A Discrete Variational Approach for Investigation of Stationary   Localized States In A Discrete Nonlinear Schr$\ddot{\rm {\bf {o}}}$dinger   Equation, Named IN-DNLS
  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. 
K. Kundu