Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

Berg, Jan Bouwe van den; Deschênes, Andréa; Lessard, Jean‐Philippe; James, J. D. Mireles

doi:10.1137/140984506

Cited by 41 publications

(16 citation statements)

References 33 publications

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“…As said in the Introduction, this work is far from being the first application of this kind of rigorous computational techniques to solve systems of PDEs, see for instance [5,13,16,18,22,23]. Particularly related to our work is the result presented in [10].…”

Section: Remark 22mentioning

confidence: 91%

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

Breden

Castelli

2018

Journal of Differential Equations

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In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.

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Section: Remark 22mentioning

confidence: 91%

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

Breden

Castelli

2018

Journal of Differential Equations

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“…Estimate (49) comes from a meticulous but rather straightforward analysis of the various terms appearing in (48) for the columns of indices l > 4K − 2, using that Estimates similar to (49) were obtained previously in [68,72].…”

Section: The Bound Zmentioning

confidence: 99%

Rigorous validation of stochastic transition paths

Breden

Kuehn

2019

Journal de Mathématiques Pures et Appliquées

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Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical Müller-Brown test potential.

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“…Firstly, the existence of standing waves between rolls and hexagonal patterns of the two‐dimensional pattern formation PDE model is given in as the evolutionary equation

u_{t} = - {(1 + normal Δ)}^{2} u + μ u - β | \nabla u |^{2} - u^{3},

where u = u ( x , y , t ) and Δ is the two‐dimensional Laplacion. This is a generalization of the Swift–Hohenberg equation given in .…”

Section: Introductionmentioning

confidence: 99%

“…This is a generalization of the Swift–Hohenberg equation given in . The term β |∇ u | 2 contributes to a break in symmetry; other finer details of the model and parameters are spelt out in the references mentioned previously, particularly in .…”

Section: Introductionmentioning

confidence: 99%