arXiv subjects

Andrew Hofstrand

Publications and source records attributed to Andrew Hofstrand.

3 recordsLinked to original sources

arXiv2026

Discrete Breathers in a Honeycomb Lattice Near a Semi-Dirac Point

We study the dynamics of discrete breathers---spatially localized and time-periodic solutions---inside the bandgap of a nonlinear honeycomb lattice where the dispersion landscape approaches a so-called semi-Dirac point in which the bands cross linearly in one direction and quadratically in the orthogonal direction. By studying breather dynamics in two opposing asymptotic regimes, near the continuum and anti-continuum limits, we capture the spatial profiles of hybrid coherent structures having central cores supported on a finite number of lattice sites and infinite decaying tails that are well approximated by exact separable solutions to an effective long-wave PDE theory at spatial infinity. We find that these breathers are dynamically stable over a wide range of parameters and find an instability transition. Finally, we analyze the Floquet stability of spatially extended nonlinear plane waves bifurcating from the zero solution at the edges of the gap and how they shape breather profiles inside the gap.

nlin.PS↗

arXiv2024

Families of Discrete Breathers on a Nonlinear Kagome Lattice

The unique geometry of the two-dimensional tripartite Kagome lattice is responsible for shaping diverse families of spatially localized and time-periodic nonlinear modes known as discrete breathers. We state conditions for the existence of breathers and compute their spatiotemporal profiles near the edges of the linear phonon spectrum. Our findings include the existence of strongly nonlinear and dynamically stable breathers inside the band gap on the infinite lattice, asymptotic expressions for breather energy thresholds in the weakly nonlinear regime, and explicit breather solutions that remain compactly supported on the lattice and undergo stability transitions.

nlin.PS↗

arXiv2024

Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators

We study a semilinear hyperbolic system of PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model introduced by Hadad, Vitelli and Alu (HVA) in \cite{HVA17}. We classify the system's traveling waves, and study their stability properties. We focus on traveling pulse solutions (``solitons'') on a nontrivial background and moving domain wall solutions (kinks); both arise as heteroclinic connections between spatially uniform equilibrium of a reduced dynamical system. We present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and spectral stability of moving domain walls. Our stability results are in terms of weighted $H^1$ norms of the perturbation, which capture the phenomenon of {\it convective stabilization}; as time advances, the traveling wave ``outruns'' the \underline{growing} disturbance excited by an initial perturbation; the non-trivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.

nlin.PS↗