37 research outputs found
A dimension-breaking phenomenon for water waves with weak surface tension
It is well known that the water-wave problem with weak surface tension has
small-amplitude line solitary-wave solutions which to leading order are
described by the nonlinear Schr\"odinger equation. The present paper contains
an existence theory for three-dimensional periodically modulated solitary-wave
solutions which have a solitary-wave profile in the direction of propagation
and are periodic in the transverse direction; they emanate from the line
solitary waves in a dimension-breaking bifurcation. In addition, it is shown
that the line solitary waves are linearly unstable to long-wavelength
transverse perturbations. The key to these results is a formulation of the
water wave problem as an evolutionary system in which the transverse horizontal
variable plays the role of time, a careful study of the purely imaginary
spectrum of the operator obtained by linearising the evolutionary system at a
line solitary wave, and an application of an infinite-dimensional version of
the classical Lyapunov centre theorem.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s00205-015-0941-
On scattering of solitons for the Klein-Gordon equation coupled to a particle
We establish the long time soliton asymptotics for the translation invariant
nonlinear system consisting of the Klein-Gordon equation coupled to a charged
relativistic particle. The coupled system has a six dimensional invariant
manifold of the soliton solutions. We show that in the large time approximation
any finite energy solution, with the initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution of
the free Klein-Gordon equation. It is assumed that the charge density satisfies
the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof
is based on an extension of the general strategy introduced by Soffer and
Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert
space onto the solitary manifold, modulation equations for the parameters of
the projection, and decay of the transversal component.Comment: 47 pages, 2 figure
Anomalous Dynamic Scaling in Locally-Conserved Coarsening of Fractal Clusters
We report two-dimensional phase-field simulations of locally-conserved
coarsening dynamics of random fractal clusters with fractal dimension D=1.7 and
1.5. The correlation function, cluster perimeter and solute mass are measured
as functions of time. Analyzing the correlation function dynamics, we identify
two different time-dependent length scales that exhibit power laws in time. The
exponents of these power laws are independent of D, one of them is apparently
the classic exponent 1/3. The solute mass versus time exhibits dynamic scaling
with a D-dependent exponent, in agreement with a simple scaling theory.Comment: 5 pages, 4 figure
Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode
An amplitude equation for an unstable mode in a collisionless plasma is
derived from the dynamics on the two-dimensional unstable manifold of the
equilibrium. The mode amplitude decouples from the phase due to the
spatial homogeneity of the equilibrium, and the resulting one-dimensional
dynamics is analyzed using an expansion in . As the linear growth rate
vanishes, the expansion coefficients diverge; a rescaling
of the mode amplitude absorbs these
singularities and reveals that the mode electric field exhibits trapping
scaling as . The dynamics for
depends only on the phase where is the derivative of the dielectric as
.Comment: 11 pages (Latex/RevTex), 2 figures available in hard copy from the
Author ([email protected]); paper accepted by Physical Review
Letter
Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the
shock wave case, we study stability of compressive, or "shock-like", boundary
layers of the isentropic compressible Navier-Stokes equations with gamma-law
pressure by a combination of asymptotic ODE estimates and numerical Evans
function computations. Our results indicate stability for gamma in the interval
[1, 3] for all compressive boundary-layers, independent of amplitude, save for
inflow layers in the characteristic limit (not treated). Expansive inflow
boundary-layers have been shown to be stable for all amplitudes by Matsumura
and Nishihara using energy estimates. Besides the parameter of amplitude
appearing in the shock case, the boundary-layer case features an additional
parameter measuring displacement of the background profile, which greatly
complicates the resulting case structure. Moreover, inflow boundary layers turn
out to have quite delicate stability in both large-displacement and
large-amplitude limits, necessitating the additional use of a mod-two stability
index studied earlier by Serre and Zumbrun in order to decide stability
The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations
We describe a basic framework for studying dynamic scaling that has roots in
dynamical systems and probability theory. Within this framework, we study
Smoluchowski's coagulation equation for the three simplest rate kernels
, and . In another work, we classified all self-similar
solutions and all universality classes (domains of attraction) for scaling
limits under weak convergence (Comm. Pure Appl. Math 57 (2004)1197-1232). Here
we add to this a complete description of the set of all limit points of
solutions modulo scaling (the scaling attractor) and the dynamics on this limit
set (the ultimate dynamics). The main tool is Bertoin's L\'{e}vy-Khintchine
representation formula for eternal solutions of Smoluchowski's equation (Adv.
