Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

# Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

This is a companion web site for the paper by Neuberger, Sieben and Swift, published in the SIAM J. Applied Dynamical systems, vol. 12, No. 3, pp. 1237-1279.
The final version is here, and the paper is also available as a preprint at the arXiv.

There are two parallel web sites. To switch between static and interactive options come back to this “PDE on Cube Home” page using the link at the top of each page, and click the other option.

static: figures that load quicker, or
interactive: 3D figures that you can rotate with the mouse.

The “static” and “interactive” links above take you to a page that lists all symmetry types at once. From there you can click a link to a separate page for each of the 99 symmetry types. You can also jump from the page for one symmetry type to the page for another. The “symmetry types” link on the second line of each page takes you to the page with all of the symmetry types.

The page for each symmetry type shows a flag diagram (either static or interactive) for a function with this symmetry type, and possible generic bifurcations with symmetry between these symmetry types. If we have actually found a solution with this symmetry type then a contour map of the solution is shown. (Sadly, the contour maps cannot rotate interactively, only the flag diagrams.)

This web site uses S4 to denote the 24-element symmetric group. The Octahedral group O and the full symmetry group of the tetrahedron Td are both isomorphic to S4, as indicated in Figure 3 of the preprint. Both O and Td are written as S4 in the web site. Similarly, the web site uses A4 to denote the 12-element alternating group. Figure 3 in the preprint.

The computations were done using MPQueue, a package for simplified parallel computing. The paper has appeared in print here, and the preprint is here.
The source code is freely available at our web site.