We study a two-dimensional (2-D) potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution ...of the Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the ‘virtual’ fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time invariants associated with the topology of the conformal plane and Kelvin's theorem for a virtual fluid are explored. We called them ‘winding’ and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double exponential and supersedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with a free surface by unifying the problem of motion of vortex sheets, and the problem of 2-D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included.
We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are ...taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable
$w$
into the fluid’s area, with the real line of
$w$
mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov (J. Appl. Mech. Tech. Phys., vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions.
We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly ...under a special choice of boundary conditions at infinity. We assume that the fluid surface at infinity is unperturbed, while the velocity increase is proportional to distance and inversely proportional to time. This means that the fluid is compressed according to a self-similar law. We consider perturbations of a self-similarly compressible fluid and show that their evolution can be accurately described analytically after a conformal map of the fluid surface to the lower half-plane and the introduction of two arbitrary functions analytic in this half-plane. If one of these functions is equal to zero, then the solution can be written explicitly. In the general case, the solution appears to be a rapidly converging series whose terms can be calculated using recurrence relations.
An analytical expression is obtained for the total scattering cross-section in the discrete source method, which uses the representation of the far field pattern of the scattered field in terms of ...functions nonorthogonal on the unit sphere. The latter makes it possible to reduce the time of calculating the integral scattering cross-section by an order of magnitude in the numerical implementation. In addition, the use of the optical theorem and analytical expressions for the extinction cross-section permits one to calculate the absorption cross-section without the procedure of integrating the near field over the scatterer surface.
The synucleinopathies are a group of neurodegenerative diseases characterized by the oligomerization of alpha-synuclein protein in neurons or glial cells. Recent studies provide data that ceramide ...metabolism impairment may play a role in the pathogenesis of synucleinopathies due to its influence on alpha-synuclein accumulation. The aim of the current study was to assess changes in activities of enzymes involved in ceramide metabolism in patients with different synucleinopathies (Parkinson’s disease (PD), dementia with Lewy bodies (DLB), and multiple system atrophy (MSA)). The study enrolled 163 PD, 44 DLB, and 30 MSA patients as well as 159 controls. Glucocerebrosidase, alpha-galactosidase, acid sphingomyelinase enzyme activities, and concentrations of the corresponding substrates (hexosylsphingosine, globotriaosylsphingosine, lysosphingomyelin) were measured by liquid chromatography tandem-mass spectrometry in blood. Expression levels of
GBA
,
GLA
, and
SMPD1
genes encoding glucoceresobridase, alpha-galactosidase, and acid sphingomyelinase enzymes, correspondently, were analyzed by real-time PCR with TaqMan assay in CD45 + blood cells. Increased hexosylsphingosine concentration was observed in DLB and MSA patients in comparison to PD and controls (
p
< 0.001) and it was associated with earlier age at onset (AAO) of DLB (
p
= 0.0008).
SMPD1
expression was decreased in MSA compared to controls (
p
= 0.015). Acid sphingomyelinase activity was decreased in DLB, MSA patients compared to PD patients (
p
< 0.0001,
p
< 0.0001, respectively), and in MSA compared to controls (
p
< 0.0001). Lower acid sphingomyelinase activity was associated with earlier AAO of PD (
p
= 0.012). Our data support the role of lysosomal dysfunction in the pathogenesis of synucleinopathies, namely, the pronounced alterations of lysosomal activities involved in ceramide metabolism in patients with MSA and DLB.
We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and ...surface tension. A time-dependent conformal mapping
$z(w,t)$
of the lower complex half-plane of the variable
$w$
into the area filled with fluid is performed with the real line of
$w$
mapped into the free fluid’s surface. We study the dynamics of singularities of both
$z(w,t)$
and the complex fluid potential
$\unicodeSTIX{x1D6F1}(w,t)$
in the upper complex half-plane of
$w$
. We show the existence of solutions with an arbitrary finite number
$N$
of complex poles in
$z_{w}(w,t)$
and
$\unicodeSTIX{x1D6F1}_{w}(w,t)$
which are the derivatives of
$z(w,t)$
and
$\unicodeSTIX{x1D6F1}(w,t)$
over
$w$
. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of
$z_{w}(w,t)$
at these
$N$
points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations,
arXiv:1206.2046
) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of
$\unicodeSTIX{x1D6F1}_{w}(w,t)$
are also the constants of motion while non-zero gravity
$g$
ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both
$z_{w}(w,t)$
and
$\unicodeSTIX{x1D6F1}_{w}(w,t)$
at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is
$4N$
for zero gravity and
$4N-1$
for non-zero gravity. For the second-order poles we found
$6N$
motion integrals for zero gravity and
$6N-1$
for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.
