Solutions of super Knizhnik–Zamolodchikov equations

Letters in Mathematical Physics - We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated with the Lie superalgebra, of infinite rank, of...     

Bosonization and Integral Representation of Solutions¶of the Knizhnik–Zamolodchikov–Bernard Equations

We construct an integral representation of solutions of the Knizhnik–Zamolodchikov–Bernard equations, using the Wakimoto modules. Received: 5 October 1998 / Accepted: 8 February 1999     

Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the modified qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit boundary quantizations, i.e., boundary solutions to the qYB equation of the form I + tr + t²r₂ +. In the last section we list and give quantizations for all classical r-matrices in sl(3) sl(3).     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.     

Exact Polynomial Solutions of Schrdinger Equation with Various Hyperbolic Potentials

The Schrdinger equation with hyperbolic potential 2V(x) =-V0sinhq(x/d)/cosh6(x/d)(q = 0, 1, 2, 3) is studied by transforming it into the confluent Heun equation. We obtain general symmetric and antisymmetric polynomial solutions of the Schrdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction's nodes decreases with the increase of potential strengths, and the particle tends to the bottom of the potential well correspondingly.     

Multipermutation Solutions of the Yang–Baxter Equation

Set-theoretic solutions of the Yang–Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r : X × X → X × X which satisfies the braid relation.     

Higher derivative corrections in holographic Zamolodchikov–Polchinski theorem

We study higher derivative corrections in holographic dual of Zamolodchikov–Polchinski theorem that states the equivalence between scale invariance and conformal invariance in unitary d-dimensional Poincaré invariant field theories. From the dual holographic perspective, we find that a sufficient condition to show the holographic theorem is the generalized strict null-energy condition of the matter sector in effective (d+1)-dimensional gravitational theory. The same condition has appeared in the holographic dual of the “c-theorem” and our theorem suggests a deep connection between the two, which was manifested in two-dimensional field theoretic proof of the both.     

New Elliptic Solutions of the Yang–Baxter Equation

We consider two finite index endomorphisms \({\rho}\), \({\sigma}\) of any AFD factor M. We characterize the condition for there being a sequence \({\{ u_n\}}\) of unitaries of the factor M with \({\mathrm{Ad}u_n \circ \rho \to \sigma}\). The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of AFD factors, obtained by Kawahiashi–Sutherland–Takesaki and Masuda–Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.     

Cosmological and Quantum Solutions of the Navier–Stokes Equations

Russian Physics Journal - It is shown that the vector Navier–Stokes equation has a variety of quantum solutions, so the scope of this equation is not limited to the field of classical...     

Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation

We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szegő and Baxter. We also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro-geometric solutions.Supported in part by the US National Science Foundation under Grants No. DMS-0200219 and DMS-0405526.The research of the second and third author was supported in part by the Research Council of Norway     

On Improving Accuracy of Finite-Element Solutions of the Effective-Mass Schr¨odinger Equation for Interdiffused Quantum Wells and Quantum Wires

We use the Galerkin approach and the finite-element method to numerically solve the effective-mass Schr¨odinger equation.The accuracy of the solution is explored as it varies with the range of the numerical domain.The model potentials are those of interdiffused semiconductor quantum wells and axially symmetric quantum wires.Also,the model of a linear harmonic oscillator is considered for comparison reasons.It is demonstrated that the absolute error of the electron ground state energy level exhibits a minimum at a certain domain range,which is thus considered to be optimal.This range is found to depend on the number of mesh nodes N approximately as α_0 log_e~(α1)(α_2N),where the values of the constants α_0,α_1,and α_2are determined by fitting the numerical data.And the optimal range is found to be a weak function of the diffusion length.Moreover,it was demonstrated that a domain range adaptation to the optimal value leads to substantial improvement of accuracy of the solution of the Schr¨odinger equation.     

Global Solutions to Gross–Neveu Equation

We prove global existence of solutions to Gross–Neveu equations. Given a local solution, we obtain a uniform L ^∞ bound of the solution by applying local form of charge conservation.     

The Soliton Solutions of A （2＋1）-Dimensional Integrable Equation of Classical Spin System

We present the bilinear equivalence expression of a （2＋1）-dimensional integrable equation of a classical spin system. Based on this, we construct its single-soliton solutions and two-soliton solutions by Hirota＇s bilinear method. Meanwhile we show the evolution and propagation manners of two-solitons of the spin system graphically.     

Generalized Knizhnik–Zamolodchikov equations,off-shell Bethe ansatz and non-skew-symmetric classical r-matrices

Using non-skew-symmetric sl(2)$\mathit{sl}(2)$-valued classical r-matrices with spectral parameters we construct a generalization of the Knizhnik–Zamolodchikov (KZ) equations. We obtain integral solutions of the constructed KZ-type equations using the “off-shell” Bethe ansatz technique. We consider several examples of the obtained generalized KZ equations and their integral solutions that correspond to the “K-twisted” non-skew-symmetric classical r-matrices parametrized by arbitrary complex parameter ξ.     

On Triple Product and Rational Solutions of Yang–Baxter Equation

The Yang–Baxter equation is reinvestigated in the framework of triple system. By requiring the rational R matrix of the Yang–Baxter equation satisfying the generalized Filippov condition, we derive a relation with respect to the rational R matrix. Moreover the case of the super Yang–Baxter equation is also investigated.     

Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations

We show that given a finite-dimensional real Lie algebra acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations. Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche Scientifique PROTARS III.     

Exact Solutions of the Klein–Gordon Equation with Hylleraas Potential

We present the exact solution of the Klein–Gordon with Hylleraas Potential using the Nikiforov–Uvarov method. We obtain explicitly the bound state energy eigenvalues and the corresponding eigen function for s-wave. The wave functions obtained are expressed in terms of Jacobi polynomials.