Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations

TL;DR

This paper justifies approximating small-amplitude weakly coupled Klein-Gordon oscillators with discrete nonlinear Schrödinger equations using two different mathematical methods, and discusses applications to breather solutions.

Contribution

It compares two methods for justifying the approximation and demonstrates their equivalence, advancing understanding of Klein-Gordon lattice dynamics.

Findings

Both methods yield equivalent results in approximation validation.

The discrete nonlinear Schrödinger equation effectively models Klein-Gordon lattice breathers.

The paper provides a rigorous foundation for using DNLS in lattice dynamics studies.

Abstract

Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrodinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss applications of the discrete nonlinear Schrodinger equation in the context of existence and stability of breathers of the Klein--Gordon lattice.