Global bifurcation for asymptotically linear Schr\"odinger equations

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr\"odinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} The method is topological, based on recent developments of degree theory. We use the inversion $u\to v:= u/\Vert u\Vert_X^2$ in an appropriate Sobolev space $X=W^{2,p}({\mathbb R}^N)$, and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables (\lambda,v) \in {\mathbb R} \x X. This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions 'bifurcating from infinity'. We believe that, for the values of $\lambda$ covered by our bifurcation approach, the existence result we obtain for positive solutions of \eqref{1} is the most general so fa

Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.Comment: To appear in Comm. PD

On a functional satisfying a weak Palais-Smale condition

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.Comment: 18 page

Long time dynamics and coherent states in nonlinear wave equations

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to appear in Fields Institute Communications: Advances in Applied Mathematics, Modeling, and Computational Science 201

Multi-solitary waves for the nonlinear Klein-Gordon equation

International audienceWe consider the nonlinear Klein-Gordon equation in $\R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates

CH4 emission estimates from an active landfill site inferred from a combined approach of CFD modelling and in situ FTIR measurements

Globally, the waste sector contributes to nearly a fifth of anthropogenic methane emitted to the atmosphere and is the second largest source of methane in the UK. In recent years great improvements to reduce those emissions have been achieved by installation of methane recovery systems at landfill sites and subsequently methane emissions reported in national emission inventories have been reduced. Nevertheless, methane emissions of landfills remain uncertain and quantification of emission fluxes is essential to verify reported emission inventories and to monitor changes in emissions. Here we present a new approach for methane emission quantification from a complex source like a landfill site by applying a Computational Fluid Dynamics (CFD) model to calibrated in situ measurements of methane as part of a field campaign at a landfill site near Ipswich, UK, in August 2014. The methane distribution for different meteorological scenarios is calculated with the CFD model and compared to methane mole fractions measured by an in situ Fourier Transform Infrared (FTIR) spectrometer downwind of the prevailing wind direction. Assuming emissions only from the active site, a mean daytime flux of 0.83 mg m−2 s−1, corresponding to 53.26 kg h−1, was estimated. The addition of a secondary source area adjacent to the active site, where some methane hotspots were observed, improved the agreement between the simulated and measured methane distribution. As a result, the flux from the active site was reduced slightly to 0.71 mg m−2 s−1 (45.56 kg h−1), at the same time an additional flux of 0.32 mg m−2 s−1 (30.41 kg h−1) was found from the secondary source area. This highlights the capability of our method to distinguish between different emission areas of the landfill site, which can provide more detailed information about emission source apportionment compared to other methods deriving bulk emissions

Studies of the Gaviota Slide Offshore Southern California

We are engaged in a study of a seafloor landslide off the coast of Santa Barbara, California. A large scar there remains from the Goleta slide, a well studied feature (1.51 km3 of failed material) that likely failed several thousand years ago. A smaller neighboring feature, the Gaviota slide (0.02 km3 of failed material), was probably triggered during the 1812 Santa Barbara earthquake. Our investigations started in 2004 with a chirp sonar survey. The survey revealed a relationship between a “crack” in the sediment propagating from the Gaviota slide’s headwall and a thrust fault clearly seen in the subsurface layers. In the next phase of our project we are applying three new time-lapse seafloor geodetic techniques that vary in spatial and temporal resolution. One method uses optical fibers stretched and buried in the sediment to monitor creep. Each cable has an optical system that measures the absolute length of the stretched optical fiber with a precision of 1 mm every hour. The cables vary in length from 250 m to 750 m. A second system consists of an array of precise acoustic transponders on the seafloor interrogated by several buoyantly suspended command nodes. Offshore engineering tests of these reveal a precision of 5 mm over baselines up to 2 km. Finally, we are developing an AUV-borne precision mapping capability that promises to provide a monitor of seafloor shape changes that occur over tracklines of many kilometers in length with a precision goal of 10 cm. We are currently preparing these geodetic monitoring tools for deployment across a presumed future headwall near the Gaviota slide in a nested fashion to provide redundancy and a means to compare resolutions

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.Comment: 9 page