The role of power law nonlinearity in the discrete nonlinear Schr\"{o}dinger equation on the formation of stationary localized states in the Cayley tree

TL;DR

This paper investigates how power law nonlinearity influences the formation and stability of stationary localized states in the discrete nonlinear Schrödinger equation on Cayley trees, using a transformation to simplify the analysis.

Contribution

It introduces a transformation reducing Cayley trees to one-dimensional chains with a defect, enabling analysis of localized states with power law nonlinearity.

Findings

Localized states depend on nonlinearity strength and impurity configuration.

Stability diagrams of localized states are provided for various cases.

The transformation simplifies the analysis of complex Cayley tree structures.

Abstract

We study the formation of stationary localized states using the discrete nonlinear Schr\"{o}dinger equation in a Cayley tree with connectivity $K$. Two cases, namely, a dimeric power law nonlinear impurity and a fully nonlinear system are considered. We introduce a transformation which reduces the Cayley tree into an one dimensional chain with a bond defect. The hopping matrix element between the impurity sites is reduced by $1/\surd K$. The transformed system is also shown to yield tight binding Green's function of the Cayley tree. The dimeric ansatz is used to find the reduced Hamiltonian of the system. Stationary localized states are found from the fixed point equations of the Hamiltonian of the reduced dynamical system. We discuss the existence of different kinds of localized states. We have also analyzed the formation of localized states in one dimensional system with a bond defect and nonlinearity which does not correspond to a Cayley tree. Stability of the states is discussed and stability diagram is presented for few cases. In all cases the total phase diagram for localized states have been presented.