Dislocations in an Anisotropic Swift-Hohenberg Equation

Abstract

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 dislocations is then established by showing that this reduced system possesses robust heteroclinic orbits. Running head: Dislocations in a modified Swift-Hohenberg equationCorresponding author: Mariana Haragus, mharagus@univ-fcomte.fr

Small‐amplitude grain boundaries of arbitrary angle in the Swift‐Hohenberg equation

“…We outline the proof which essentially follows the strategy in [5,6]. We substitute into (2.23) together with its conjugate the ansatz In fact, T is smooth on the weighted spaces Y r η (here we use the particular form of the ansatz) and we have…”

Grain boundaries in the Swift–Hohenberg equation

“…(see also [9] for further properties). In particular, the heteroclinic orbit (0, C * + , C * − ) is reversible.…”

“…The linear operator M r has been studied in detail in [9]. The results in [9,Section 4.2] show that M r is a Fredholm operator with index 0 in (L 2 ) 2 and has a one-dimensional kernel spanned by (C + , C − ).…”

“…The results in [9,Section 4.2] show that M r is a Fredholm operator with index 0 in (L 2 ) 2 and has a one-dimensional kernel spanned by (C + , C − ). Consequently, M r is a Fredholm operator with index 0 in X r η , for sufficiently small η > 0, with a trivial kernel, since (C + , C − ) does not belong to X r η .…”

An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects

“…the Swift-Hohenberg equation (1). It would be of significant interest to explore the connections with the considerable body of work on the rigorous bifurcation analysis of defects [31,49,37], that start directly with the microscopic evolution equations and use tools from dynamical systems theory/center manifold reductions and functional analysis to prove the existence of defects and characterize their mutual interactions.…”