  For a one-dimensional linear lattice, earlier work has shown how to
systematically construct a slowly-decaying linear potential bearing a localized
eigenmode embedded in the continuous spectrum. Here, we extend this idea in two
directions: The first one is in the realm of the discrete nonlinear Schrodinger
equation, where the linear operator of the Schrodinger type is considered in
the presence of a Kerr focusing or defocusing nonlinearity and the embedded
linear mode is continued into the nonlinear regime as a discrete solitary wave.
The second case is the Klein-Gordon setting, where the presence of a cubic
nonlinearity leads to the emergence of embedded-in-the-continuum discrete
breathers. In both settings, it is seen that the stability of the modes near
the linear limit turns into instability as nonlinearity is increased past a
critical value, leading to a dynamical delocalization of the solitary wave (or
breathing) state. Finally, we suggest a concrete experiment to observe these
embedded modes using a bi-inductive electrical lattice.
