlying topological Landau-Ginzburg models of D-type is presented. Like the dispersionless Toda hierarchy, the new hierarchy has two distinct (positive and negative) sets of flows. Special solutions corresponding to topological Landau-Ginzburg models of D-type are characterized by a Riemann-Hilbert problem, which can be converted into a generalized hodograph transformation. This construction gives an embedding of the finite-dimensional small phase space of these models into the full space of flows of this hierarchy. One of flat coordinates in the small phase space turns out to be identical to the first negative time variable of the hierarchy, whereas the others belong to the positive flows.     

Explicit integration of the full symmetric Toda hierarchy and the sorting property

We give an explicit formula for the solution to the initial-value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridagonal formulae are given for the case of matrices with 2M+1 nonzero diagonals.     

On Segal-Wilson's construction for the τ-functions of the constrained KP hierarchies

In this Letter, we study the constrained KP hierarchies by employing Segal-Wilson's theory on the -functions of the KP hierarchy. We first describe the elements of the Grassmannian which correspond to solutions of the constrained KP hierarchy, and then we show how to construct its rational and soliton solutions from these elements of the Grassmannian.     

On finite-zone solutions of relativistic Toda lattices

The finite-zone solutions of relativistic Toda lattices are investigated using the recurrence relations method. As a result, a nonlinear bundle of relativistic Toda lattices is with corresponding stationary and dynamical systems. New Poisson and Hamiltonian structures are introduced. Then the problem of integrating the obtained canonical systems are reduced to the Jacobi problem of inversion.     

Differential algebras andD-modules in super toda lattice hierarchy

A new super Toda lattice hierarchy is proposed and formulated in the language of differential algebra. AD-module structure is shown to exist behind this nonlinear system and to play the same role as a similarD-module for the super KP hierarchy. From the structure of thisD-module, one can indeed see a direct connection with a set of affine coordinates on an infinite-dimensional super Grassmannian manifold. These affine coordinates are the basic ingredients of an intrinsic construction of functions as well as symmetries.     

Multi-component KdV hierarchy,V-algebra and non-Abelian toda theory

We prove the recently conjectured relation between the 2 × 2-matrix differential operatorL = ² –U and a certain nonlinear and nonlocal Poisson bracket algebra (V-algebra), containing a Virasoro subalgebra, which appeared in the study of a non-Abelian Toda field theory. In particular, we show that thisV-algebra is precisely given by the second Gelfand-Dikii bracket associated withL. The Miura transformation that relates the second to the first Gelfand-Dikii bracket is given. The two Gelfand-Dikii brackets are also obtained from the associated (integro-) differential equation satisfied by fermion bilinears. The asymptotic expansion of the resolvent of (L -) = 0 is studied and its coefficientsR _l yield an infinite sequence of Hamiltonians with mutually vanishing Poisson brackets. We recall how this leads to a matrix KdV hierarchy, which here are flow equations for the three component fieldsT,V ⁺,V ^– ofU. ForV ^± = 0, they reduce to the ordinary KdV hierarchy. The corresponding matrix mKdV equations are also given, as well as the relation to the pseudo-differential operator approach. Most of the results continue to hold ifU is a Hermitiann ×n matrix. Conjectures are made aboutn ×n-matrix,mth-order differential operatorsL and associatedV _(n,m)-algebras.     

The generalized KP hierarchy

The KP hierarchy has been extended to a large set of integrable hierarchies. The main idea is to utilize the additional symmetry of the KP hierarchy, although not commuting among themselves. But we found that their proper combinations do commute with the original KP flows and among themselves, so as to give rise to an enlarged integrable hierarchy, which we call the generalized KP hierarchy.     

The algebra of differential operators on the circle and W KP (q)

Radul has recently introduced a map from the Lie algebra of differential operators on the circle of W _n . In this Letter, we extend this map to W _KP ^(q) , a recently introduced one-parameter deformation of W_KP - the second Hamiltonian structure of the KP hierarchy. We use this to give a short proof that W is the algebra of additional symmetries of the KP equation.     

Dynamical symmetries and Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.