  The dynamics of coupled nonlinear oscillator systems is often described by
the classical discrete nonlinear Schrodinger equation (DNLSE). In its simplest
version, the DNLSE is made up of two terms -- a nearest-neighbor hopping term
and an on-site cubic nonlinear term. Each of the terms is preceded by a
coefficient that can take on either a positive or a negative sign. Each of the
DNLSE versions is derived from a corresponding equivalent Hamiltonian. The
result is a small family of four versions of the DNLSE Hamiltonian, each with
its own associated ground state, all indeed scattered in myriad of scientific
publications. Here we present a comprehensive picture for the ground states of
DNLSE systems, summarize existing results and provide new insights. First, we
classify the four DNLSE Hamiltonians into two pairs according to the sign of
the nonlinear term -- a "positive/negative Hamiltonian pair" if the sign of the
nonlinear term is positive/negative respectively. Ground states of the positive
Hamiltonian pair are discrete plane waves in either a ferromagnetic-like or an
antiferromagnetic-like configuration, depending on the sign of the hopping
term. Ground states of the negative Hamiltonian pair are either unstaggered or
staggered site-centered discrete breathers. Next, we discuss some properties of
the ground states associated with the positive-Hamiltonian pair -- entropy,
temperature, correlations and stability. We extend our ground state stability
discussion to include excited plane waves. We propose to engineer a
density-energy conserving perturbation and show that under such perturbation,
all excited plane waves are entropy-unstable. For site-centered discrete
breathers -- the ground states of the negative-Hamiltonian pair -- we have
divided system nonlinearity into two ranges and wrote very good analytic
approximations for the breathers in each range.