—The paper inverstigates variation in the parameters of the complex amplitude vector of narrowband signals along ray trajectories as applied to short waves in a three-dimensionally inhomogeneous ...anisotropic ionosphere plasma. Numerical experiments are conducted, as well as analysis of the polarization and amplitude characteristics of ordinary and extraordinary waves depending on the position of the source, the carrier frequency of the signal, the angular coordinates of the radiation and geophysical conditions.
Generalized Primitive Potentials Zakharov, V. E.; Zakharov, D. V.
Doklady. Mathematics,
03/2020, Letnik:
101, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Recently, we introduced a new class of bounded potentials of the one-dimensional stationary Schrödinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These ...potentials, which we call primitive, are obtained as limits of rapidly decreasing reflectionless potentials, or multisoliton solutions of KdV. In this note, we introduce generalized primitive potentials, which are obtained as limits of all rapidly decreasing potentials of the Schrödinger operator. These potentials are constructed by solving a contour problem, and are determined by a pair of positive functions on a finite interval and a functional parameter on the real axis.
We report results of paleomagnetic studies of mafic dikes and sills from the Tas-Yuryakh magmatic complex on the Olenek uplift in the northeast of the Siberian platform. The paleomagnetic record in ...the rocks corresponds to an episode of anomalous state of the geomagnetic field that persisted from the Ediacaran period (~580 Ma and younger) to the end of the Fortunian age. Paleointensity measurements indicate an extremely low value of the virtual dipole moment during this time. This presumably caused a disruption of the normal Geocentric Axial Dipole model, so much so that the world magnetic anomalies made a substantial contribution. We propose that the Antarctic anomaly influenced the magnetization of the Siberian craton rocks during this period of very low dipole moment. The high latitudes corresponding to the observed paleopole do not correspond to the actual paleogeography of Siberia and can be used for paleoreconstructions only after adjusting for this anomaly. The true position of the Olenek uplift at the Precambrian–Paleozoic boundary was close to 30° S above the southeastern periphery of the African (Tuzo) mantle hot field.
•The Cu2−XSe-Sb2Se3 system phase diagram was revisited and reconstructed.•The samples of CuSbSe2 and Cu3SbSe3 annealed at 450 °C 4320 h are single-phase.•Cu3SbSe3 and CuSbSe2 melt incongruently, ...ΔН(Cu3SbSe3) = 67.6 J/g ΔН(CuSbSe2) = 85.1 J/g.•CuSb3Se5 is formed at 445 °C, ΔНf = 21.5 J/g, and decomposes at 527 °C, ΔНd = 36.7 J/g.•Thermal expansion anisotropy for CuSbSe2 and Sb2Se3 is similar to As2S3 and AsS.
The phase diagram of the Cu2−XSe-Sb2Se3 system is revisited to clarify ambiguity/disagreement in previously reported data. Ternary Cu3SbSe3 and CuSbSe2 compounds were obtained. In order to confirm that the phases have been identified correctly, crystal structures were solved, and the energy band gaps measured. For the sample containing 75 mol% Sb2Se3 and 25 mol% Cu1.995Se the temperature range of the stability of the high-temperature CuSb3Se5 phase was determined for the first time. This phase is formed at 445 °С, decomposes following a peritectic reaction at 527 °С, and can be quenched. A high-temperature X-ray diffraction study of a sample containing 75 mol% Sb2Se3 and 25 mol% Cu2Se allowed us to measure the thermal expansion of the CuSbSe2 and Sb2Se3 phases present in the sample. The anisotropy of thermal expansion of CuSbSe2 is similar to that of As2S3 (orpiment); thermal expansion of Sb2Se3 is similar to that of AsS (realgar). The 6 balance equations of the invariant phase transformations involving all the ternary compounds existing in the Cu2−XSe-Sb2Se3 system were suggested for the first time. The temperature and the enthalpies of all these transformations were measured. A phase diagram of the Cu2−XSe-Sb2Se3 system was found for the first time in all the range of concentrations at temperatures from ambient to the complete melting. This diagram takes into consideration the phase equilibria that involve all the ternary compounds that are possible in this system. The liquidus of the Cu2−XSe-Sb2Se3 system was calculated according to Redlich-Kister equation; it agrees with the experimental data within 1–17 °С.
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