Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
1995-09-26
Nonlinear Sciences
Pattern Formation and Solitons
62 pages (with figures), uuencoded, compressed postscript file
Scientific paper
We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.
Dionne B.
Silber Mary
Skeldon Anne C.
No associations
LandOfFree
Stability Results for Steady, Spatially--Periodic Planforms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Stability Results for Steady, Spatially--Periodic Planforms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stability Results for Steady, Spatially--Periodic Planforms will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-443250