Phase Separation Patterns from Directional Quenching

Monteiro, Rafael; Scheel, Arnd

doi:10.1007/s00332-017-9361-x

Cited by 18 publications

(58 citation statements)

References 23 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The instability may however be propagating at a slower speed than the quenching, such that unstable patterns can be observed in a large region in the wake of the quenching line. Similar formation of unstable, or merely metastable patterns in the wake of the quenching line has been observed in phase separation processes; see and references therein. It is worth noting that our perturbation analysis is insensitive to these instabilities, as the main non‐degeneracy assumption only relies on coperiodic stability of the periodic patterns.…”

Section: Applications and Discussionsupporting

confidence: 62%

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Goh

Scheel

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We study the effect of domain growth on the orientation of striped phases in a Swift–Hohenberg equation. Domain growth is encoded in a step‐like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern‐forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern‐forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill‐posed, infinite‐dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional‐analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling‐wave problem. We resolve the former difficulty using a farfield‐core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot‐strapping.

show abstract

Section: Applications and Discussionsupporting

confidence: 62%

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Goh

Scheel

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In light of theorem 4.3, solutions of the modified equation (37) have gradient-like behavior. Since the modified equation(37) locally coincides with the original equation(35), we conclude that for small values of µ, small amplitude solutions of (35) (with c ‰ 0) exhibit gradient-like behavior.…”

mentioning

confidence: 56%

“…Combining these calculations, we obtain Λ c,µ pA, Bq "´1 4 αA 4´1 4 κ 2 κ 2 0 B 2`o´`A2`| B|˘2¯`O´`|c|`|µ|˘`|A|`|B|˘2¯, as pA, B, c, µq Ñ p0, 0, 0, 0q. Interestingly, (53) is the only place in the computation of this expansion where we used that M 0 consists of solutions of (37). Furthermore, we did not need to compute the Taylor expansion of Ψ itself.…”

Section: A11 Constructing a Trapping Regionmentioning

confidence: 99%

Spatial Hamiltonian Identities for Nonlocally Coupled Systems

Bakker

Scheel

2018

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.Differential equations of gradient form u t "´∇Epuq or of Hamiltonian form u t " J∇Epuq arise throughout mathematical modeling from maximal energy dissipation first principles, or from least action principles, respectively. Mathematically, the energy E is simply a function defined over a finite or infinite-dimensional space. The associated gradient and Hamiltonian flows are differential equations with equilibria given by the critical points of the energy. Energies over infinite-dimensional spaces are usually defined on function spaces through integrals of nonlinear functions of state variables and their derivatives. Critical points then solve Euler-Lagrange equations, commonly of elliptic type. Dependence of the energy on derivatives of the state variables encodes local interactions, often derived in various types of continuum limits. We are interested here in cases where this continuum limit retains nonlocal interaction terms. More specifically, we are interested in the somewhat specific class of energies E that contain nonlocal interaction term of the formmodeling phenomena such as long-range interactions of agents in social interaction, between particles through nonlocal force fields, or of neurons, labeled in a feature space x through synaptic connections. In all those cases, the convolution structure embodies the modeling assumption of translational invariance of physical space. We are

show abstract

“…The most physically relevant scenario to be considered consists of the case c 0. According to results in [MS17b], whenever the speed c of the quenching front is small the equation (1.2) admits a rich family of patterns , as we now briefly describe. Horizontal patterns: H κ , π < κ ∞.…”

Section: Introductionmentioning

confidence: 93%

Horizontal patterns from finite speed directional quenching

Monteiro¹

2018

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

View full text Add to dashboard Cite

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in our case the interface moves with speed c in such a way that the bistable region grows. According to results from [MS17a, MS17b], several patterns exist when c 0, and here we investigate their persistence for finite c > 0, clarifying the pattern selection mechanism related to the speed c of the quenching front.

show abstract

Phase Separation Patterns from Directional Quenching

Cited by 18 publications

References 23 publications

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Spatial Hamiltonian Identities for Nonlocally Coupled Systems

Horizontal patterns from finite speed directional quenching

Contact Info

Product

Resources

About