Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

A new phase in the production of quality-controlled sea level data

Sea level is an essential climate variable (ECV) that has a direct effect on many people through inundations of coastal areas, and it is also a clear indicator of climate changes due to external forcing factors and internal climate variability. Regional patterns of sea level change inform us on ocean circulation variations in response to natural climate modes such as El Niño and the Pacific Decadal Oscillation, and anthropogenic forcing. Comparing numerical climate models to a consistent set of observations enables us to assess the performance of these models and help us to understand and predict these phenomena, and thereby alleviate some of the environmental conditions associated with them. All such studies rely on the existence of long-term consistent high-accuracy datasets of sea level. The Climate Change Initiative (CCI) of the European Space Agency was established in 2010 to provide improved time series of some ECVs, including sea level, with the purpose of providing such data openly to all to enable the widest possible utilisation of such data. Now in its second phase, the Sea Level CCI project (SL_cci) merges data from nine different altimeter missions in a clear, consistent and well-documented manner, selecting the most appropriate satellite orbits and geophysical corrections in order to further reduce the error budget. This paper summarises the corrections required, the provenance of corrections and the evaluation of options that have been adopted for the recently released v2.0 dataset (https://doi.org/10.5270/esa-sea_level_cci-1993_2015-v_2.0-201612). This information enables scientists and other users to clearly understand which corrections have been applied and their effects on the sea level dataset. The overall result of these changes is that the rate of rise of global mean sea level (GMSL) still equates to ∼ 3.2 mm yr−1 during 1992–2015, but there is now greater confidence in this result as the errors associated with several of the corrections have been reduced. Compared with v1.1 of the SL_cci dataset, the new rate of change is 0.2 mm yr−1 less during 1993 to 2001 and 0.2 mm yr−1 higher during 2002 to 2014. Application of new correction models brought a reduction of altimeter crossover variances for most corrections

Excitation Thresholds for Nonlinear Localized Modes on Lattices

Breathers are spatially localized and time periodic solutions of extended Hamiltonian dynamical systems. In this paper we study excitation thresholds for (nonlinearly dynamically stable) ground state breather or standing wave solutions for networks of coupled nonlinear oscillators and wave equations of nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously characterized by variational methods. The excitation threshold is related to the optimal (best) constant in a class of discr ete interpolation inequalities related to the Hamiltonian energy. We establish a precise connection among $d$, the dimensionality of the lattice, $2\sigma+1$, the degree of the nonlinearity and the existence of an excitation threshold for discrete nonlinear Schr\"odinger systems (DNLS). We prove that if $\sigma\ge 2/d$, then ground state standing waves exist if and only if the total power is larger than some strictly positive threshold, $\nu_{thresh}(\sigma, d)$. This proves a conjecture of Flach, Kaldko& MacKay in the context of DNLS. We also discuss upper and lower bounds for excitation thresholds for ground states of coupled systems of NLS equations, which arise in the modeling of pulse propagation in coupled arrays of optical fibers.Comment: To appear in Nonlinearit

Collapse of an Instanton

We construct a two parameter family of collapsing solutions to the 4+1 Yang-Mills equations and derive the dynamical law of the collapse. Our arguments indicate that this family of solutions is stable. The latter fact is also supported by numerical simulations.Comment: 17 pages, 1 figur

Traveling waves for nonlinear Schr\"odinger equations with nonzero conditions at infinity, II

We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schr\"odinger equations with nonzero conditions at infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic" equations) in space dimension $N \geq 2$. We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy.Comment: Final version, accepted for publication in the {\it Archive for Rational Mechanics and Analysis.} The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-017-1131-

An Improved and Homogeneous Altimeter Sea Level Record from the ESA Climate Change Initiative

Sea Level is a very sensitive index of climate change since it integrates the impacts of ocean warming and ice mass loss from glaciers and the ice sheets. Sea Level has been listed as an Essential Climate Variable (ECV) by the Global Climate Observing System (GCOS). During the past 25 years, the sea level ECV has been measured from space by different altimetry missions that have provided global and regional observations of sea level variations. As part of the Climate Change Initiative (CCI) program of the European Space Agency (ESA) (established in 2010), the Sea Level project (SL_cci) aimed at providing an accurate and homogeneous long-term satellite-based sea level record. At the end of the first phase of the project (2010-2013), an initial version (v1.1) of the sea level ECV has been made available to users (Ablain et al., 2015). During the second phase (2014-2017), improved altimeter standards have been selected to produce new sea level products (called SL_cci v2.0) based on 9 altimeter missions for the period 1993-2015 (https://doi.org/10.5270/esa-sea_level_cci-1993_2015-v_2.0-201612). Corresponding orbit solutions, geophysical corrections and altimeter standards used in this v2.0 dataset are described in details in Quartly et al. (2017). The present paper focuses on the description of the SL_cci v2.0 ECV and associated uncertainty and discusses how it has been validated. Various approaches have been used for the quality assessment such as internal validation, comparisons with sea level records from other groups and with in-situ measurements, sea level budget closure analyses and comparisons with model outputs. Compared to the previous version of the sea level ECV, we show that use of improved geophysical corrections, careful bias reduction between missions and inclusion of new altimeter missions lead to improved sea level products with reduced uncertainties at different spatial and temporal scales. However, there is still room for improvement since the uncertainties remain larger than the GCOS requirements. Perspectives for subsequent evolutions are also discussed

Stable directions for small nonlinear Dirac standing waves

We prove that for a Dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues close enough, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schr\"{o}dinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation.Comment: 62 page

An effective mass theorem for the bidimensional electron gas in a strong magnetic field

We study the limiting behavior of a singularly perturbed Schr\"odinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane $(x,y)$ and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schr\"odinger equations in the plane $(x,y)$, coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator

Continental mass change from GRACE over 2002-2011 and its impact on sea level

Present-day continental mass variation as observed by space gravimetry reveals secular mass decline and accumulation. Whereas the former contributes to sea-level rise, the latter results in sea-level fall. As such, consideration of mass accumulation (rather than focussing solely on mass loss) is important for reliable overall estimates of sea-level change. Using data from the Gravity Recovery And Climate Experiment satellite mission, we quantify mass-change trends in 19 continental areas that exhibit a dominant signal. The integrated mass change within these regions is representative of the variation over the whole land areas. During the integer 9-year period of May 2002 to April 2011, GIA-adjusted mass gain and mass loss in these areas contributed, on average, to −(0.7 ± 0.4) mm/year of sea-level fall and + (1.8 ± 0.2) mm/year of sea-level rise; the net effect was + (1.1 ± 0.6) mm/year. Ice melting over Greenland, Iceland, Svalbard, the Canadian Arctic archipelago, Antarctica, Alaska and Patagonia was responsible for + (1.4±0.2) mm/year of the total balance. Hence, land-water mass accumulation compensated about 20 % of the impact of ice-melt water influx to the oceans. In order to assess the impact of geocentre motion, we converted geocentre coordinates derived from satellite laser ranging (SLR) to degree-one geopotential coefficients. We found geocentre motion to introduce small biases to mass-change and sea-level change estimates; its overall effect is + (0.1 ± 0.1) mm/year. This value, however, should be taken with care owing to questionable reliability of secular trends in SLR-derived geocentre coordinates

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model