Schlesinger transforms for the discrete painlevé IV equation

We derive an auto-Bäcklund transformation for the discrete Painlevé IV equation and use it in order to derive Schlesinger transformations for the same equation as well as particular solutions in perfect analogy to the continuous case.     

A study of the discrete Pv equation: Miura transformations and particular solutions

We derive the form of the Miura transformation of the discrete P_v equation and show that it is indeed an auto-Bäcklund transformation, i.e. it relates the discrete P_v to itself. Using this auto-Bäcklund, we obtain the Schlesinger transformations of discrete P_v which relate the solution for one set of the parameters of the equation to that of another set of neighbouring parameters. Finally, we obtain particular solutions of the discrete P_v (i.e. solutions that exist only for some specific values of the parameters). These solutions are of two types: solutions involving the confluent hypergeometric function (on codimension-one submanifold of parameters) and rational solutions (on codimension-two submanifold of parameters).     

On stationary topological solitons in a two-dimensional anisotropic Heisenberg model

We study stationary two-dimensional solitons in an easy-axis Heisenberg magnet with the Hamiltonian density wherei=1, 2,a=1, 2, 3, and (x _i ) is the angle between unit vector s(x _i ) and the easy axis, 0<p<. Stable solitons with a topological chargeQ=1 and localized distributionss ^a (x _i ) withQ=2 are found. The existence of the bound states of two solitons withQ=1 is shown numerically for 0<p<.     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

Invariants of linear combinations of transpositions

It is shown that for a one-parameter set H _N of linear combinations of N(N–1)/2 elementary transpositions {P _jk } (1j at arbitrary natural N3, one can construct a variety {I _m } (3mN) of operators which commute with H _N . Being applied to SU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction.     

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.     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

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.     

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.     

Loop algebras and their relation to the conformal structure of integrable systems

We use the theorem of Kostant, Adler and Symes to construct an infinite set of local polynomials in involution with respect to the Poisson bracket realisation of the Neveu-Schwartz sector of the N=1 superconformal algebra.     

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.     

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.     

Darboux transformations for supersymmetric korteweg-de vries equations

We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin-Radul sKdV. Several Bäcklund transformations are derived for the MRM sKdV and MR sKdV systems.     

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.     

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.     

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.     

Eigensystem and full character formula of theW 1+∞ algebra withc = 1

By using the free field realizations, we analyze the representation theory of theW ₁₊ algebra withc = 1. The eigenvectors for the Cartan subalgebra ofW ₁₊ are parametrized by Young diagrams, and explicitly written down byW ₁₊ generators. Moreover, their eigenvalues and full character formula are also obtained.     

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 the relation between weak and strong invariance of differential equations

Some concepts of Lie algebra cohomology are used to systematize the search for differential equations invariant under a given Lie groupG. In particular, it is shown that if a strongly invariant equation exists, then all weakly invariant equations differ from it only by an arbitrary multiplicative factor. If no strongly invariant equation exists, then cohomology theory can be used to simplify the search for weakly invariant equations.