Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations

Abstract

The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without specific weight functions, respectively. Under stationary reductions, matrix discrete Painlevé I and matrix asymmetric discrete Painlevé I equations are derived separately not only from the noncommutative nonisospectral lattices themselves, but also from their Lax pairs. The rationality of the stationary reduction has been justified in the sense that quasideterminant solutions are provided for the corresponding matrix discrete Painlevé equations.

After we posted our paper arXiv:submit/5730163 on arXiv, we got an email from Xiaolu Yue. She told us that they also worked on this topic and sent us their results. Since we have achieved the results at the same time individually, we decided to collaborate after discussion. We'd like to withdraw this paper if possible and submit it after revision. I'm sorry for the inconvenience. Thank you