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

SIGMA 9 (2013), 068, 39 pages      arXiv:1311.2675      https://doi.org/10.3842/SIGMA.2013.068

### Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations

Kenichi Kondo
5-13-12-207 Matsubara, Setagaya-ku, Tokyo 156-0043, Japan

Received January 08, 2013, in final form October 31, 2013; Published online November 12, 2013

Abstract
Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.

Key words: ultradiscrete sine-Gordon equation; symmetrized max-plus algebra; noncommutative discrete sine-Gordon equation; noncommutative ultradiscrete sine-Gordon equation.

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