On the nonexistence of degenerate phase-shift multibreathers in Klein-Gordon models with interactions beyond nearest neighbors

In this work, we study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in KleinGordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice or coupled chains. We examine initially the necessary persistence conditions of the system derived by the so-called Effective Hamiltonian Method, in order to seek for unperturbed solutions whose continuation is feasible. Although this approach provides useful insights, in the presence of degeneracy, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. By means of a Lyapunov-Schmidt decomposition, we are able to establish that the bifurcation equation for our models can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding, beyond nearest-neighbor, discrete nonlinear Schr¨odinger equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by an additional Lyapunov-Schmidt decomposition, and translated to our original problem on the Klein-Gordon system. In this way, among other results, we can prove nonexistence of four-sites vortex-like waveforms in the zigzag Klein-Gordon model. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration.