Ricerc@Sapienza

The reviewed article concerns the combined KdV-negative-order KdV (KdV-nKdV) equation.
Via truncated Painleve´ expansion, multiple residual symmetries of such an equation are
constructed.

The KdV eigenfunction equation is considered: some explicit solutions are constructed. These, to the best of the authors’ knowledge, new solutions represent an example of the powerfulness of the method de- vised. Specifically, Ba ̈cklund transformation are applied to reveal alge- braic properties enjoyed by nonlinear evolution equations they connect.

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here considered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as rt = rxx + 2rxr. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be hereditary.

Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equations. Matrix equation can be viewed as a specialisation of operator equations in the finite dimensional case when operators are finite dimensional and, hence, admit a matrix representation. Baecklund transformations allow to reveal structural properties [S. Carillo and C. Schiebold, J. Math. Phys.