### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 053, 14 pages      arXiv:1704.00043      https://doi.org/10.3842/SIGMA.2017.053
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Symmetries of the Hirota Difference Equation

Andrei K. Pogrebkov ab
a) Steklov Mathematical Institute of Russian Academy of Science, Moscow, Russia
b) National Research University Higher School of Economics, Moscow, Russia

Received March 31, 2017, in final form July 02, 2017; Published online July 07, 2017

Abstract
Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as ''times'' of the nonlinear integrable partial differential-difference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented.

Key words: Hirota difference equation; symmetries; integrable differential-difference and differential equations; additional symmetries.

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