Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection.A key tool is the application of Bäcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1 + 1-dimensions are considered, a "Bäcklund chart" depicts links relating, in particular, the Korteweg-de Vries ͑KdV͒ to the modified KdV ͑mKdV͒ hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinitedimensional parameter. To start with, the potential KdV hierarchy is considered. Then Bäcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.

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Three classes of soliton systems associated with scalar Lax operators are considered. They represent, in turn, the 2 q 1-dimensional hierarchies of the KP equation, the modified KP equation and the Dym equation. They are related via gauge transformations and reciprocal links. For each class an algebraic construction of Ž . the symmetry transformation generated by products of adjoint eigenfunctions is given. The links between the soliton hierarchies are extended to these symmetries.

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ["Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods," J. Math. Phys. 50, 073510 (2009)]. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies. (C) 2011 American Institute of Physics. [doi:10.1063/1.3576185

Nonlinear integrable evolution equations in 1 q 1 dimensions arise from constraints of the 2 q 1-dimensional hierarchies associated with the Kadomtsev᎐ Ž . Petviashvili KP equation, the modified KP equation and the Dym equation, respectively. Links of Backlund type and of reciprocal type are known to exisẗ between the 2 q 1-dimensional systems. The corresponding links between the constrained flows are discussed in a general framework. In particular, squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered. The links between the soliton hierarchies are extended to these symmetries. ᮊ 1998 Academic Press

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Abstract. Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in [4]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [22, Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived. 1991 Mathematics Subject Classification. 35Q53; 46L55; 37K35.