On the (non)existence of degenerate phase-shift localised solution in KG (and dNLS) nonlocal lattices – Simone Paleari (Università degli Studi di Milano)

Venue

Sala Seminari (Dip. Matematica).

Abstract

We study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in Klein-Gordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, theconsideredlattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensionalsquare lattice or coupled chains. The usual strategy to deal with such a bifurcation problem consists in the examination of the necessary persistence conditions of the systemderived by the so-calledEffective Hamiltonian Method, in order to seek for unperturbed solutions whose continuation is feasible. Although this approachprovides useful insights, in the presence of degeneracy, it does not allow us to determine if theyconstitute true solutions of oursystem. In order to overcome this obstacle, we follow a different route. By means of a Lyapunov-Schmidt decomposition, we areable to establish that the bifurcation equation for our models can be considered, in thesmall energy and small coupling regime, asa perturbation of a corresponding, beyond nearest-neighbor, discrete nonlinear Schrödinger equation. There, nonexistence resultsof degenerate phase-shift discrete solitons can be demonstrated by anadditional Lyapunov-Schmidt decomposition, and translatedto our original problem on the Klein-Gordon system. In this way, among other results, we can prove nonexistence of four-sitesvortex-like waveforms in the zigzag Klein-Gordon model.

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