Bidifferential calculus, matrix SIT and sine-Gordon equations

We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework. An infinite family of exact solutions is then obtained by application of a general result that generates exact solutions from solutions of a linear system of arbitrary matrix size. A side result is a solution formula for the sine-Gordon equation.Comment: 7 pages, 2 figures, 19th International Colloquium on Integrable Systems and Quantum Symmetries (ISQS19), Prague, Czech Republic, June 201

Exact vortex solutions of the complex sine-Gordon theory on the plane

We construct explicit multivortex solutions for the first and second complex sine-Gordon equations. The constructed solutions are expressible in terms of the modified Bessel and rational functions, respectively. The vorticity-raising and lowering Backlund transformations are interpreted as the Schlesinger transformations of the fifth Painleve equation.Comment: 10 pages, 1 figur

Bicomplexes and Integrable Models

We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are presented, including the nonlinear Schrodinger and sine-Gordon equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st

Darboux Transformations for SUSY Integrable Systems

Several types of Darboux transformations for supersymmetric integrable systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations are considered. We also present solutions such as supersolitons and superkinks.Comment: 13 pages. LaTeX209 with LamuPhys and EPSF packages, 3 figures. Contribution to the proceedings of the "Integrable Models and Supersymmetry" meeting held at Chicago on July'9

Note on Backlund transformations

The method of obtaining Backlund transformations proposed by Chern and Tenenblat (1986) fits completely the approach of obtaining Backlund transformations by prolongation techniques. For KdV, MKdV and sine-Gordon equations the only difference consists in the application of a non-linear representation of the prolongation algebra other than the usual one. This representation can be obtained by a coordinate transformation of the prolongation variabl

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.Comment: 19 page