40 research outputs found
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 -bound for the local phase
shift of modulating line solitons for polynomially localized perturbations
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
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
Global well-posedness for the KP-I equation on the background of a non localized solution
We prove that the Cauchy problem for the KP-I equation is globally well-posed
for initial data which are localized perturbations (of arbitrary size) of a
non-localized (i.e. not decaying in all directions) traveling wave solution
(e.g. the KdV line solitary wave or the Zaitsev solitary waves which are
localized in and periodic or conversely)
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
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media
Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations (survey)
Analysis and Stochastic