Multi-component matrix KP hierarchies as symmetry-enhanced scalar KP hierarchies and their Darboux-Bäcklund solutions (Q2781677)

The authors of this interesting paper show that, any multi-component matrix KP hierarchy is equivalent to the standard one-component (scalar) KP hierarchy endowed with a special infinite set of Abelian additional symmetries, generated by squared eigenfunction potentials. Given an ordinary one-component KP hierarchy (KP\(_1\)), one can always construct a \(N-\)component matrix KP hierarchy, embedding the original one, in the following way. There exist \(N-1\) infinite sets of adjoint eigenfunctions of the initial KP\(_1\). It is constructed by KP\(_1\) an infinite-dimensional Abelian algebra of additional (``ghost'') symmetries. The one-component KP hierarchy equipped with such additional symmetry structure turns out to be equivalent to the standard \(N\)-component matrix KP hierarchy. By a subset of the ``ghost'' symmetry generating adjoint eigenfunctions of KP\(_1\), new tau-functions which satisfy Hirota bilinear identities of \(N\)-component matrix KP hierarchy are defined. The authors employ a special version of the known DB transformation technique within the ordinary scalar KP hierarchy in the Sato formulation for a symmetric derivation of explicit multiple-Wronskian tau-function solutions of all multi-component matrix KP hierarchies.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].