Nonlinear waves in lattices and metamaterialsDuran, Henry A. (2022) Nonlinear waves in lattices and metamaterials. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractThe combination of dispersion and nonlinearity often leads to the formation of nonlinear waves with complex bifurcation structure. This thesis focuses on the existence, stability and dynamic evolution of several different types of these waveforms in spatially discrete nonlinear systems. In the first part of the thesis, we consider traveling solitary waves in a lattice where the competition between nonlinear short-range interactions and all-to-all harmonic long-range interactions yields two parameter regimes. We compute exact traveling waves for both cases and investigate their stability. Perturbing the unstable solution along the corresponding eigenvector, we identify two scenarios of the dynamics of their transition to stable branches. In the first case, the perturbed wave slows down after expelling a dispersive shock wave, and in the second case, it speeds up and is accompanied by the formation of a slower small-amplitude traveling solitary wave. In the second part, we explore the existence, stability and dynamical properties of moving discrete breathers in a nonlinear lattice. We propose a numerical procedure that allows us to systematically construct breathers traveling more than one lattice site per period. We explore their stability spectrum and connect it to the energy-frequency bifurcation diagrams. We illustrate in this context examples of the energy being a multivalued function of the frequency. Finally, we probe the moving breather dynamics and observe how In the third part, we turn to stationary discrete breathers. We seek such solutions in a discrete model that describes an engineered structure consisting of a chain of pairs of rigid cross-like units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions. We then compute numerically exact solutions and investigate their properties and stability. Our findings demonstrate that the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling, symmetry-breaking and other types of bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system. Share
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