  IN-DNLS, considered here is a countable infinite set of coupled one
dimensional nonlinear ordinary differential difference equations with a tunable
nonlinearity parameter, $\nu$. This equation is continuous in time and discrete
in space with lattice translational invariance and has global gauge invariance.
When $\nu = 0$, it reduces to the famous integrable Ablowitz - Ladik (AL)
equation. Otherwise it is nonintegrable. The formation of unstaggered and
staggered stationary localized states (SLS) in IN-DNLS is studied here using
discrete variational method. The appropriate functional is derived and its
equivalence to the effective Lagrangian is established. From the physical
consideration, the ansatz of SLS is assumed to have the functional form of
stationary soliton of AL equation. So, the ansatz contains three optimizable
parameters, defining width ($\beta^{-1})$, maximum amplitude and its position
($\sqrt\Psi$, $x_{0}$). Four possible situations are considered. An unstaggered
SLS can be either on-site peaked $(x_{0} = 0.0)$ or inter-site peaked $(x_{0} =
0.5)$. On the other hand, a staggered SLS can be either Sievers-Takeno (ST)
like mode $(x_{0} = 0.0)$, or Page(P) like mode $(x_{0} = 0.5)$. It is shown
here that unstable SLS arises due to incomplete consideration of the problem.
In the exact calculation, there exists no unstable mode. The width of an
unstaggered SLS of either type decreases with increasing $\nu > 0$.
Furthermore, on-site peaked state is found to be energetically stable. These
results are explained using the effective mass picture. For the staggered SLS,
the existence of ST like mode and P like mode is shown to be a fundamental
property of a system, described by IN-DNLS. Their properties are also
investigated. For large width and small amplitude SLS, the known asymptotic
result for the amplitude is obtained. Further scope and possible extensions of
this work are discussed.
