This manuscript describes harmonic generation in semiconductor superlattices, starting from a nonequilibrium Green's functions input to relaxation rate-type analytical approximations for the Boltzmann equation in which imperfections in the structure lead to asymmetric current flow and scattering processes under forward and reverse bias. The resulting current-voltage curves and the predicted consequences on harmonic generation, notably the development of even harmonics, are in good agreement with experiments. Significant output for frequencies close to 1 THz (7th harmonic) at room temperature, after excitation by a 141-GHz input signal, demonstrate the potential of superlattice devices for gigahertz to terahertz applications.

The Kuramoto-Sakaguchi system of coupled phase oscillators, where interaction between oscillators is determined by a single harmonic of phase differences of pairs of oscillators, has very simple emergent dynamics in the case of identical oscillators that are globally coupled: there is a variational structure that means the only attractors are full synchrony (in-phase) or splay phase (rotating wave/full asynchrony) oscillations and the bifurcation between these states is highly degenerate. Here we show that nonpairwise coupling-including three and four-way interactions of the oscillator phases-that appears generically at the next order in normal-form based calculations, can give rise to complex emergent dynamics in symmetric phase oscillator networks. In particular, we show that chaos can appear in the smallest possible dimension of four coupled phase oscillators for a range of parameter values.In this paper, we show that symmetrically coupled phase oscillators with generic (but nonpairwise) interactions yield rich dynamics even for as few as N = 4 oscillators. Although the lowest order approximation of a phase reduction of symmetrically coupled oscillators close to a Hopf bifurcation has only the Kuramoto-Sakaguchi first harmonic interaction terms, the next order includes generic terms with both second harmonic pairwise interactions and interactions of up to four phases 9 . The symmetries we consider imply the existence of invariant subspaces such that the ordering of phases is preserved 10 . In contrast to the Kuramoto-Sakaguchi equations, the additional nonpairwise interaction terms mean we can find attracting chaos for a range of normal form parameter values. As a consequence, the phase dynamics of generic weakly coupled oscillators will be quite rich and chaos can occur even for the phase dynamics in the weak coupling limit without amplitude degrees of freedom 30 .

Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry S N are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive -and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small > 0. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with S N symmetry. For fixed N and taking the limit 0 < λ 1, we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of N . Using a normalization that maintains nontrivial interactions in the limit N → ∞, we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.

In this paper, we examine the possibility of generating bright and dark solitons, as well as multi-pulse solitary waves in Bose–Einstein condensates. Starting from the linear limit of the problem, where such solutions are known explicitly, we use ‘dynamic continuation’ through the temporal variation of the scattering length, which can be realized in experiments upon using external time-dependent magnetic fields. We examine the stability of the resulting solutions both through direct numerical simulations and through numerical continuation of the relevant steady states and linear stability analysis. Analytical results, illustrating the nature of the bifurcation of the nonlinear solutions, are found to be in good agreement with the numerical findings.

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