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.     

Generalized string equations for double Hurwitz numbers

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov–Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of c=1$c=1$ string theory except that the Orlov–Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermion bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so-called Lambert curve emerges in a specialization of its solution. This seems to be another way of deriving the spectral curve of the random matrix approach to Hurwitz numbers.     

The Riccati and Ermakov-Pinney hierarchies

Abstract

The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315–337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201–225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.     

Universal Characters and an Extension of the KP Hierarchy

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we define vertex operators which play roles of raising operators for the universal character. By means of the vertex operators, we obtain a series of non-linear partial differential equations of infinite order, called the UC hierarchy; we regard it as an extension of the KP hierarchy. We investigate also solutions of the UC hierarchy; the totality of the space of solutions forms a direct product of two infinite-dimensional Grassmann manifolds, and its infinitesimal transformations are described in terms of the Lie algebra .     

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators.      

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.     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

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.     

An extended integrable fractional-order KP soliton hierarchy

In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.     

Master symmetries and R-matrices for the Toda lattice

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. These brackets may also be obtained by using r-matrices.     

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.     

Multi-component hierarchies of double bracket equations

A new integrable hierarchy, with equations defined by double brackets of two matrix pseudo-differential operators (Lax pairs), is constructed. Some algebraic properties are demonstrated. It is also shown that each equation is equivalent to a certain gradient flow. A new version of the Zakharov-Shabat type equations is proved. Formal solutions of this hierarchy are constructed using a matrix “double bracket bilinear identity”.     

Microscopic derivation of hydrodynamic balance equations in the presence of an electromagnetic field

Using field-theoretic methods we derive balance equations for a charged fluid in an external electromagnetic field the effects of which are included by minimal coupling. An infinite hierarchy of balance equations for tensor operators is derived. A macroscopic velocity field is introduced by a unitary transformation on the field operators. Suitable statistical averages in the local equilibrium approximation yield macroscopic balance equations. The significance of new terms is discussed.     

Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

The gauge transformation of the q-deformed modified KP hierarchy

In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.     

Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type

A new approach to soliton equations, based on τ functions (or Hirota's dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy). The infinite-dimensional orthogonal group acts on the space of BKP τ-functions. The Sawada-Kotera equation is obtained as a reduction of BKP. Its infinitesimal transformations constitute the Euclidean Lie Algebra A₂⁽²⁾.