TY - JOUR
T1 - Stationary Coexistence of Hexagons and Rolls via Rigorous Computations
AU - van den Berg, G.J.B.
AU - Deschênes, A.
AU - Lessard, J.P.
AU - Mireles-James, J.
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
UR - https://www.scopus.com/pages/publications/84937826727
UR - https://www.scopus.com/inward/citedby.url?scp=84937826727&partnerID=8YFLogxK
U2 - 10.1137/140984506
DO - 10.1137/140984506
M3 - Article
SN - 1536-0040
VL - 14
SP - 942
EP - 979
JO - SIAM Journal on Applied Dynamical Systems
JF - SIAM Journal on Applied Dynamical Systems
IS - 2
ER -