Searcharxiv⌕ Search

SEARCH · Searcharxiv

Results for “solv-int”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 559 records · Page 31Linked to original sources
arXiv1999

Backlund transformations for many-body systems related to KdV

We present Backlund transformations (BTs) with parameter for certain classical integrable n-body systems, namely the many-body generalised Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Backlund parameter.

solv-int↗
arXiv1999

Baxter's Q-operator for the homogeneous XXX spin chain

Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator for the homogeneous XXX model as integral operator in standard representation of SL(2). The connection between Q-operator and local Hamiltonians is discussed. It is shown that operator of Lipatov's duality symmetry arises naturally as leading term of the asymptotic expansion of Q-operator for large values of spectral parameter.

solv-int↗
arXiv1998

Quantum 2+1 evolution model

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.

solv-int↗
arXiv1998

Yangians, finite W-algebras and the Non Linear Schrodinger hierarchy

We show an algebra morphism between Yangians and some finite W-algebras. This correspondence is nicely illustrated in the framework of the Non Linear Schrodinger hierarchy. For such a purpose, we give an explicit realization of the Yangian generators in terms of deformed oscillators.

hep-th↗
arXiv1999

An exact solution method for 1D polynomial Schrödinger equations

Stationary 1D Schrödinger equations with polynomial potentials are reduced to explicit countable closed systems of exact quantization conditions, which are selfconsistent constraints upon the zeros of zeta-regularized spectral determinants, complementing the usual asymptotic (Bohr--Sommerfeld) constraints. (This reduction is currently completed under a certain vanishing condition.) In particular, the symmetric quartic oscillators are admissible systems, and the formalism is tested upon them. Enforcing the exact and asymptotic constraints by suitable iterative schemes, we numerically observe geometric convergence to the correct eigenvalues/functions in some test cases, suggesting that the output of the reduction should define a contractive fixed-point problem (at least in some vicinity of the pure $q^4$ case).

math-ph↗
arXiv1999

Spontaneous polarization of the Kondo problem associated with the higher-spin six-vertex model

We study the multi-channel Kondo model associated with an integrable higher-spin analogue of the anti-ferroelectric six-vertex model, which is constructed by inserting spin 1/2 to spin 1 lines: $... C^3 \otimes C^3 \otimes C^2 \otimes C^3 \otimes C^3 ... $. We formulate the problem in terms of representation theory of quantum affine algebra $U_q(hat{sl}_2)$. We derive an exact formula for the spontaneous staggered polarization for our model, which corresponds to Baxter`s formula for the six-vertex model.

solv-int↗
arXiv1999

Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation

We consider the polynomials $ϕ_n(z)= κ_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrtλ (z+ 1/z)) dz/2 πi z$ on the unit circle in the complex plane. The leading coefficient $κ_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlevé II equation. The leading coefficient and second leading coefficient of $ϕ_n(z)$ can be expressed asymptotically in terms of the Painlevé II function.

solv-int↗
arXiv1998

Duality between integrable Stackel systems

For the Stackel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.

solv-int↗
arXiv1999

Boundary breathers in the sinh-Gordon model

We present an investigation of the boundary breather states of the sinh-Gordon model restricted to a half-line. The classical boundary breathers are presented for a two parameter family of integrable boundary conditions. Restricting to the case of boundary conditions which preserve the ϕ--> -ϕsymmetry of the bulk theory, the energy spectrum of the boundary states is computed in two ways: firstly, by using the bootstrap technique and subsequently, by using a WKB approximation. Requiring that the two descriptions of the spectrum agree with each other allows a determination of the relationship between the boundary parameter, the bulk coupling constant, and the parameter appearing in the reflection factor derived by Ghoshal to describe the scattering of the sinh-Gordon particle from the boundary.

hep-th↗
arXiv1999

Non-additive fusion, Hubbard models and non-locality

In the framework of quantum groups and additive R-matrices, the fusion procedure allows to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here fusion is shown to generalize naturally to non-additive R-matrices, which therefore do not have a quantum group symmetry. This method is then applied to the generalized Hubbard models. Although the resulting integrable models are not as simple as the starting ones, the general structure is that of two spin-(s times s') sl(2) models coupled at the free-fermion point. An important issue is the probable lack of regular points which give local Hamiltonians. This problem is related to the existence of second order zeroes in the unitarity equation, and arises for the XX models of higher spins, the building blocks of the Hubbard models. A possible connection between some Lax operators L and R-matrices is noted.

solv-int↗
arXiv1999

Integrable deformations of oscillator chains from quantum algebras

A family of completely integrable nonlinear deformations of systems of N harmonic oscillators are constructed from the non-standard quantum deformation of the sl(2,R) algebra. Explicit expressions for all the associated integrals of motion are given, and the long-range nature of the interactions introduced by the deformation is shown to be linked to the underlying coalgebra structure. Separability and superintegrability properties of such systems are analysed, and their connection with classical angular momentum chains is used to construct a non-standard integrable deformation of the XXX hyperbolic Gaudin system.

solv-int↗
arXiv1999

Nonlinear Evolution Equations Invariant Under Schroedinger Group in three-dimensional Space-time

A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space $\R^3\times\C$ is presented. Using this result a variety of general second order evolution equations invariant under the corresponding groups are constructed and their physical significance are discussed.

solv-int↗
arXiv1998

New solutions to the Reflection Equation and the projecting method

New integrable boundary conditions for integrable quantum systems can be constructed by tuning of scattering phases due to reflection at a boundary and an adjacent impurity and subsequent projection onto sub-spaces. We illustrate this mechanism by considering a gl(m<n)-impurity attached to an open gl(n)-invariant quantum chain and a Kondo spin S coupled to the supersymmetric t-J model.

cond-mat↗
arXiv2000

The structure of quotients of the Onsager algebra by closed ideals

We study the Onsager algebra from the ideal theoretic point of view. A complete classification of closed ideals and the structure of quotient algebras are obtained. We also discuss the solvable algebra aspect of the Onsager algebra through the use of formal Lie algebras.

math.QA↗
arXiv1999

Nonstandard coproducts and the Izergin-Korepin open spin chain

Corresponding to the Izergin-Korepin (A_2^(2)) R matrix, there are three diagonal solutions (``K matrices'') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K=1 is known to have U_q(o(3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U_q(o(3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry.

hep-th↗