Related papers: Localised patterns in a generalised Swift--Hohenbe…
We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swift-Hohenberg equation that include coupling to a mean flow. A projection operator is included in our…
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…
We investigate the pattern dynamics of the one-dimensional nonreciprocal Swift-Hohenberg model. Characteristic spatiotemporal patterns such as disordered, aligned, swap, chiral-swap, and chiral phases emerge depending on the parameters. We…
Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the…
Third order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up- and down-hexagons in pattern forming systems. These amplitude equations predict the nonexistence of bistable ranges…
We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry $x \mapsto -x$. This system exhibits a Turing instability and we…
Non-topological defects such as grain boundaries abound in pattern forming systems, arising from local variations of pattern properties such as amplitude, wavelength, orientation, etc. We introduce the idea of treating such non-topological…
The characteristic size for spatial structure, that emerges when the bifurcation parameter in model partial differential equations is slowly increased through its critical value, depends logarithmically on the size of added noise. Numerics…
Numerical studies of the May-Leonard model for cyclically competing species exhibit spontaneous spatial structures in the form of spirals. It is desirable to obtain a simple coarse-grained evolution equation describing spatio-temporal…
Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with…
We study the excitation of spatial patterns by resonant, multi-frequency forcing in systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. Using weakly nonlinear analysis we show that for small amplitudes only stripe…
The classical Cahn-Hilliard (CH) equation corresponds to a gradient dynamics model that describes phase decomposition in a binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via a large-scale…
We present results on stripe formation in the Swift-Hohenberg equation with a directional quenching term. Stripes are "grown" in the wake of a moving parameter step line, and we analyze how the orientation of stripes changes depending on…
We show the appearance of spatiotemporal stochastic resonance in the Swift-Hohenberg equation. This phenomenon emerges when a control parameter varies periodically in time around the bifurcation point. By using general scaling arguments and…
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…
Integral asymptotics play an important role in the analysis of differential equations and in a variety of other settings. In this work, we apply an integral asymptotics approach to study spatially localized solutions of a heterogeneous…
The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the…
Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast…
We investigate the snaking of localised patterns, seen in numerous physical applications, using a variational approximation. This method naturally introduces the exponentially small terms responsible for the snaking structure, that are not…
Ultrafast laser irradiation can induce spontaneous self-organization of surfaces into dissipative structures with nanoscale reliefs. These surface patterns emerge from symmetry-breaking dynamical processes that occur in…