
Stationary coexistence
of hexagons and rolls via rigorous computations
J.B. van den Berg, A.
Deschênes, J.P. Lessard and J.D. Mireles James
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 twodimensional 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 (1), 85110, 2003] about the coexistence
of hexagons and rolls.
