Fractional nonlinear surface impurity in a 2D lattice

TL;DR

This paper investigates how fractional Laplacians affect localized modes around nonlinear impurities at the edges of a 2D lattice, revealing bound state behaviors dependent on fractional order and impurity strength.

Contribution

It introduces a formalism using lattice Green functions to analyze bound states in fractional 2D lattices with nonlinear impurities, extending understanding of impurity modes.

Findings

Single bound state exists for linear impurities above a threshold strength.

Up to two bound states can occur in the nonlinear (cubic) case.

Impurity mode energy scales with impurity strength at small fractional exponents.

Abstract

We study the formation of localized modes around a generalized nonlinear impurity which is located at the boundary of a semi-infinite square lattice, and where we replace the standard discrete Laplacian by a fractional one, characterized by a fractional exponent $0<\alpha<1$ where $\alpha=1$ marks the standard, non-fractional case. We specialize to two impurity cases: impurity at an "edge" and impurity at a "corner" and use the formalism of lattice Green functions to obtain in closed form the bound state energy and its mode amplitude. It is found that, for any fractional exponent and for impurity strengths above a certain threshold, there is always a single bound state for the linear impurity, while for the nonlinear (cubic) case, up to two bound states are possible. At small fractional exponents, the energy of the impurity mode becomes directly proportional to the impurity strength.