Noncommutative Korteweg—de Vries and modifiedKorteweg—de Vries hierarchies via recursion methods

摘要

Here, noncommutative hierarchies of nonlinear equations are studied. They repre-sent a generalization to the operator level of corresponding hierarchies of scalarequations, which can be obtained from the operator ones via a suitable projection.A key tool is the application of Backlund transformations to relate differentoperator-valued hierarchies. Indeed, in the case when hierarchies in1 + 1-dimensions are considered, a "Backlund chart" depicts links relating, in par-ticular, the Korteweg-de Vries (KdV) to the modified KdV (mKdV) hierarchy.Notably, analogous links connect the hierarchies of operator equations. The mainresult is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered.Then Backlund transformations are exploited to derive solution formulas in thecase of KdV and mKdV hierarchies. It is remarked that hierarchies of matrixequations, of any dimension, are also incorporated in the present framework.