The generalized Yablonskii-Vorob'ev polynomials and their properties

Rational solutions of the generalized second Painlevé hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painlevé hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials.     

Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.     

Infinite hierarchies of exact solutions of the einstein and einstein-maxwell equations for interacting waves and inhomogeneous cosmologies

For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so-called "monodromy transform" approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases.     

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K?hler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K?hler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling–Tod (Eguchi–Hansen) solution. An extended space-time ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? is shown to be foliated by four dimensional hyper-K?hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator. Received: 27 January 2000 / Accepted: 20 March 2000     

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.     

Solutions of the nonlocal(2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy

The(2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are presented.Solutions in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced hierarchies.The advantage of our method is that we start from the known solutions of the unreduced bilinear equations,and obtain solitons and multiple-pole solutions for the variety of classical and nonlocal reductions.Dynamical behaviors of some obtained solutions are illustrated.It is remarkable that for some real nonlocal equations,amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions.     

On the Dispersionless Dym Hierarchy: Dressing Formulation and Twistor Construction

We discuss the dressing formulation and twistor construction for the dispersionless Dym (dDym) hierarchy. In particular, we investigate one-variable and two-variables reductions of the dDym hierarchy to illustrate the formalism. We derive the associated string equations of the reduced dDym hierarchies and obtain their hodograph solutions.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

A Note on the Third Family of N=2 Supersymmetric KdV Hierarchies

Abstract

We propose a hamiltonian formulation of the N = 2 supersymmetric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular, the third family of N = 2 KdV hierarchies is recovered. We also give an easy construction of Wronskian solutions of the KP and KdV type equations.     

Exact solutions for KdV system equations hierarchy

Exact solutions for KdV system equations hierarchy are obtained by using the inverse scattering transform. Exact solutions of isospectral KdV hierarchy, nonisospectral KdV hierarchies and τ$\tau$-equations related to the KdV spectral problem are obtained by reduction. The interaction of two solitons is investigated.     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Two hierarchies of multi-component Kaup-Newell equations and theirs integrable couplings

Two hierarchies of multi-component Kaup-Newell equations are derived from an arbitrary order matrix spectral problem, including positive non-isospectral Kaup-Newell hierarchy and negative non-isospectral Kaup-Newell hierarchy. Moreover, new integrable couplings of the resulting Kaup-Newell soliton hierarchies are constructed by enlarging the associated matrix spectral problem.     

On the singular sector of the Hermitian random matrix model in the large N limit

The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations.     

Two Hierarchies of New Differential-Difference Equations Related to the Darboux Transformations of the Kaup–Newell Hierarchy

Two hierarchies of new nonlinear differential-difference equations with one continuous variable and one discrete variable are constructed from the Darboux transformations of the Kaup–Newell hierarchy of equations. Their integrable properties such as recursion operator, zero-curvature representations, and bi-Hamiltonian structures are studied. In addition, the hierarchy of equations obtained by Wu and Geng is identified with the hierarchy of two-component modified Volterra lattice equations.     

A theoretical analysis of mode-mixing in optical waveguides with nearest-neighbour mode coupling

Properties of the solutions of the coupled equations describing the flow of average mode power in optical waveguides are investigated. All of the important properties and their theoretical foundations are first reviewed for systems with arbitrary coupling and loss coefficients. It is then shown that, for the special class of systems with nearest-neighbour coupling, the solutions are expressible in terms of orthogonal polynomials. In particular, for systems obeying a quasi-uniform loss model and having coupling coefficients with simple dependence on mode number (uniform or linear), the solutions are associated with the classical polynomials. As illustrations of this analysis, the characteristics of optical power flow in three such systems are studied in detail. These include a slab-waveguide with uniform coupling, the corresponding cylindrical waveguide with uniform coupling, and a parabolic-index fibre with linear coupling dependence.     

Hierarchies of multi-component mKP equations and theirs integrable couplings

First, a new multi-component modified Kadomtsev-Petviashvill (mKP) spectral problem is constructed by k-constraint imposed on a general pseudo-differential operator. Then, two hierarchies of multi-component mKP equations are derived, including positive non-isospectral mKP hierarchy and negative non-isospectral mKP hierarchy. Moreover, new integrable couplings of the resulting mKP soliton hierarchies are constructed by enlarging the associated matrix spectral problem.     

Linear integral transformations and hierarchies of integrable nonlinear evolution equations

Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations.Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.     

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.     

On multivortex solutions of the weierstrass representation

The connection between the complex sine-Gordon equation on the plane associated with a Weierstrass-type system and the possibility of constructing several classes of multivortex solutions is discussed in detail. It is shown that the amplitudes of these vortex solutions represented in polar coordinates satisfy the fifth Painlevé equation. We perform the analysis using the known relations for the Painlevé equations and construct explicit formulas in terms of the Umemura polynomials, which are τ functions for rational solutions to the third Painlevé equation. New classes of multivortex solutions to the Weierstrass system are obtained through the use of this proposed procedure.     

Positive and Negative Hierarchies of Integrable Lattice Models Associated with a Hamiltonian Pair

A difference Hamiltonian operator involving two arbitrary constants is presented, and it is used to construct a pair of nondegenerate Hamiltonian operators. The resulting Hamiltonian pair yields two difference hereditary operators, and the associated positive and negative hierarchies of nonlinear integrable lattice models are derived through the bi-Hamiltonian formulation. Moreover, the two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. The use of zero curvature equation leads us to conclude that all resulting integrable lattice models are local and that the integrable lattice models in the positive hierarchy are of polynomial type and the integrable lattice models in the negative hierarchy are of rational type.