Abstract. Stable localized roll structures have been observed in many physical problems and model equations, notably in the 1D Swift-Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches, so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher-Kolmogorov-Petrovskii-Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.

We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples, including reaction-diffusion systems with non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local cubic nonlinearity. In each case, we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

PurposeThe purpose of this paper is to examine the family dinner in Los Angeles County, California, focusing on the role of commercial foods and the time invested in food preparation. Popular media emphasize the increasing use of processed commercial foods in the USA.Design/methodology/approachA total of 64 dinner preparation and consumption events were videotaped and observed (32 families, two weeknights each). Observations determined the source of food served (restaurant, take‐out, or home‐cooked), the ingredients and dishes in each meal prepared at home, and the time required to prepare it.FindingsThe findings in this paper showed that, even when prepared at home, most evening meals included processed commercial foods in at least moderate amounts. Home‐cooked meals required an average of 34 minutes' “hands‐on time” and 52 minutes' “total time” to prepare. Heavy use of commercial foods saved, on average, ten to 12 minutes, hands‐on time but did not reduce total preparation time. Commercial foods require more limited cooking skills and permit more complex dishes or meals to be prepared within a given time‐frame than do raw ingredients. They may also reduce time investment at stages other than meal preparation, such as shopping.Originality/valueThis paper provides a rare glimpse of food preparation and meal consumption behavior on the family level. Most reports on US food habits are based on survey and purchasing data, rather than direct observation of household activities as used here.