Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

G.J.B. van den Berg, A. Deschênes, J.P. Lessard, J. Mireles-James

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

In this work we introduce a rigorous computational method for finding heteroclinic solutions of a system of two second order differential equations. These solutions correspond to standing waves between rolls and hexagonal patterns of a two-dimensional pattern formation PDE model. After reformulating the problem as a projected boundary value problem (BVP) with boundaries in the stable/unstable manifolds, we compute the local manifolds using the parameterization method and solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85-110] about the coexistence of hexagons and rolls.
Original languageEnglish
Pages (from-to)942-979
JournalSIAM Journal on Applied Dynamical Systems
Volume14
Issue number2
DOIs
Publication statusPublished - 2015

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