Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.     

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.     

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.     

A tri-Hamiltonian formulation of the full Kostant-Toda lattice

We define Poisson structures which lead to a tri-Hamiltonian formulation for the full Kostant-Toda lattice. In addition, a hierarchy of vector fields called master symmetries are constructed and they are used to generate the nonlinear Poisson brackets and other invariants. Various deformation relations are investigated. The results are analogous to results for the finite nonperiodic Toda lattice.     

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.     

Integrability of linear and nonlinear evolution equations and the associated nonlinear fourier transforms

The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.     

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.     

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.     

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.     

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 the moduli space of SU(2) monopoles and Toda flow

Rational functions of degreek which parametrize the moduli space of SU(2)k-monopoles, can be regarded as transfer functions for certain linear dynamical systems. It is shown that a time shift for linear systems induces a finite nonperiodic complexk×k Toda flow on the parameter space of generic SU(2)k-monopoles. Thus, there exists an integrable flow of the Toda type over the moduli space of SU(2) monopoles.This research was supported in part by National Science Foundation under Grant ECS-8718026.     

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.     

On the euler equation: Bi-Hamiltonian structure and integrals in involution

We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the physical phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.     

Hirota polynomials for the KP and BKP hierarchies

Using symmetric function techniques, we derive closed-form expressions for the Hirota polynomials for thepth modified KP and BKP hierarchies in terms of Schur and SchurQ-polynomials, respectively. The Hirota polynomials for the BKP hierarchy can also be expressed as Pfaffians while those for thepth modified KP hierarchies can, under certain conditions, be expressed as determinants.     

Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.     

Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra

Introducing the notion of an admissible graded Lie subalgebra A of the Nijenhui-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex C _A of the Chevalley-Eilenberg complex (C ⁰ M), extends in a cononical way as a graded cohomology class of weight — 1 of A. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight — 1 of the Schouten graded Lie algebra of M. This allows us to construct explicitly all 1-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.     

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.     

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 interaction of solitons for a class of integrable systems in the spacetime R n+1

A class of integrable systems of nonlinear partial differential equations in the spacetime R ⁿ⁺¹ is introduced. Single and multi-soliton solutions are constructed by using the Darboux matrix method. It is proved that as t±, a k multi-soluton solution splits asymptotically into k single solitons. Moreover, the interaction between solitons is elastic if we consider their magnitudes only.The work is C. H. Gu supported by the Chinese National Program for fundamental research nonlinear science.