589 research outputs found
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
(finite energy) solutions to the focusing 3d cubic nonlinear
Schr\"odinger equation (NLS) scatter,
i.e. approach the solution to a linear Schr\"odinger equation as . The criteria is expressed in terms of the scale-invariant quantities
and , where denotes the
initial data, and and denote the (conserved in time) mass and
energy of the corresponding solution . The focusing NLS possesses a
soliton solution , where is the ground-state solution to a
nonlinear elliptic equation, and we prove that if and
, then the
solution is globally well-posed and scatters. This condition is sharp in
the sense that the soliton solution , for which equality in these
conditions is obtained, is global but does not scatter. We further show that if
, 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
Illustration of birds and flowershttps://scholarsjunction.msstate.edu/cht-sheet-music/12326/thumbnail.jp
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 . The threshold is given by the
ground state for the energy-critical NLS: . 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, -subcritical perturbation 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 interaction
We determine and study the ground states of a focusing Schr\"odinger equation
in dimension one with a power nonlinearity and a strong
inhomogeneity represented by a singular point perturbation, the so-called
(attractive) 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) . 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 does not exceed a value that lies
between 2 and 2.5, and become unstable for . Finally, for 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 (> 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
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