Breathers for the Discrete Nonlinear Schr\"odinger equation with nonlinear hopping

TL;DR

This paper studies the existence and power bounds of breathers in nonlinear Schrödinger lattices with nonlinear hopping, providing theoretical thresholds and numerical validation for breather solutions.

Contribution

It introduces new lower bounds on breather power using variational and fixed point methods, extending understanding of breather existence in nonlinear lattices.

Findings

Lower bounds serve as thresholds for breather existence.

Numerical results confirm the accuracy of theoretical bounds.

Bounds depend on lattice parameters, dimension, and breather frequency.

Abstract

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schr\"odinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.