15,749 research outputs found
Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: Matter-wave breathers
We demonstrate that stabilization of solitons of the multidimensional
Schrodinger equation with a cubic nonlinearity may be achieved by a suitable
periodic control of the nonlinear term. The effect of this control is to
stabilize the unstable solitary waves which belong to the frontier between
expanding and collapsing solutions and to provide an oscillating solitonic
structure, some sort of breather-type solution. We obtain precise conditions on
the control parameters to achieve the stabilization and compare our results
with accurate numerical simulations of the nonlinear Schrodinger equation.
Because of the application of these ideas to matter waves these solutions are
some sort of matter breathers
Localized solutions for the finite difference semi-discretization of the wave equation
We study the propagation properties of the solutions of the finite-difference
space semi-discrete wave equation on an uniform grid of the whole Euclidean
space. We provide a construction of high frequency wave packets that propagate
along the corresponding bi-characteristic rays of Geometric Optics with a group
velocity arbitrarily close to zero. Our analysis is motivated by control
theoretical issues. In particular, the continuous wave equation has the
so-called observability property: for a sufficiently large time, the total
energy of its solutions can be estimated in terms of the energy concentrated in
the exterior of a compact set. This fails to be true, uniformly on the
mesh-size parameter, for the semi-discrete schemes and the observability
constant blows-up at an arbitrarily large polynomial order. Our contribution
consists in providing a rigorous derivation of those wave packets and in
analyzing their behavior near that ray, by taking into account the subtle added
dispersive effects that the numerical scheme introduces.Comment: 7 pages, 1 figur
A finite element data assimilation method for the wave equation
We design a primal-dual stabilized finite element method for the numerical
approximation of a data assimilation problem subject to the acoustic wave
equation. For the forward problem, piecewise affine, continuous, finite element
functions are used for the approximation in space and backward differentiation
is used in time. Stabilizing terms are added on the discrete level. The design
of these terms is driven by numerical stability and the stability of the
continuous problem, with the objective of minimizing the computational error.
Error estimates are then derived that are optimal with respect to the
approximation properties of the numerical scheme and the stability properties
of the continuous problem. The effects of discretizing the (smooth) domain
boundary and other perturbations in data are included in the analysis.Comment: 23 page
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