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Nonlinear waves in lattices and metamaterials

Duran, Henry A. (2022) Nonlinear waves in lattices and metamaterials. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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The 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
the associated instabilities change their speed, typically slowing them down over long-time simulations.

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.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Duran, Henry A.had46@pitt.eduhad460000-0001-8761-1717
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairVainchtein, Anna A.aav4@pitt.eduaav4
Committee MemberWang, Dehuadhwang@pitt.edudhwang
Committee MemberChen, Mingmingchen@pitt.edumingchen
Committee MemberKevrekidis, Panayotis
Date: 30 September 2022
Date Type: Publication
Defense Date: 23 June 2022
Approval Date: 30 September 2022
Submission Date: 28 July 2022
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 116
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: solitary traveling waves, discrete breathers, Fermi-Pasta-Ulam, long-range interactions, metamaterials, Floquet spectrum, instability, nonlinear resonances, bifurcations
Date Deposited: 30 Sep 2022 18:23
Last Modified: 30 Sep 2022 18:23


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