Related papers: Spontaneous motion of localized structures induced…
In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organised by a codimension-three point at…
We report on a novel behavior of solitary localized structures in a real Swift-Hohenberg equation subjected to a delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to nontrivial…
The Swift-Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for $x<0$ and…
Pattern formation often occurs in spatially extended physical, biological and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and…
We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional…
Theories of localised pattern formation are important to understand a broad range of natural patterns, but are less well-understood than more established mechanisms of domain-filling pattern formation. Here, we extend recent work on pattern…
I argue that ``good'' mathematical models of spatio-temporal dynamics in two-dimensions require non-local operators in the nonlinear terms. Consequently, the often used Swift-Hohenberg equation requires modification as it is purely local.…
A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one…
Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure --- it is…
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a…
The stability of a circular localized spot with respect to azimuthal perturbations is studied in in a variational Swift-Hohenberg model equation. The conditions under which the circular shape undergoes an elliptical deformation that…
A great variety of nonlinear dissipative systems are known to host structures having a correlation range much shorter than the size of the system. The dynamics of these Localized Structures (LSs) have been investigated so far in situations…
In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally…
The effect of external fluctuations on the formation of spatial patterns is analysed by means of a stochastic Swift-Hohenberg model with multiplicative space-correlated noise. Numerical simulations in two dimensions show a shift of the…
Using the supersymmetry technique, we study the localization-delocalization transition in quasi-one-dimensional non-Hermitian systems with a direction. In contrast to chains, our model captures the diffusive character of carriers' motion at…
The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with non-trivial…
The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic…
We study the Swift-Hohenberg equation - a paradigm model for pattern formation - with "large" spatially periodic coefficients and find a Turing bifurcation that generates patterns whose leading order form is a Bloch wave modulated by…
The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing related but time-dependent structures may result. These may consist of breathing…
We consider instabilities of a single mode with finite wavenumber in inversion symmetric spatially one dimensional systems, where the character of the bifurcation changes from sub- to supercritical behaviour. Starting from a general…