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The Estabrook-Wahlquist prolongation method is applied to the (compact and noncompact) continuous isotropic Heisenberg model in 1 + 1 dimensions. Using a special realization (an algebra of the Kac-Moody type) of the arising incomplete prolongation Lie algebra, a whole family of nonlinear field equations containing the original Heisenberg system is generated.
Recent results are surveyed pertaining to the complete integrability of some novel n-particle models in dimension one. These models generalize the Calogero-Moser systems related to classical root systems. Quantization leads to difference operators instead of differential operators.
By introducing generalized Backlund Transformations depending on arbitrary functions, wave and localized soliton solutions of the Davey- Stewartson equations are generated. Moreover explicit soliton solutions of the Hamiltonian DSI and DSIII equations are obtained.
We first review the properties of the conventional $τ$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $τ$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.
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 and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. 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.
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation of variables. Though much simpler than two others, it has important relations to the quantum integrability. Separation of variables is constructed for the $SL(3)$ magnetic chain --- an example of integrable model associated to a nonhyperelliptic algebraic curve.
The algebra of monodromy matrices for sl(n) trigonometric R-matrices is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product of L-operators. Cocommutativity of representations is discussed. A special class of representations - factorizable representations is introduced and intertwiners for cocommuting factorizable representations are written through the Boltzmann weights of the sl(n) chiral Potts model. Let us consider an algebra generated by noncommutative entries of the matrix $T(u)$ satisfying the famous bilinear relation originated from the quantum inverse scattering method: $R(\la-μ)T(\la)T(μ)=T(μ)T(\la) R(\la-μ)$ where $R(\la)$ is R-matrix. For historical reasons this algebra is called the algebra of monodromy matrices. If $\g$ is a simple finite-dimensional Lie algebra and $R(\la)$ is $\g$-invariant R-matrix the algebra of monodromy matrices after a proper specialization gives the Yangian $Y(\g)$ introduced by Drinfeld. If $R(\la)$ is corresponding trigonometric R-matrix this algebra is closely connected with $U_q(\g)$ and $U_q(\hat\g)$ at zero level. If $R(\la)$ is $sl(2)$ elliptic R-matrix the algebra of monodromy matrices gives rise to Sklyanin's algebra. In this paper we shall study algebras of monodromy matrices for $sl(n)$ trigonometric R-matrices at roots of 1. Finite-dimensional cyclic irreducible polynomial representations and their intertwiners are discussed.
The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K$_2$, which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of Bäcklund transformations to evolve new solutions.
The focus of this paper is on the probability, $E_β(0;J)$, that a set $J$ consisting of a finite union of intervals contains no eigenvalues for the finite $N$ Gaussian Orthogonal ($β=1$) and Gaussian Symplectic ($β=4$) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary ($β=2$) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function.
For the two-dimensional Schrödinger equation $$ [- Δ+v(x)]ψ=Eψ,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed positive energy with a fast decaying at infinity potential $v(x)$ dispersion relations on the scattering data are given.Under "small norm" assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization of scattering data for the potentials from the Schwartz class $S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zero scattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parameterized by a function of one variable) of two-dimensional spherically-symmetric real potentials from the Schwartz class $S$ transparent at a given energy. For the two-dimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing at infinity, potentials transparent at a fixed energy. For any dimension greater or equal 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdV-type equations in dimension 2+1 related with the scattering problem $(*)$ (the Novikov-Veselov equations) do not preserve, in general, these dispersion relations starting from the second one. As a corollary these equations do not preserve, in general , the decay rate faster then $|x|^{-3}$ for initial data from the Schwartz class.
We study a 12-parameter stochastic process involving particles with two-site interaction and hard-core repulsion on a $d$-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values $n_{\vec{x}}=0,1$. We show that on a 10-parameter submanifold the $k$-point equal-time correlation functions $\exval{n_{\vec{x}_1} \cdots n_{\vec{x}_k}}$ satisfy linear differential- difference equations involving no higher correlators. In particular, the average density $\exval{n_{\vec{x}}} $ satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian $H$, the model becomes equivalent to a lattice model in thermal equilibrium in $d+1$ dimensions. We show that the spectrum of $H$ is identical to the spectrum of the quantum Hamiltonian of a $d$-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system.
We prove polynomial identities for the N=1 superconformal model SM(2,4ν) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4ν) and M(2ν-1,4ν). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eight-fold symmetric phase. The solution is obtained using Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling, which were found by M. Widom.
The main purpose of the paper is to demonstrate that condition of invariance with respect to the Legendre transformations allows effectively isolate the class of integrable difference equations on the triangular lattice, which can be considered as discrete analogues of relativistic Toda type lattices. Some of obtained equations are new, up to the author knowledge. As an example, one of them is studied in more details, in particular, its higher continuous symmetries and zero curvature representation are found.
A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schrödinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.
These notes provide an introduction to the theory of random matrices. The central quantity studied is $τ(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/π} {\sinπ(x-y)\over x-y} χ_I(y)$. Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $χ_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in $I$ is equal to $τ(a)$. Also $τ(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$'s are the independent variables) that were first derived by Jimbo, Miwa, M{ô}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{é} V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$.
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious damping or excitation. Methods that do preserve all the Hamiltonian properties, i.e., for which the time-forward map is symplectic, are called symplectic integration algorithms or SIAs. Although such integrators are symplectic in theory, they are not symplectic if implemented using finite-precision arithmetic. This paper explains how to eliminate this problem by using ``Lattice SIAs'' and shows that these methods yield significant advantages when the computational error is dominated by roundoff. Using a lattice SIA is equivalent to evolving the exact solution of a problem with a Hamiltonian that is slightly different from the original. Lattice methods are useful for studies of the long-term evolution of Hamiltonian dynamical systems.
By relating the two-dimensional U(N) Principal Chiral Model to a simple linear system we obtain a free-field parametrisation of solutions. Obvious symmetry transformations on the free-field data give symmetries of the model. In this way all known `hidden symmetries' and Bäcklund transformations, as well as a host of new symmetries, arise.