A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form. We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature

Experimental demonstration of the supersonic-subsonic bifurcation in the circular jump: A hydrodynamic white hole

We provide an experimental demonstration that the circular hydraulic jump represents a hydrodynamic white hole or gravitational fountain (the time-reverse of a black hole) by measuring the angle of the Mach cone created by an object in the "supersonic" inner flow region. We emphasise the general character of this gravitational analogy by showing theoretically that the white hole horizon constitutes a stationary and spatial saddle-node bifurcation within dynamical-systems theory. We also demonstrate that the inner region has a "superluminal" dispersion relation, i.e., that the group velocity of the surface waves increases with frequency, and discuss some possible consequences with respect to the robustness of Hawking radiation. Finally, we point out that our experiment shows a concrete example of a possible "transplanckian distortion" of black/white holes.Comment: 5 pages, 5 figures. New "transplanckian effect" described. Several clarifications, additional figures and references. Published versio

Transverse instability of gravity-capillary line solitary water waves

The gravity-capillary water-wave problem concerns the irrotational flow of a perfect fluid in a domain bounded below by a rigid bottom and above by a free surface under the influence of gravity and surface tension. In the case of large surface tension the system has a travelling line solitary-wave solution for which the free surface has a localised profile in the direction of propagation and is homogeneous in the transverse direction. In this note we show that this line solitary wave is linearly unstable under spatially inhomogeneous perturbations which are periodic in the direction transverse to propagation

State-of-the-art report: Intergenerational linkages in families

__Abstract__ We present a state-of-the-art of the literature on linkages between generations within families. We focus specifically on intergenerational coresidence, upward and downward intergenerational transfers in families and the relationship between norms of family obligation and intergenerational transfers. An overview of the academic literature on these topics is provided, as well as suggestions for future research

Normalizations with exponentially small remainders for nonautonomous analytic periodic vector fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case

A new device used in the restoration of kinematics after total facet arthroplasty

Facet degeneration can lead to spinal stenosis and instability, and often requires stabilization. Interbody fusion is commonly performed, but it can lead to adjacent-segment disease. Dynamic posterior stabilization was performed using a total facet arthroplasty system. The total facet arthroplasty system was originally intended to restore the natural motion of the posterior stabilizers, but follow-up studies are lacking due to limited clinical use. We studied the first 14 cases (long-term follow-up) treated with this new device in our clinic. All patients were diagnosed with lumbar stenosis due to hypertrophy of the articular facets on one to three levels (maximum). Disk space was of normal height. The design of this implant allows its use only at levels L3-L4 and L4-L5. We implanted nine patients at the L4-L5 level and four patients at level L3-L4. Postoperative follow-up of the patients was obtained for an average of 3.7 years. All patients reported persistent improvement of symptoms, visual analog scale score, and Oswestry Disability Index score. Functional scores and dynamic radiographic imaging demonstrated the functional efficacy of this new implant, which represents an alternative technique and a new approach to dynamic stabilization of the vertebral column after interventions for spine decompression. The total facet arthroplasty system represents a viable option for dynamic posterior stabilization after spinal decompression. For the observed follow-up, it preserved motion without significant complications or apparent intradisk or adjacent-disk degeneration. © 2014 Vermesan et al

The phase shift of line solitons for the KP-II equation

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations

Global Hopf bifurcation in the ZIP regulatory system

Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells.Comment: 22 pages, 5 figures, uses svjour3.cl

Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed