Plantation Lullaby

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On the 2d Zakharov system with L^2 Schr\"odinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change

A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the solution to a linear Schr\"odinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solution $e^{it}Q(x)$, where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} < \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{it}Q(x)$, for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M[u]E[u] \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS

Plantation Lullaby

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The dynamics of the 3D radial NLS with the combined terms

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\"{o}dinger equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.Comment: 46page

Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ′\delta^\prime interaction

We determine and study the ground states of a focusing Schr\"odinger equation in dimension one with a power nonlinearity $|\psi|^{2\mu} \psi$ and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) $\delta^\prime$ interaction, located at the origin. The time-dependent problem turns out to be globally well posed in the subcritical regime, and locally well posed in the supercritical and critical regime in the appropriate energy space. The set of the (nonlinear) ground states is completely determined. For any value of the nonlinearity power, it exhibits a symmetry breaking bifurcation structure as a function of the frequency (i.e., the nonlinear eigenvalue) $\omega$. More precisely, there exists a critical value \om^* of the nonlinear eigenvalue \om, such that: if \om_0 < \om < \om^*, then there is a single ground state and it is an odd function; if \om > \om^* then there exist two non-symmetric ground states. We prove that before bifurcation (i.e., for \om < \om^*) and for any subcritical power, every ground state is orbitally stable. After bifurcation (\om =\om^*+0), ground states are stable if $\mu$ does not exceed a value $\mu^\star$ that lies between 2 and 2.5, and become unstable for $\mu > \mu^*$. Finally, for $\mu > 2$ and \om \gg \om^*, all ground states are unstable. The branch of odd ground states for \om \om^*, obtaining a family of orbitally unstable stationary states. Existence of ground states is proved by variational techniques, and the stability properties of stationary states are investigated by means of the Grillakis-Shatah-Strauss framework, where some non standard techniques have to be used to establish the needed properties of linearization operators.Comment: 46 pages, 5 figure

Capturing of organic carbon and nitrogen in eelgrass sediments of southern Scandinavia

The ability of seagrass meadows to filter nutrients and capture and store CO2 and nutrients in the form of organic carbon (OC) and nitrogen (N) in their sediments may help to mitigate local eutrophication as well as climate change via meadow restoration and protection. This study assesses OC and N sediment stocks (top 50 cm) and sequestration rates within Danish eelgrass meadows. At four locations, eelgrass-vegetated and nearby unvegetated plots were studied in protected and exposed areas. The average OC and N sediment 50 cm stocks were 2.6 ± 0.3 kg OC m − 2 and 0.23 ± 0.01 kg N m − 2, including vegetated and unvegetated plots. In general, OC and N stocks did not differ significantly between eelgrass meadows and unvegetated sediments. Lack of accumulation of excess 210Pb suggested sediment erosion or low rates of sediment accumulation at most sites. OC accumulation rates ranged from 6 to 134 g m − 2 yr − 1 and N from 0.7 to 14 g m − 2 yr − 1. Generalized additive models showed that ≥ 80 % of the variation in sediment OC and N stocks was explained by sediment grain size, organic matter source, and hydrodynamic exposure. Long cores, dated with 210Pb, showed declining OC and N densities toward present time, suggesting long-term declines in eelgrass OC and N pools. Estimates of potential nation-wide OC and N accumulation in eelgrass sediments show that they could annually capture up to 0.7 % ± 0.5 % of CO2 emissions and 6.9 % ± 5.2 % of the total terrestrial N load

On the nonlinear stability of mKdV breathers

A mathematical proof for the stability of mKdV breathers is announced. This proof involves the existence of a nonlinear equation satisfied by all breather profiles, and a new Lyapunov functional which controls the dynamics of small perturbations and instability modes. In order to construct such a functional, we work in a subspace of the energy one. However, our proof introduces new ideas in order to attack the corresponding stability problem in the energy space. Some remarks about the sine-Gordon case are also considered.Comment: 7 p

Region-specific drivers cause low organic carbon stocks and sequestration rates in the saltmarsh soils of southern Scandinavia

Saltmarshes are known for their ability to act as effective sinks of organic carbon (OC) and their protection and restoration could potentially slow down the pace of global warming. However, regional estimates of saltmarsh OC storage are often missing, including for the Nordic region. To address this knowledge gap, we assessed OC storage and accumulation rates in 17 saltmarshes distributed along the Danish coasts and investigated the main drivers of soil OC storage. Danish saltmarshes store a median of 10 kg OC m−2 (interquartile range, IQR: 13.5–7.6) in the top meter and sequester 31.5 g OC m−2 yr−1 (IQR: 41.6–15.7). In a global context, these values are comparatively low. Soils with abundant clay (&gt; 20%), older and stable saltmarshes in mesohaline settings, and with low proportion of algal organic material showed higher OC densities, stocks, and accumulation rates. Grazing led to significantly higher OC stocks than neighboring ungrazed locations, likely due to trampling modifying soil abiotic conditions (higher erosion-resistance and higher clay content) that slow carbon decay. Scaling up, Danish saltmarsh soils, comprising about 1% of the country's area, have the potential to yearly capture up to 0.1% of Denmark's annual consumption-based CO2 emissions. Our research expands the baseline data needed to advance blue carbon research and management in the Nordic region while highlighting the need for a more comprehensive approach to saltmarsh management that considers the full range of services of these ecosystems and does not only focus on climate benefits.</p

Nonlinear coherent states and Ehrenfest time for Schrodinger equation

We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.Comment: 27 page