Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law

Traveling waves and defects in the complex Swift-Hohenberg equation.

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves investigated numerically. A number of distinct dynamical regimes is identi...

Dislocations in an anisotropic Swift-Hohenberg equation

We study the existence of dislocations in an anisotropic Swift-Hohenberg equation. We find dislocations as traveling or standing waves connecting roll patterns with different wavenumbers in an infinite strip. The proof is based on a bifurcation analysis. Spatial dynamics and center-manifold reduction yield a reduced, coupled-mode system of differential equations. Existence of traveling dislocat...

Modulation Equation for Stochastic Swift-Hohenberg Equation

The purpose of this paper is to study the influence of large or unbounded domains on a stochastic PDE near a change of stability, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider the stochastic Swift-Hohenberg equation and derive rigorously the Ginzburg-Landau equation as a modulation equation for the amplitudes of...

Traveling Fronts of a Conservation Law with Hyper-dissipation

Studied here are traveling-front solutions φ (x−ct) of a conservation law with hyper-dissipation appended. The evolution equation in question is a simple conservation law with a fourth-order dissipative term, namely ut + 2uux + uxxxx = 0, where > 0. The traveling front is restricted by the asymptotic conditions φ (x) → L± as x → ±∞, where L+ < L−, and the symmetry condition φ (x)+φ (−x) = L−+L+...

Fronts between hexagons and squares in a generalized Swift-Hohenberg equation.