`Formation of singularities in natural
systems'
VIDI-grant from the Dutch Science Organisation NWO
Summary
The main theme of this proposal is singularity formation in natural
systems.
Singularities arise when nonlinear effects dominate the dispersive
ones, up to the formation of the singularity. Singularity formation,
also
called blowup, has received a considerable amount of
attention
in problems ranging from nonlinear optics, plasma physics and
combustion to hydrodynamics, and from stellar dynamics
to chemotaxis in bacteria.
We focus on projects that are motivated by concrete applications.
The equations that are used to model the applications can be divided
into two classes: amplitude equations and systems of reaction-diffusion
equations.
In the study of blowup solutions for these equations, we will
combine
various mathematical methods. The methods that will be used have been
developed during the last couple of years,
however, the techniques are much broader applicable.
In the study, we heavily rely on the combination of numerical,
asymptotical
and geometrical methods. One method does not stand on its own, the
interplay
between them is essential. Moreover, each method leads, apart from
conclusions
common to all of them, to results specific for each method,
contributing to
the overall picture.
The projects
We study singularity formation in amplitude equations and in systems of
reaction-diffusion equations where the focus lies on three different
projects:
Blowup in the Korteweg-de Vries equation
Most literature related to collapse in the Korteweg-de Vries equation
is devoted to
understanding the stability of solitary waves. These solitons
can become unstable and
blow up. Hardly
any studies handling the blowup
structures themselves are known for the Korteweg-de Vries equation.
Therefore, in this project, the focus lies on blowup
structures for the Korteweg-de Vries equation.
Blowup in systems of reaction-diffusion equations
We will study blowup solutions for a generalised system of
reaction-diffusion
equations that incorporates the Gierer-Meinhardt and other models.
Very recently, it has been observed that, for certain choices of the
parameters,
pulse solutions can become unstable in which a pulse starts to grow
without
bound and blowup. The aim of this project is to analyse this phenomenon
further.
Nonlinear Schrödinger equation with higher order dispersive terms
In this project, we address the question of blowup solutions for
a
Nonlinear Schrödinger equation to which higher order terms
describing
dispersive corrections are
added. Although the Nonlinear Schrödinger equation
describes the leading order dynamics of self-focusing,
it is seen
in experiments that as the pulse undergoes collapse, its intensity
eventually
becomes sufficiently large for higher order nonlinearities to halt
this collapse.
Hence, extra nonlinear terms must be added to the Nonlinear Schrödinger
equation
to give a more
accurate description of the behaviour near the singularity formation.
We take higher order dispersive terms into account to analyse whether
these terms indeed do arrest collapse or only postpone it as was
suggested
in the physics literature.
PhD- and Postdoc
positions
Within the VIDI-grant, a PhD- and two Postdoc positions are available
for
the three
projects. Applicants should have a background in (applied) analysis
and dynamical systems and should have knowledge of partial differential
equations,
asymptotic analysis (perturbation methods) and related topics.
Informal inquiries are welcome by email.
To apply please send a CV together with a letter of motivation by email.
Contact: vivi[at]math.leidenuniv.nl