Authors:Hirose S; Inoguchi J, Kajiwara K, et al. Abstract: AbstractThe local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the non-linear Schrödinger equation. In this article, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete non-linear Schrödinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $\tau$ functions of the two-component KP hierarchy.Communicated by: Alexander I. Bobenko PubDate: Sun, 09 Jun 2019 00:00:00 GMT DOI: 10.1093/integr/xyz003 Issue No:Vol. 4, No. 1 (2019)
Authors:Vieira R; Lima-Santos A. Abstract: AbstractWe study in this article the algebraic Bethe Ansatz from a combinatorial perspective. The commutation relations are represented by combinatorial diagrams so that the action of the transfer matrix over the $n$th excited state can be found through the analysis of some combinatorial trees. This allows us to compute the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations, in a straightforward way. As examples, we consider the six-vertex model with both periodic and non-periodic boundary conditions. In parallel, we also present a comprehensive introduction to the usual algebraic Bethe Ansatz (for both the periodic and the non-periodic cases).Communicated by: Atsuo Kuniba PubDate: Mon, 20 May 2019 00:00:00 GMT DOI: 10.1093/integr/xyz002 Issue No:Vol. 4, No. 1 (2019)
Authors:Ferrari A; Mason L. Abstract: AbstractWe study a twistor correspondence based on the Joukowski map reduced from one for stationary-axisymmetric self-dual Yang–Mills and adapt it to the Painlevé III equation. A natural condition on the geometry (axis-simplicity) leads to solutions that are meromorphic at the fixed singularity at the origin. We show that it also implies a quantization condition for the parameter in the equation. From the point of view of generalized monodromy data, the condition is equivalent to triviality of the Stokes matrices and half-integral exponents of formal monodromy. We obtain canonically defined representations in terms of a Birkhoff factorization whose entries are related to the data at the origin and the Painlevé constants.Communicated by: Dr. Maciej Dunajski PubDate: Mon, 04 Mar 2019 00:00:00 GMT DOI: 10.1093/integr/xyz001 Issue No:Vol. 4, No. 1 (2019)
Authors:Vermeeren M. Abstract: AbstractA pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of integrability that has mainly been studied in the context of multidimensionally consistent lattice equations. It unifies multidimensional consistency with the variational character of the equations. An analogous continuous structure exists for integrable hierarchies of differential equations. We present a continuum limit procedure for pluri-Lagrangian systems. In this procedure, the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives. Then, we seek differential equations whose solutions interpolate the embedded discrete solutions. The continuous systems found this way are hierarchies of differential equations. We show that this continuum limit can also be applied to the corresponding pluri-Lagrangian structures. We apply our method to the discrete Toda lattice and to equations H1 and Q1$_{\delta = 0}$ from the ABS list. PubDate: Mon, 18 Feb 2019 00:00:00 GMT DOI: 10.1093/integr/xyy020 Issue No:Vol. 4, No. 1 (2019)
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