Orbital stability of periodic waves for the nonlinear Schroedinger equation

The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure

Global dynamic topography observations reveal limited influence of large-scale mantle flow

Convective circulation of the Earth’s mantle maintains some fraction of surface topography that varies with space and time. Most predictive models show that this dynamic topography has peak amplitudes of about ±2 km, dominated by wavelengths of 104 km. Here, we test these models against our comprehensive observational database of 2,120 spot measurements of dynamic topography that were determined by analysing oceanic seismic surveys. These accurate measurements have typical peak amplitudes of ±1 km and wavelengths of approximately 103 km, and are combined with limited continental constraints to generate a global spherical harmonic model, the robustness of which has been carefully tested and benchmarked. Our power spectral analysis reveals significant discrepancies between observed and predicted dynamic topography. At longer wavelengths (such as 104 km), observed dynamic topography has peak amplitudes of about ±500 m. At shorter wavelengths (such as 103 km), significant dynamic topography is still observed. We show that these discrepancies can be explained if short-wavelength dynamic topography is generated by temperature-driven density anomalies within a sub-plate asthenospheric channel. Stratigraphic observations from adjacent continental margins show that these dynamic topographic signals evolve quickly with time. More rapid temporal and spatial changes in vertical displacement of the Earth’s surface have direct consequences for fields as diverse as mantle flow, oceanic circulation and long-term climate change.This research was supported by a BP-Cambridge collaboration. We are grateful to ION for permission to publish partial seismic reflection profiles shown in Fig. 2 from their IndiaSPAN and Greater BrasilSPAN data sets