Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

1991 Mathematics Subject Classification. Primary 37K55, 35B10, 35J10, 35Q40, 35Q55.In this paper, one-dimensional (1D) nonlinear Schrödinger equation [equation omitted] with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small-amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkho ff normal form reduction and an improved KAM method.Partially supported by NSF grant DMS0204119

Quasi-Periodic Breathers in Hamiltonian Networks of Long-Range Coupling

1991 Mathematics Subject Classification. Primary 37K60, 37K55This work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.The first author is partially supported by NSFC grant 10771098, 973 projects of China and NSFJP grant BK2007134. The third author is partially supported by NSFC grant 10428101 and NSF grants DMS0204119, DMS0708331

Characterisation of the relationship between surface texture and surface integrity of superalloy components machined by grinding

The surface texture of a machined component is influenced largely by the processing parameters used during machining and hence, there is a relationship between both the formation of the surface texture and surface integrity of the machined component. In the study to be reported in this paper, GH4169, a hard-to-cut superalloy, widely used in aero-engines, was selected for a detailed investigation into the relationship between the surface texture and the component-performance (surface integrity) of the machined components for which a series of grinding experiments with different grinding-wheels and grinding parameter-values was carried out in order to quantitatively analyze variations of the surface roughness with processing parameters. Further, considering that the features of the ground-surfaces measured are of a random nature, statistic properties of the produced surfaces were revealed and characterised with power spectral density function (PSD) and auto-covariance function(ACV) method respectively

Probing gravitational non-minimal coupling with dark energy surveys

We investigate observational constraints on a specific one-parameter extension to the minimal quintessence model, where the quintessence field acquires a quadratic coupling to the scalar curvature through a coupling constant $\xi$. The value of $\xi$ is highly suppressed in typical tracker models if the late-time cosmic acceleration is driven at some field values near the Planck scale. We test $\xi$ in a second class of models in which the field value today becomes a free model parameter. We use the combined data from type-Ia supernovae, cosmic microwave background, baryon acoustic oscillations and matter power spectrum, to weak lensing measurements and find a best-fit value $\xi > 0.289$ where $\xi = 0$ is excluded outside the 95 per cent confidence region. The effective gravitational constant $G_{\rm eff}$ subject to the hint of a non-zero $\xi$ is constrained to $-0.003 < 1- G_{\rm eff}/G < 0.033$ at the same confidence level on cosmological scales, and can be narrowed down to $1- G_{\rm eff}/G < 2.2 \times 10^{-5}$ when combining with Solar System tests.Comment: Context extended, figures and references added, title changed to match with accepted version for publicatio