Topological Invariant and Anomalous Edge Modes of Strongly Nonlinear Systems

Abstract

Despite the extensive studies of topological states, their characterization in strongly nonlinear classical systems has been lacking. In this work, we identify the proper definition of Berry phase for nonlinear bulk modes and characterize topological phases in one-dimensional (1D) generalized nonlinear Schrödinger equations in the strongly nonlinear regime. We develop an analytic strategy to demonstrate the quantization of nonlinear Berry phase due to reflection symmetry. Mode amplitude itself plays a key role in nonlinear modes and controls topological phase transitions. We then show bulk-boundary correspondence by identifying the associated nonlinear topological edge modes. Interestingly, anomalous topological modes decay away from lattice boundaries to plateaus governed by fixed points of nonlinearities. We propose passive photonic and active electrical systems that can be experimentally implemented. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.

28 pages, 15 figures

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• Material vs. structure: Topological origins of band-gap truncation resonances in periodic structures
• Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial
• Strongly Nonlinear Topological Phases of Cascaded Topoelectrical Circuits
• Fully-Polarized Topological Isostatic Metamaterials in Three Dimensions
• Odd elasticity of a catalytic micromachine
• Hopf Bifurcation of Nonlinear Non-Hermitian Skin Effect
• Protected chaos in a topological lattice
• Nonlinear Corner States in Topologically Nontrivial Kagome Lattice
• Standing and Traveling Waves in a Model of Periodically Modulated One-dimensional Waveguide Arrays
• Hermitian and non-Hermitian topology in active matter
• Reservoir Engineering for Classical Nonlinear Fields
• Opening and closing a bandgap via alternating softening and hardening nonlinearities
• Quantum theory of fractional topological pumping of lattice solitons
• Theory of anomalous Landau-Zener tunneling induced by nonlinear coupling
• Soliton Pumping in the Rice-Mele Model with On-Cell Kerr Nonlinearity
• Dark solitons in nonlinear Su-Schrieffer-Heeger lattices