5 research outputs found
Darboux transformations, finite reduction groups and related Yang-Baxter maps
In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which
are invariant under the action of finite reduction groups. We present
6-dimensional YB maps corresponding to Darboux transformations for the
Nonlinear Schr\"odinger (NLS) equation and the derivative Nonlinear
Schr\"odinger (DNLS) equation. These YB maps can be restricted to
dimensional YB maps on invariant leaves. The former are completely
integrable and they also have applications to a recent theory of maps
preserving functions with symmetries \cite{Allan-Pavlos}. We give a
dimensional YB-map corresponding to the Darboux transformation for a
deformation of the DNLS equation. We also consider vector generalisations of
the YB maps corresponding to the NLS and DNLS equation.Comment: 18 pages, revised version. The format of the paper has changed, we
added one sectio
Partially integrable nonlinear equations with one higher symmetry
In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function