Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi$ with periodic boundary conditions. Our result includes all values of $\theta$ and $\gamma$ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0 0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= \mu |x|^{-\frac {2} {\alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution

A Fujita-type blowup result and low energy scattering for a nonlinear Schr\"o\-din\-ger equation

In this paper we consider the nonlinear Schr\"o\-din\-ger equation $i u_t +\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac {2} {N}$ and $\Im \kappa <0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $\alpha >\frac {2} {N}$ and $\kappa \in {\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. More precisely, we prove that if $\frac {8} {N + \sqrt{ N^2 +16N }} < \alpha \le \frac {4} {N}$, then small data give rise to global, scattering solutions in $H^1$

Tropical Pacific spatial trend patterns in observed sea level: internal variability and/or anthropogenic signature?

In this study we focus on the sea level trend pattern observed by satellite altimetry in the tropical Pacific over the 1993–2009 time span (i.e. 17 yr). Our objective is to investigate whether this 17-yr-long trend pattern was different before the altimetry era, what was its spatio-temporal variability and what have been its main drivers. We try to discriminate the respective roles of the internal variability of the climate system and of external forcing factors, in particular anthropogenic emissions (greenhouse gases and aerosols). On the basis of a 2-D past sea level reconstruction over 1950–2009 (based on a combination of observations and ocean modelling) and multi-century control runs (i.e. with constant, preindustrial external forcing) from eight coupled climate models, we have investigated how the observed 17-yr sea level trend pattern evolved during the last decades and centuries, and try to estimate the characteristic time scales of its variability. For that purpose, we have computed sea level trend patterns over successive 17-yr windows (i.e. the length of the altimetry record), both for the 60-yr long reconstructed sea level and the model runs. We find that the 2-D sea level reconstruction shows spatial trend patterns similar to the one observed during the altimetry era. The pattern appears to have fluctuated with time with a characteristic time scale of the order of 25–30 yr. The same behaviour is found in multi-centennial control runs of the coupled climate models. A similar analysis is performed with 20th century coupled climate model runs with complete external forcing (i.e. solar plus volcanic variability and changes in anthropogenic forcing). Results suggest that in the tropical Pacific, sea level trend fluctuations are dominated by the internal variability of the ocean–atmosphere coupled system. While our analysis cannot rule out any influence of anthropogenic forcing, it concludes that the latter effect in that particular region is stillhardly detectable

On the propagation of an optical wave in a photorefractive medium

The aim of this paper is first to review the derivation of a model describing the propagation of an optical wave in a photorefractive medium and to present various mathematical results on this model: Cauchy problem, solitary waves

Predictable hydrodynamic conditions explain temporal variations in the density of benthic foraging seabirds in a tidal stream environment

Tidal stream turbines could have several direct impacts upon pursuit-diving seabirds foraging within tidal stream environments (mean horizontal current speeds > 2 ms−1), including collisions and displacement. Understanding how foraging seabirds respond to temporally variable but predictable hydrodynamic conditions immediately around devices could identify when interactions between seabirds and devices are most likely to occur; information which would quantify the magnitude of potential impacts, and also facilitate the development of suitable mitigation measures. This study uses shore-based observational surveys and Finite Volume Community Ocean Model outputs to test whether temporally predictable hydrodynamic conditions (horizontal current speeds, water elevation, turbulence) influenced the density of foraging black guillemots Cepphus grylle and European shags Phalacrocorax aristotelis in a tidal stream environment in Orkney, United Kingdom, during the breeding season. These species are particularly vulnerable to interactions with devices due to their tendency to exploit benthic and epi-benthic prey on or near the seabed. The density of both species decreased as a function of horizontal current speeds, whereas the density of black guillemots also decreased as a function of water elevation. These relationships could be linked to higher energetic costs of dives in particularly fast horizontal current speeds (>3 ms−1) and deeper water. Therefore, interactions between these species and moving components seem unlikely at particularly high horizontal current speeds. Combining this information, with that on the rotation rates of moving components at lower horizontal current speeds, could be used to assess collision risk in this site during breeding seasons. It is also likely that moderating any device operation during both lowest water elevation and lowest horizontal current speeds could reduce the risk of collisions for these species in this site during this season. The approaches used in this study could have useful applications within Environmental Impact Assessments, and should be considered when assessing and mitigating negative impacts from specific devices within development sites

Is anthropogenic sea level fingerprint already detectable in the Pacific ocean ?

Sea level rates up to three times the global mean rate are being observed in the western tropical Pacific since 1993 by satellite altimetry. From recently published studies, it is not yet clear whether the sea level spatial trend patterns of the Pacific Ocean observed by satellite altimetry are mostly due to internal climate variability or if some anthropogenic fingerprint is already detectable. A number of recent studies have shown that the removal of the signal corresponding to the Pacific Decadal Oscillation (PDO)/Interdecadal Pacific Oscillation (IPO) from the observed altimetry sea level data over 1993–2010/2012 results in some significant residual trend pattern in the western tropical Pacific. It has thus been suggested that the PDO/IPO-related internal climate variability alone cannot account for all of the observed trend patterns in the western tropical Pacific and that the residual signal could be the fingerprint of the anthropogenic forcing. In this study, we investigate if there is any other internal climate variability signal still present in the residual trend pattern after the removal of IPO contribution from the altimetry-based sea level over 1993–2013. We show that subtraction of the IPO contribution to sea level trends through the method of linear regression does not totally remove the internal variability, leaving significant signal related to the non-linear response of sea level to El Niño Southern Oscillation (ENSO). In addition, by making use of 21 CMIP5 coupled climate models, we study the contribution of external forcing to the Pacific Ocean regional sea level variability over 1993–2013, and show that according to climate models, externally forced and thereby the anthropogenic sea level fingerprint on regional sea level trends in the tropical Pacific is still too small to be observable by satellite altimetry

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^\theta$ as $\theta \to \pm \pi /2$. It turns out that $T_{max}^\theta$ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /2$ in the case where the solution of the limiting nonlinear Schr\"odinger equation blows up in finite time (respectively, is global).Comment: 22 page

Past terrestrial water storage (1980–2008) in the Amazon Basin reconstructed from GRACE and in situ river gauging data

Terrestrial water storage (TWS) composed of surface waters, soil moisture, groundwater and snow where appropriate, is a key element of global and continental water cycle. Since 2002, the Gravity Recovery and Climate Experiment (GRACE) space gravimetry mission provides a new tool to measure large-scale TWS variations. However, for the past few decades, direct estimate of TWS variability is accessible from hydrological modeling only. Here we propose a novel approach that combines GRACE-based TWS spatial patterns with multi-decadal-long in situ river level records, to reconstruct past 2-D TWS over a river basin. Results are presented for the Amazon Basin for the period 1980–2008, focusing on the interannual time scale. Results are compared with past TWS estimated by the global hydrological model ISBA-TRIP. Correlations between reconstructed past interannual TWS variability and known climate forcing modes over the region (e.g., El Niño-Southern Oscillation and Pacific Decadal Oscillation) are also estimated. This method offers new perspective for improving our knowledge of past interannual TWS in world river basins where natural climate variability (as opposed to direct anthropogenic forcing) drives TWS variations