A Completeness Study on Certain $2\times 2$ Lax Pairs Including Zero Terms

Abstract

We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times 2$ 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.

References in corpus (2)

• Classification of 3D consistent quad-equations
• Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation

Cited by in corpus (5)

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• Simple identification of fake Lax pairs
• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• Algebraic entropy for systems of quad equations
• Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation