Editorial to Homoclinic snaking at 21: in memory of Patrick Woods

This editorial serves as an extended introduction to the Special Issue. It gives the context to homoclinic snaking, especially the contribution of Patrick Woods. A very brief summary of more recent developments serves as a motivation to each paper that follows.

Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

Curvature effects and radial homoclinic snaking

In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

Spatial localization beyond steady states in the neighbourhood of the Takens–Bogdanov bifurcation

Double-zero eigenvalues at a Takens–Bogdanov (TB) bifurcation occur in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in 1D with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves (MW) and travelling waves, and describes the transitions and bifurcations between these states. Values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr, G. & Knobloch, E. (1987) The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A, 322, 243-279). In this work we develop a new and simple pattern-forming partial differential equation (PDE) model, based on the Swift–Hohenberg equation, adapted to have the TB normal form at onset. This model allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localized solutions by examining pattern formation in wide domains. We are able to recover two types of localized states, that of a localized steady state (LSS) in the background of the trivial state (TS) and that of a spatially localized travelling wave (LTW) in the background of the TS, which have previously been observed in other systems. Additionally, we identify two new types of spatially localized states: that of a LSS in a MW background and that of a LTW in a steady state (SS) background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond SSs.

Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance

Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.

A validated asymptotic thermomechanical model for air-gap formation in tapered moulds in the continuous casting of steel

A recent asymptotics-based thermomechanical model is adapted and applied to the mould region in the continuous casting of round steel billets, with a view to describing the complex interplay between air-gap formation, mould taper, cooling channel width and cooling water velocity. Although the situation is steady state, the analysis leads to what is mathematically a dual moving-boundary problem for the solid–melt and solid–air interfaces, where the distance from the top of the mould region is the time-like variable in the problem. Moreover, the two interfaces are initiated at different locations. In addition, the thermal and mechanical problems are found to decouple and it is possible to solve the first ahead of the second. The model equations are solved numerically using a finite-difference method, and the approach is subsequently successfully validated against a previous finite-element model and experimental data from temperature measurements taken within the mould.