Generalized nonlinear impurity in a linear chain

We study the problem of one nonlinear impurity embedded in a linear tight-binding host. The impurity is of the type found in the generalized discrete nonlinear Schr\"odinger equation. We obtain analytically a phase diagram that describes the presence of bound states for different nonlinearity parameter values and nonlinearity exponents. We find that two impurity states are possible in some parameter regimes. From the numerical solution of the complete dynamical problem we obtain information on the nonlinear site survival probability that shows a dynamical self-trapping that is compatible with the findings of the stationary-state analysis.