Appl. Prob. 12 (2002) 547--64). This representation linearizes the dynamics on
the scaling attractor, revealing these dynamics to be conjugate to a continuous
dilation, and chaotic in a classical sense. Furthermore, our study of scaling
limits explains how Smoluchowski dynamics ``compactifies'' in a natural way
that accounts for clusters of zero and infinite size (dust and gel)
Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns
We analyze a recent experiment of Sharon \textit{et al.} (2003) on the
coarsening, due to surface tension, of fractal viscous fingering patterns
(FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw
model, a natural model for that experiment, belongs to the same universality
class as model B of phase ordering. Two series of numerical simulations with
model B are performed, with the FVFPs grown in the experiment, and with
Diffusion Limited Aggregates, as the initial conditions. We observed
Lifshitz-Slyozov scaling at intermediate distances and very slow
convergence to this scaling at small distances. Dynamic scale invariance breaks
down at large distances.Comment: 4 pages, 4 eps figures; to appear in Phys. Rev.
Effects of starch/polycaprolactone-based blends for spinal cord injury regeneration in neurons/glial cells viability and proliferation
Spinal cord injury (SCI) leads to drastic alterations on the quality of life of afflicted individuals. With the advent of Tissue Engineering and Regenerative Medicine where approaches combining biomaterials, cells and growth factors are used, one can envisage novel strategies that can adequately tackle this problem. The objective of this study was to evaluate a blend of starch with poly(Δ-caprolactone) (SPCL) aimed to be used for the development of scaffolds spinal cord injury (SCI) repair. SPCL linear parallel filaments were deposited on polystyrene coverslips and assays were carried out using primary cultures of hippocampal neurons and glial cells. Light and fluorescence microscopy observations revealed that both cell populations were not negatively affected by the SPCL-based biomaterial. MTS and total protein quantification indicated that both cell viability and proliferation rates were similar to controls. Both neurons and astrocytes occasionally contacted the surface of SPCL filaments through their dendrites and cytoplasmatic processes, respectively, while microglial cells were unable to do so. Using single cell [Ca2+ ]i imaging, hippocampal neurons were observed growing within the patterned channels and were functional as assessed by the response to a 30 mM KCl stimulus. The present data demonstrated that SPCL-based blends are potentially suitable for the development of scaffolds in SCI regenerative medicine.Portuguese Foundation for Science and Technology through funds from POCTI and/or FEDER programs (Funding to ICVS, 3B's Research Group and post doctoral fellowship to A.J. Salgado-SFRH/BPD/17595/2004)
Existence and stability of viscoelastic shock profiles
We investigate existence and stability of viscoelastic shock profiles for a
class of planar models including the incompressible shear case studied by
Antman and Malek-Madani. We establish that the resulting equations fall into
the class of symmetrizable hyperbolic--parabolic systems, hence spectral
stability implies linearized and nonlinear stability with sharp rates of decay.
The new contributions are treatment of the compressible case, formulation of a
rigorous nonlinear stability theory, including verification of stability of
small-amplitude Lax shocks, and the systematic incorporation in our
investigations of numerical Evans function computations determining stability
of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure
Asymptotic stability of solitary waves
We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case where f(u)=u p+1 / (p+1) , p =1, 2, 3 (and 30, with f â C 4 ). The same asymptotic stability result for KdV is also proved for the case p =2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values of p between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameter p increases beyond the critical value p =4.) The solution is decomposed into a modulating solitary wave, with time-varying speed c(t) and phase Îł( t ) ( bound state part ), and an infinite dimensional perturbation ( radiating part ). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. As p â4 â , the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46489/1/220_2005_Article_BF02101705.pd