Some breathers and multi-breathers for FPU-type chains

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are close to (possibly infinite) sums of breather solutions

Most stable structure for hard spheres

The hard sphere model is known to show a liquid-solid phase transition, with the solid expected to be either face centered cubic or hexagonal close packed. The difference in free energy between the two structures is very small and various attempts have been made to determine which one is the more stable. We contrast the different approaches and extend one.Comment: 5 pages, 1 embedded figure, to appear in Phys Rev

Some symmetric boundary value problems and non-symmetric solutions

We consider the equation −Delta u = wf ′(u) on a symmetric bounded domain in Rn with Dirichlet boundary conditions. Here w is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions u that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric solutions and the non-existence of symmetric solutions. In particular, we construct a solution u for the disk in R2 that has index 2 and whose modulus |u| has only one reflection symmetry. We also provide a corrected proof of [12, Theorem 1]