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

SIGMA 11 (2015), 087, 22 pages      arXiv:1505.02833      https://doi.org/10.3842/SIGMA.2015.087

### Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity

Alex Kasman
Department of Mathematics, College of Charleston, USA

Received May 13, 2015, in final form October 28, 2015; Published online November 01, 2015

Abstract
A wave function of the $N$-component KP Hierarchy with continuous flows determined by an invertible matrix $H$ is constructed from the choice of an $MN$-dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in $x$ having eigenvalues that are matrix functions of the spectral parameter $z$. If the space of distributions is invariant under left multiplication by $H$, then a matrix coefficient differential-translation operator in $z$ is shown to share this eigenfunction and have an eigenvalue that is a matrix function of $x$. This paper not only generates new examples of bispectral operators, it also explores the consequences of non-commutativity for techniques and objects used in previous investigations.

Key words: bispectrality; multi-component KP hierarchy; Darboux transformations; non-commutative solitons.

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