Stationary Localized States Due to a Nonlinear Dimeric Impurity Embedded in a Perfect 1-D Chain

TL;DR

This paper investigates stationary localized states caused by a nonlinear dimeric impurity in a perfect 1-D chain using the Discrete Nonlinear Schrödinger Equation, revealing phase diagrams and maximum localized states.

Contribution

It introduces a novel analysis of localized states with a nonlinear impurity, deriving phase diagrams and identifying critical parameters for state formation.

Findings

Maximum of six localized states identified.

Phase diagram mapped in the (χ, σ) parameter space.

Solutions include symmetric and asymmetric localized states.

Abstract

The formation of Stationary Localized states due to a nonlinear dimeric impurity embedded in a perfect 1-d chain is studied here using the appropriate Discrete Nonlinear Schr$\ddot{o}$dinger Equation. Furthermore, the nonlinearity has the form, $\chi |C|^\sigma$ where $C$ is the complex amplitude. A proper ansatz for the Localized state is introduced in the appropriate Hamiltonian of the system to obtain the reduced effective Hamiltonian. The Hamiltonian contains a parameter, $\beta = \phi_1/\phi_0$ which is the ratio of stationary amplitudes at impurity sites. Relevant equations for Localized states are obtained from the fixed point of the reduced dynamical system. $|\beta|$ = 1 is always a permissible solution. We also find solutions for which $|\beta| \ne 1$. Complete phase diagram in the $(\chi, \sigma)$ plane comprising of both cases is discussed. Several critical lines separating various regions are found. Maximum number of Localized states is found to be six. Furthermore, the phase diagram continuously extrapolates from one region to the other. The importance of our results in relation to solitonic solutions in a fully nonlinear system is discussed.