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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
of the lower complex half-plane of the variable
into the area filled with fluid is performed with the real line of
mapped into the free fluid’s surface. We study the dynamics of singularities of both
and the complex fluid potential
in the upper complex half-plane of
. We show the existence of solutions with an arbitrary finite number
of complex poles in
which are the derivatives of
. 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
points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results.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
are also the constants of motion while non-zero gravity
ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both
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
for zero gravity and
for non-zero gravity. For the second-order poles we found
motion integrals for zero gravity and
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.
This article considers how legal systems capture different cultural perceptions of work in an individual’s life. We inquire how two models—“human capital,” based on the works of Adam Smith; and “vocation,” based on the works of G. W. F. Hegel—are reflected in legal regulations and judicial rhetoric in the United States and Germany. Specifically, we examine how these two legal systems treat the practice of using personal names—the most direct referents to individuals’ identities—in business. We discuss three sets of cases: cases involving the use of personal names as trademarks, cases involving conflicts between parties with similar names, and cases involving the transfer of rights in personal names. The article demonstrates that the US legal system treats work as a commercial asset, as “human capital” in Smith’s sense, whereas German law perceives work as an integral part of one’s identity, echoing the Hegelian line of “vocation.”
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
into the fluid’s area, with the real line of
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.
A complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluids. Resonance three-wave interaction processes – decay instability and confluence of two waves – are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach to inertial wave theory are investigated and discussed.
Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (Dysthe Proc. R. Soc. Lond. A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.
The results of an initial technical design of the AIST spacecraft and payload are presented. The spacecraft design is subjected to the main requirement: to supply the smooth rotation. Two telescopes are symmetrically placed around central beam combiner unit. The fine adjustment of the light pressure centre position by means of special-purpose sails is proposed. Scientific objectives of the project are the implementation of astrometric survey of the sky and obtaining a series of star position catalogues with milliarcsecond accuracy, including a second epoch of Hipparcos/Tycho catalogues. The survey list will consist of nearly 10–15 million stars brighter than 15th magnitude.
Observations of Seyfert galaxies in X-ray region reveal the wide emissive lines in their spectra, which can arise in inner parts of accretion disks, where the effects of General Relativity (GR) must be taken into account. A spectrum of a solitary emission line of a hot spot in Kerr accretion disk is simulated, depending on the radial coordinate r and the angular momentum a = J/M of a black hole, under the assumption of equatorial circular motion of a hot spot. It is shown that the characteristic two-peak line profile with the sharp edges arises at a large distance, (about r ≈ (3 − 10) rg). The inner regions emit the line, which is observed with one maximum and extremely wide red wing. We present results of simulations for the isothermal and Shakura – Sunayev disks.
The first results of observations of microlensing have discovered a phenomenon, predicted in the papers of Byalko (1969) and Paczynsky (1986). A character of the gravitational microlens is unknown till now, although the most widespread hypothesis assumes that they are compact dark objects as brown dwarfs. Nevertheless, they could be presented by another objects, in particular, an existence of the dark objects consisting of the super symmetrical weakly interacting particles (neutralino) has been recently discussed in the papers of Gurevich et al. We consider microlensing by a neutralino star in framework of a rough model which is rather clear and we obtain analytical expressions for results. We approximate the density of distribution mass of a neutralino star in formwhere r is the current value of a distant from star's center, ρo is mass density of a neutralino star for distance a0 from a center, an is a “radius” neutralino star. The detailed analysis of the model is presented in the papers of Zakharov & Sazhin (1996a, 1996b). This research has been supported in part by Russian Foundation of Fundamental Research (grant N 96-02-17434).
A radiochemical 71Ga−71 Ge experiment to determine the integral flux of neutrinos from the sun has been constructed at the Baksan Neutrino Observatory in the USSR. Measurements have begun with 30 tonnes of gallium. The experiment is being expanded with the addition of another 30 tonnes. The motivation, experimental procedures, and present status of this experiment are presented.