2023 Theses Doctoral

Study of Traveling Waves in a Nonlinear Continuum Dimer Model

Li, Huaiyu

We study a system of semilinear hyperbolic PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model of SSH type of Hadad, Vitelli and Alú in [1]. We classify the system’s traveling waves, and study their stability properties. We focus on pulse solutions (solitons) on a nontrivial background and moving domain wall solutions (kinks and antikinks), corresponding to heteroclinic orbits for a reduced two-dimensional dynamical system. We further present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and the spectral stability of moving domain walls.

Our result for nonlinear stability is expressed in terms of appropriately weighted ?1-norms of the perturbation, which captures the phenomenon of convective stabilization; as time advances, the traveling wave “outruns” the growing disturbance excited by an initial perturbation. We use our analytical results to interpret phenomena observed in numerical simulations. The results for (linear) spectral stability are studied in appropriately-weighted ?2-norms.