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Non-classical symmetry reduction: example of the Boussinesq equation
A symmetry of an equation will leave the set of all solutions invariant. A 'conditional symmetry' will leave only a subset of solutions, defined by some differential condition, invariant. The authorsExpand
On higher-order Riccati equations as Bäcklund transformations
In this short paper we illustrate by few examples the special role played by higher-order Riccati equations in the construction of Backlund transformations for integrable systems.
Conditions for the existence of higher symmetries of evolutionary equations on the lattice
In this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential-difference equations depending only on nearest neighboringExpand
The generalized symmetry method for discrete equations
The generalized symmetry method is applied to a class of completely discrete equations including the Adler–Bobenko–Suris list. Assuming the existence of a generalized symmetry, we derive a fewExpand
Bäcklund transformations and nonlinear differential difference equations.
  • D. Levi, R. Benguria
  • Mathematics, Medicine
  • Proceedings of the National Academy of Sciences…
  • 1 September 1980
It is shown that any Bäcklund transformation of a nonlinear differential equation integrable by the multichannel Schrödinger eigenvalue problem can be written in the form V(x) = U'V - VU. This allowsExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Nonlinear differential difference equations as Backlund transformations
Shows that the best known nonlinear differential difference equations associated with the discrete Schrodinger spectral problem and also with the discrete Zakharov-Shabat spectral problem can beExpand
Painlevé transcendents : their asymptotics and physical applications
I: Asymptotics of Painleve Transcendents, Connection Formulas, New Mathematical Features.- Integral Equations and Connection Formulae for the Painleve Equations.- Continuous and Discrete PainleveExpand
Umbral calculus, difference equations and the discrete Schrödinger equation
In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. TheExpand
Symmetries and conditional symmetries of differential difference equations
Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined withExpand