SIAM Journal on Applied Dynamical Systems · 2015 · 52 citations · 31 references
Natural SciencesMathematicsPhysicsComputational PhysicsDynamical SystemsPattern FormationBifurcation TheoryBoundary ConditionsBoundary-value ProblemsNumerical Methods For Differential EquationsStability TheoryBoundary ProblemBoundary Value Problem SolversChebyshev PolynomialsHeteroclinic BifurcationHeteroclinic OrbitsSecond-order Differential EquationsStable ManifoldsUnstable Manifolds
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.
31