Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation

Abstract

The discrete nonlinear Schr\"odinger equation is used to study the formation of stationary localized states due to a single nonlinear impurity in a Caley tree and a dimeric nonlinear impurity in the one-dimensional system. The rotational nonlinear impurity and the impurity of the form -\ensuremath{\chi}|C${\mathrm{|}}^{\mathrm{\ensuremath{\sigma}}}$ where \ensuremath{\sigma} is arbitrary, and \ensuremath{\chi} is the nonlinearity parameter are considered. Furthermore, |C| represents the absolute value of the amplitude. Altogether four cases are studied. The usual Green's-function approach and the ansatz approach are coherently blended to obtain phase diagrams showing regions of different number of states in the parameter space. Equations of critical lines separating various regions in phase diagrams are derived analytically. For the dimeric problem with the impurity -\ensuremath{\chi}|C${\mathrm{|}}^{\mathrm{\ensuremath{\sigma}}}$, three values of |${\mathrm{\ensuremath{\chi}}}_{\mathrm{cr}}$|, namely, |${\mathrm{\ensuremath{\chi}}}_{\mathrm{cr}}$|=2, at \ensuremath{\sigma}=0 and |${\mathrm{\ensuremath{\chi}}}_{\mathrm{cr}}$|=1 and 8/3 for \ensuremath{\sigma}=2 are obtained. The last two values are lower than the existing values. The energy of the states as a function of parameters is also obtained. A model derivation of the impurities is presented. The implication of our results in relation to disordered systems comprising nonlinear impurities and perfect sites is discussed.