`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