Gradient catastrophe of nonlinear topological edge states

Highlighting our research published in Physical Review Research a few days ago: Gradient catastrophe of nonlinear photonic valley-Hall edge pulses

In this paper we studied the propagation dynamics of topological edge wavepackets in the presence of Kerr nonlinearity, describing light-induced shifts to the refractive index of the medium. Prior works studying the nonlinear dynamics of topological edge states largely focused on the analysis of plane wave-like edge states using the nonlinear Schrodinger equation, which can describe their self-focusing dynamics and modulational instability. However, the nonlinear Schrodinger equation is a purely scalar wave equation, in which the origin of the edge states (e.g. whether they are topologically protected) does not qualitatively affect the dynamics.

We carried out a theoretical analysis based on the more complete nonlinear Dirac equation, which takes into account the spin-like degrees of freedom required to create topological band gaps and edge states. Assuming the ambient medium exhibits a simple Kerr nonlinearity, we derive effective nonlinear wave equations governing the dynamics of edge wavepackets. Interestingly, a pure Kerr nonlinearity gives rise to a novel higher-order self-steepening term in the effective nonlinear wave equation, which means that an initially-symmetric wavepacket will become increasingly asymmetric as it propagates, eventually acquiring a shock-like discontinuity (gradient catastrophe).

We confirmed the validity of our effective nonlinear wave equations using direct numerical simulations of the paraxial wave equation describing 2D topological waveguide lattices. Similar phenomena may also be observed in related nonlinear wave systems, including exciton-polariton condensates in microcavities.