Supersolidity around a Critical Point in Dipolar Bose-Einstein Condensates.

We explore spatial symmetry breaking of a dipolar Bose-Einstein condensate in the thermodynamic limit and reveal a critical point in the phase diagram at which crystallization occurs via a second-order phase transition. This behavior is traced back to the significant effects of quantum fluctuations in dipolar condensates, which moreover stabilize a new supersolid phase, namely a regular honeycomb pattern with high modulational contrast and near-perfect superfluidity.

Threaded Rings that Swim in Excitable Media.

Cardiac tissue and the Belousov-Zhabotinsky reaction provide two notable examples of excitable media that support scroll waves, in which a filament core is the source of spiral waves of excitation. Here we consider a novel topological configuration in which a closed filament loop, known as a scroll ring, is threaded by a pair of counterrotating filaments that are perpendicular to the plane of the ring and end on the boundary of a thin medium. We simulate the dynamics of this threaded ring (thring) in the photosensitive Belousov-Zhabotinsky excitable medium, using the modified Oregonator reaction-diffusion equations. These computations reveal that the threading topology induces an exotic motion in which the thring swims in the plane of the ring. We propose a light templating protocol to create a thring in the photosensitive Belousov-Zhabotinsky medium and provide experimental confirmation that this protocol indeed yields a thring.

Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings.

We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.

Azimuthons in weakly nonlinear waveguides of different symmetries.

We show that weakly guiding nonlinear waveguides support stable propagation of rotating spatial solitons (azimuthons). We investigate the role of waveguide symmetry on the soliton rotation. We find that azimuthons in circular waveguides always rotate rigidly during propagation and the analytically predicted rotation frequency is in excellent agreement with numerical simulations. On the other hand, azimuthons in square waveguides may experience spatial deformation during propagation. Moreover, we show that there is a critical value for the modulation depth of azimuthons above which solitons just wobble back and forth, and below which they rotate continuously. We explain these dynamics using the concept of energy difference between different orientations of the azimuthon.

Colonies of threaded rings in excitable media.

A thring is a recent addition to the zoo of spiral wave phenomena found in excitable media and consists of a scroll ring that is threaded by a pair of counter-rotating scroll waves. This arrangement behaves as a particle that swims through the medium. Here, we present results on the dynamics, interaction, and collective behavior of several thrings via numerical simulation of the reaction-diffusion equations that model thrings created in chemical experiments. We reveal an attraction between two thrings that leads to a stable bound pair that thwarts their individual locomotion. Furthermore, such a pair emits waves at a higher frequency than a single thring, which protects the pair from the advances of any other thring and rules out the formation of a triplet bound state. As a result, the long-term evolution of a colony of thrings ultimately yields an unusual frozen nonequilibrium state consisting of a collection of pairs accompanied by isolated thrings that are inhibited from further motion by the waves emanating from the pairs.

Length of excitable knots.

In this paper, we present extensive numerical simulations of an excitable medium to study the long-term dynamics of knotted vortex strings for all torus knots up to crossing number 11. We demonstrate that FitzHugh-Nagumo evolution preserves the knot topology for all the examples presented, thereby providing a field theory approach to the study of knots. Furthermore, the evolution yields a well-defined minimal length for each knot that is comparable to the ropelength of ideal knots. We highlight the role of the medium boundary in stabilizing the length of the knot and discuss the implications beyond torus knots. We also show that there is not a unique attractor within a given knot topology.