Partial differential equations from integrable vertex models

In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations. This mechanism is illustrated for the $U_q[\widehat{\mathfrak{sl}}(2)]$ invariant six-vertex model and the resulting partial differential equation is studied for particular values of the lattice length.Comment: 19 pages. v2: affiliation and references updated, minor changes, accepted for publication in J. Math. Phy

Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor correction

New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``$XXX$ spin models --- quasi-$XXX$ spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.Comment: 49 pages, LaTe