2,145 research outputs found
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]
Traveling wave solutions for the FPU chain: a constructive approach
Traveling waves for the FPU chain are constructed by solving the associated
equation for the spatial profile of the wave. We consider solutions whose
derivatives need not be small, may change sign several times, but decrease
at least exponentially. Our method of proof is computer-assisted. Unlike other
methods, it does not require that the FPU potential has an attractive
(positive) quadratic term. But we currently need to restrict the size of that
term. In particular, our solutions in the attractive case are all supersonic
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