11,628 research outputs found
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
Riesz potentials and nonlinear parabolic equations
The spatial gradient of solutions to nonlinear degenerate parabolic equations
can be pointwise estimated by the caloric Riesz potential of the right hand
side datum, exactly as in the case of the heat equation. Heat kernels type
estimates persist in the nonlinear cas
Functional Inequalities: New Perspectives and New Applications
This book is not meant to be another compendium of select inequalities, nor
does it claim to contain the latest or the slickest ways of proving them. This
project is rather an attempt at describing how most functional inequalities are
not merely the byproduct of ingenious guess work by a few wizards among us, but
are often manifestations of certain natural mathematical structures and
physical phenomena. Our main goal here is to show how this point of view leads
to "systematic" approaches for not just proving the most basic functional
inequalities, but also for understanding and improving them, and for devising
new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a
pre-publication pdf cop
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates
We consider the stability and dynamics of multiple dark solitons in
cigar-shaped Bose-Einstein condensates (BECs). Our study is motivated by the
fact that multiple matter-wave dark solitons may naturally form in such
settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First,
we study the dark soliton interactions and show that the dynamics of
well-separated solitons (i.e., ones that undergo a collision with relatively
low velocities) can be analyzed by means of particle-like equations of motion.
The latter take into regard the repulsion between solitons (via an effective
repulsive potential) and the confinement and dimensionality of the system (via
an effective parabolic trap for each soliton). Next, based on the fact that
stationary, well-separated dark multi-soliton states emerge as a nonlinear
continuation of the appropriate excited eigensates of the quantum harmonic
oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the
stability of such structures. We find that for a sufficiently large number of
atoms, multiple soliton states may be dynamically stable, while for a small
number of atoms, we predict a dynamical instability emerging from resonance
effects between the eigenfrequencies of the soliton modes and the intrinsic
excitation frequencies of the condensate. Finally we present experimental
realizations of multi-soliton states including a three-soliton state consisting
of two solitons oscillating around a stationary one.Comment: 17 pages, 11 figure
- …