We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion qn = V (q n+1 − qn) − V (qn − q n−1 ) with generic nearest-neighbour potential V . We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength −3 suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of [12] to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.

We consider atomic chains with nonlocal particle interactions and prove the existence of nearsonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.

No abstract

This Rapid Communication reports on Mössbauer-spectroscopy measurements of the kinetics of the -to-␣ phase transition in the system Fe-Cr. By isothermally annealing samples of the compositions Fe 53.8 Cr 46.2 and Fe 51 Cr 49 at temperatures between 820 and 855°C the authors gained information about the kinetics of the phase transition in terms of the Johnson-Mehl-Avrami-Kolmogorov equation. The obtained values for the Avrami exponent allow to draw conclusions about the type of nucleation mechanism. The behavior of the Avrami exponent for near-critical temperatures and for higher temperatures indicates a change in the nucleation mechanism with T-T c .

Abstract. We investigate trace formulas for one-dimensional Schrödinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg-de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface.