Explicit Bäcklund transformations for multifield Schrödinger equations. Jordan generalizations of the Toda chain

Bäcklund transformations for multifield analogs of the nonlinear Schrödinger equation that correspond to unital Jordan algebras are found. These Bäcklund transformations are explicit invertible autotransformations and as a result they are very convenient for the construction of exact solutions. It is established that to these Bäcklund transformations there correspond integrable multifield discrete—differential equations that generalize the infinite Toda chain. A simple construction is given by means of which multifield analogs of the infinite Toda chain can be constructed from every unital Jordan algebra. New examples of such chains are given.Institute of Mathematics, Ural Scientific Center, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 2, pp. 207–219, February, 1994.     

Symmetry Approach to the Integrability Problem

We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are central in this approach. Moreover, we consider equations of the Painlevé type, master symmetries, and the problem of integrability criteria for (2+1)-dimensional models. We present the list of canonical forms for (1+1)-dimensional integrable systems. We elaborate the effective tests for integrability and the algorithms for reduction to the canonical form.     

Relativistic Toda Chains and Schlesinger Transformations

We construct the auto-Schlesinger transformations for all equations in the known list of integrable relativistic Toda chains. Our construction is essentially based on the equations being Lagrangian and on a standard transition to their Hamiltonian form; in this case, the transition is described by the changes of variables that are invertible but not pointwise. We discuss two examples of another type that has similar properties; these are also integrable Lagrangian equations allowing the Schlesinger transformation.     

Invertible changes of variables generated by Bäcklund transformations

Institute of Mathematics, Bashkir Scientific Center, Urals Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 3, pp. 368–375, December, 1990.     

Near-field intensity correlations in semicontinuous metal-dielectric films

Spatial intensity correlation functions are obtained from near-field scanning optical microscope measurements of semicontinuous metal-dielectric films. The concentration of metal particles on a dielectric surface is varied over a wide range to control the scattering strength. At low and high metal coverages where scattering is weak, the intensity correlation functions exhibit oscillations in the direction of incident light due to excitation of propagating surface waves. In the intermediate regime of metal concentration, the oscillatory behavior is replaced by a monotonic decay as a result of strong scattering and anomalous absorption. Significant differences in the near-field intensity correlations between metallic and dielectric random systems are demonstrated.     

Synthesis of (2S, 3S, 4E, 6E)-3-O-β-D-galactopyranosyl-2-N-hexadecanoyl-8-oxo-4,6-sphingadienine pentaacetate

Institute of Organic Chemistry, Ural Branch, Russian Academy of Sciences, 450054 Ufa. Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 5, pp. 1226–1227, May, 1992.     

Disorder-immune coupled resonator optical waveguide

We demonstrate that a photonic lattice with short- and long-range harmonic modulations of the refractive index facilitates formation of flat photonic bands and leads to slow propagation of light. The system can be considered a coupled-resonator optical waveguide (CROW): photonic bands with abnormally small dispersion are created due to the interaction of long-lived states in the cavity regions via weak coupling across tunneling barriers. Unlike previous CROW implementations, the proposed structures can be fabricated with interference photolithography (holography), sidestepping the issue of resonator-to-resonator fluctuation of the system parameters. The proposed holography-based approach enables fabrication of arrays with a large number of coupled optical resonators, which is necessary for practical applications.     

An Unusual Series of Autonomous Discrete Integrable Equations on a Square Lattice

Theoretical and Mathematical Physics - We present an infinite series of autonomous discrete equations on a square lattice with hierarchies of autonomous generalized symmetries and conservation laws...