59 research outputs found
On a deformation of the nonlinear Schr\"odinger equation
We study a deformation of the nonlinear Schr\"odinger equation recently
derived in the context of deformation of hierarchies of integrable systems.
This systematic method also led to known integrable equations such as the
Camassa-Holm equation. Although this new equation has not been shown to be
completely integrable, its solitary wave solutions exhibit typical soliton
behaviour, including near elastic collisions. We will first focus on standing
wave solutions, which can be smooth or peaked, then, with the help of numerical
simulations, we will study solitary waves, their interactions and finally rogue
waves in the modulational instability regime. Interestingly the structure of
the solution during the collision of solitary waves or during the rogue wave
events are sharper and have larger amplitudes than in the classical NLS
equation
On a Lagrangian reduction and a deformation of completely integrable systems
We develop a theory of Lagrangian reduction on loop groups for completely
integrable systems after having exchanged the role of the space and time
variables in the multi-time interpretation of integrable hierarchies. We then
insert the Sobolev norm in the Lagrangian and derive a deformation of the
corresponding hierarchies. The integrability of the deformed equations is
altered and a notion of weak integrability is introduced. We implement this
scheme in the AKNS and SO(3) hierarchies and obtain known and new equations.
Among them we found two important equations, the Camassa-Holm equation, viewed
as a deformation of the KdV equation, and a deformation of the NLS equation
Covariant un-reduction for curve matching
The process of un-reduction, a sort of reversal of reduction by the Lie group
symmetries of a variational problem, is explored in the setting of field
theories. This process is applied to the problem of curve matching in the
plane, when the curves depend on more than one independent variable. This
situation occurs in a variety of instances such as matching of surfaces or
comparison of evolution between species. A discussion of the appropriate
Lagrangian involved in the variational principle is given, as well as some
initial numerical investigations.Comment: Conference paper for MFCA201
Bridge Simulation and Metric Estimation on Landmark Manifolds
We present an inference algorithm and connected Monte Carlo based estimation
procedures for metric estimation from landmark configurations distributed
according to the transition distribution of a Riemannian Brownian motion
arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric.
The distribution possesses properties similar to the regular Euclidean normal
distribution but its transition density is governed by a high-dimensional PDE
with no closed-form solution in the nonlinear case. We show how the density can
be numerically approximated by Monte Carlo sampling of conditioned Brownian
bridges, and we use this to estimate parameters of the LDDMM kernel and thus
the metric structure by maximum likelihood
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