A Study of The Formation of Stationary Localized States Due to Nonlinear Impurities Using The Discrete Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates how stationary localized states form due to nonlinear impurities in discrete systems using the Discrete Nonlinear Schrödinger Equation, deriving phase diagrams and critical parameters for different impurity configurations.

Contribution

It provides analytical phase diagrams and critical impurity strength values for localized states in systems with nonlinear impurities, extending previous results with new lower critical values.

Findings

Derived phase diagrams for impurity-induced localized states.

Calculated critical impurity strengths for different nonlinearities.

Analyzed energy states and implications for disordered nonlinear systems.

Abstract

The Discrete Nonlinear Schr$\ddot{o}$dinger 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 $-\chi \mid C \mid^{\sigma}$ where $\sigma$ is arbitrary and $\chi$ is the nonlinearity parameter are considered. Furthermore, $\mid C \mid$ represents the absolute value of the amplitude. Altogether four cases are studies. The usual Greens 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 $-\chi \mid C \mid^{\sigma}$, three values of $\mid \chi_{cr} \mid$, namely, $\mid \chi_{cr} \mid = 2$, at $\sigma = 0$ and $\mid \chi_{cr} \mid = 1$ and $\frac{8}{3}$ for $\sigma = 2$ are obtained. Last two values are lower than the existing values. Energy of the states as a function of parameters is also obtained. A model derivation for the impurities is presented. The implication of our results in relation to disordered systems comprising of nonlinear impurities and perfect sites is discussed.