Stability of intrinsic localized modes on the lattice with competing power nonlinearities

TL;DR

This paper analyzes the stability of intrinsic localized modes in the discrete nonlinear Schrödinger equation with competing power nonlinearities, identifying stable configurations through spectral analysis and numerical methods.

Contribution

It provides a classification of localized modes and their stability properties for specific nonlinear power combinations in the lattice.

Findings

Larger state codes are spectrally and nonlinearly stable.

Smaller, alternating sign codes are spectrally stable but have negative Krein signature eigenvalues.

Numerical identification of stable stacked configurations.

Abstract

We study the discrete nonlinear Schrodinger equation with competing powers (p,q) satisfying 2 <= p < q. The physically relevant cases are given by (p,q) = (2,3), (p,q) = (3,4), and (p,q) = (3,5). In the anticontinuum limit, all intrinsic localized modes are compact and can be classified by their codes, which record one of two nonzero (smaller and larger) states and their sign alternations. By using the spectral stability analysis, we prove that the codes for larger states of the same sign are spectrally and nonlinearly (orbitally) stable, whereas the codes for smaller states of the alternating signs are spectrally stable but have eigenvalues of negative Krein signature. We also identify numerically the spectrally stable codes which consist of stacked combinations of the sign-definite larger states and the sign-alternating smaller states.