Nonlinear edge waves in mechanical topological insulators

Snee, David Douglas John Martin (2021) Nonlinear edge waves in mechanical topological insulators. Doctoral thesis, Northumbria University.

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Abstract

We show theoretically that the classical 1D nonlinear Schrödinger (NLS) and coupled nonlinear Schrödinger (CNLS) equations govern the envelope(s) of localised and unidirectional nonlinear travelling edge waves in a 2D mechanical topological insulator (MTI). The MTI consists of a collection of pendula with weak Duffing nonlinearity connected by linear springs that forms a mechanical analogue of the quantum spin Hall effect (QSHE). It is found, through asymptotic analysis and dimension reduction, that the NLS and CNLS respectively describe the unimodal and bimodal properties of the nonlinear system.

The governing bimodal CNLS is found to be non-integrable by nature and as such we discover new solutions by exploring the spatial dynamics of the reduced travelling wave ODE with general parameters. Such solutions include travelling fronts and, by numerically continuing these fronts, one can find vector soliton (VS) in non integrable CNLS equations. The equilibria can also undergo both pitchfork and Turing bifurcation in the reversible spatial dynamical system and we discuss relevant conditions for the existence and consequences of such critical values.

We briefly discuss the necessity of the developed front condition in forming such structures and present an analytical framework for front-grey soliton collisions by utilising conserved quantities of the non-integrable CNLS. The existence/stability of front and VS solutions can be inferred by spatial hyperbolicity and linear stability of the background fields, with the criteria presented here. VS solutions are considered in the form of bright-bright, bright-dark, and dark-dark solitons and their collision dynamics are explored qualitatively in the non-integrable regime. The Turing analysis presents the existence of periodic and localised patterned states in the CNLS, and we compare these solutions to those found in the analysis of the Swift-Hohenberg equation.

Theoretical predictions from the 1D (C)NLS are confirmed by numerical simulations of the original 2D MTI for various types of travelling waves and rogue waves. As a result of topological protection the edge solitons persist over long time intervals and through irregular boundaries. Due to the robustness of topologically
protected edge solitons (TPES) it is suggested that their existence may have significant implications on the design of acoustic devices. Spacetime simulations show a clear possibility of utilising MTIs in acoustical cloaking with TPES a vital player in such processes.

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