Solutions of the discrete nonlinear Schrödinger equation with a trap

We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave function as a series in a certain parameter. In the second, approximate method, we obtain the wave function in the continuum limit. We compare the obtained solutions with numerical results, and the relative accuracy of the solution in the form of a series does not exceed $10^{-15}$ in order of magnitude.