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Register a new userq-Discrete Toda Molecule Equation
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A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Backlund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed.
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In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: - Time discretization based on the notion of Backlund transformation; - Symplectic realizations of multi-Hamiltonian structures; - Interrelations between discrete 1D systems and lattice 2D systems; - Multi-dimensional consistency as integrability of discrete systems; - Interrelations between integrable systems of quad-equations and integrable systems of Laplace type; - Pluri-Lagrangian structure as integrability of discrete variational systems. All these concepts are illustrated by the discrete time Toda lattices and their relativistic analogs.
Bilinear structure for the discrete Painleve I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painleve I equation is naturally extended to a discrete coupled system. Corresponding matrix model is also mentioned.
An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.
We show how to lift solutions of Euclidean Einstein-Maxwell equations with non-zero cosmological constant to solutions of eleven-dimensional supergravity theory with non-zero fluxes. This yields a class of 11D metrics given in terms of solutions to $SU(infty)$ Toda equation. We give one example of a regular solution and analyse its supersymmetry. We also analyse the integrability conditions of the Killing spinor equations of N=2 minimal gauged supergravity in four Euclidean dimensions. We obtain necessary conditions for the existence of additional Killing spinors, corresponding to enhancement of supersymmetry. If the Weyl tensor is anti-self-dual then the supersymmetric metrics satisfying these conditions are given by separable solutions to the $SU(infty)$ Toda equation. Otherwise they are ambi-Kahler and are conformally equivalent to Kahler metrics of Calabi type or to product metrics on two Riemann surfaces.
We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein-Gordon models. The multi-kinks are constructed using Lins method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an $m$-structure multi-kink, there will be $m$ eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.
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