Singular-boundary reductions of type-Q ABS equations

We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2 and Q1). Singular boundary problems for these equations are systematised in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimensional consistency and uses new insights into these equations related to the singularity structure in multidimensions and the identification of an associated tau-function. The obtained special solutions can be identified with open boundary problems of the associated Toda-type systems, and have interesting application to the construction of periodic solutions.Comment: 24 pages, 5 figure

Numerical Methods for Unconstrained Minimization : An Integrated Computational Environment

This study discusses the methods, algorithms and implementation techniques involved in the computational solution of unconstrained minimization problem : min x ∈ Rn f : Rn −! R Where Rn denotes the n-dimensional Euclidean space. The main goal in this study was to implement an easy-to-use software package running in personal computers for unconstrained minimization of multidimensional functions. This software package includes C language implementations of six minimization methods (listed below), an user-interface for entering each minimization problem, and an interface to a general software system called MathematicaTM which is used for plotting the problem function and the minimization route. The following minimization methods are discussed here : - Parabolic interpolation in one-dimension - Downhill simplex method in multidimensions - Direction set method in multidimensions - Variable metric method in multidimensions - Conjugate gradients method in multidimensions - Modified steepest descent method in multidimensions The first part of this study discusses the theoretical background of the minimization algorithms to be implemented in the software package. The second part introduces the overall design of the minimization software and in greater detail describes the individual software modules, which, as a whole, implement the software package. The third part introduces the techniques for testing the minimization algorithms, describes the set of test problems, and discusses the test results

Singularities of Type-Q ABS Equations

The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.Comment: contribution to the SIDE-9 special issue of SIGM