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

SIGMA 7 (2011), 089, 12 pages      arXiv:1104.0084      https://doi.org/10.3842/SIGMA.2011.089
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

A Completeness Study on Certain 2×2 Lax Pairs Including Zero Terms

Mike C. Hay
Institute of Mathematics for Industry, Kyushu University, Ito Campus, 744 motooka, nishi-ku, Fukuoka, 819-0395, Japan

Received April 09, 2011, in final form September 04, 2011; Published online September 14, 2011

Abstract
We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all 2×2 Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems that can be associated with them. Here we allow some of the terms within the Lax matrices to be zero. We cover all possible Lax pairs of this type and find a new third order equation that can be reduced to special cases of the non-autonomous lattice KdV and lattice modified KdV equations among others.

Key words: discrete integrable systems; Lax pairs.

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