D.E. Pelinovsky, T. Penati, and S. Paleari

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

Reviews in Mathematical Physics 28 (2016), 1650015 (25 pages)

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.

discrete Klein-Gordon equation, discrete nonlinear Schrodinger equation, multi-site breathers, instability, justification of amplitude equations, energy methods, normal form theorem.