Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices

Abstract

We construct small amplitude breathers in one-dimensional (1D) and two-dimensional (2D) Klein–Gordon (KG) infinite lattices. We also show that the breathers are well-approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the KG lattice and the discrete nonlinear Schrödinger model. The proof is based on a Lyapunov–Schmidt decomposition and continuum approximation techniques introduced in [Bambusi and Penati, Continuous approximation of ground states in DNLS lattices, Nonlinearity 23 (2010), pp. 143–157], actually using its main result as an important lemma.

Keywords:

• breathers
• small amplitude solutions
• 1D and 2D
• Klein–Gordon lattices
• approximation and interpolation

AMS Subject Classifications::

• 34C25
• 37K50
• 37K60
• 65N30

Acknowledgement

This research was partially supported by PRIN 2007B3RBEY ‘Dynamical Systems and Applications’.

Notes

Note

1. Due to the autonomous and reversible nature of (1), it is rather natural to look for solutions even in time, thus, with a Fourier expansion in cosine only.