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A new approach to deformation equations of noncommutative KP hierarchies


Müller-Hoissen,  Folkert
Max Planck Institute for Dynamics and Self-Organization, Max Planck Society;

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Dimakis, A., & Müller-Hoissen, F. (2007). A new approach to deformation equations of noncommutative KP hierarchies. Journal of Physics A: Mathematical and Theoretical, 40, 7573-7596. doi:10.1088/1751-8113/40/27/010.

Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly nonassociative' (WNA) algebra structure, from which we can conclude, referring to previous work, that any solution of the Riccati system also solves the potential KP hierarchy (in the corresponding matrix algebra). We then turn to the case where the components of the matrices are multiplied using a (generalized) star product. Associated with the deformation parameters, there are additional symmetries (flow equations) which enlarge the respective KP hierarchy. They have a compact formulation in terms of the WNA structure. We also present a formulation of the KP hierarchy equations themselves as deformation flow equations.