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I work in an area of mathematics called **integrable systems**.
Very roughly speaking, an integrable system is a system of
differential equations that can be solved exactly with analytical
methods.

For example, the Kepler problem of calculating the orbit of a planet around the sun (ignoring the influence of other planets) is an integrable system, while the three-body problem is not.

In my Ph.D. thesis (2001) I studied **cofactor systems**,
a class of integrable systems related to classical mechanics.
My supervisor was Stefan Rauch,
and I also worked with Krzysztof Marciniak.

I spent the academic year 2002–2003 as a postdoc
with Jacek Szmigielski
at the University of Saskatchewan, Saskatoon, Canada,
and we have continued working together since then.
We investigate **peakons**
(peaked solitons);
this is a particular type of weak solution admitted by
some integrable nonlinear wave equations, the most well-known of which are the
**Camassa–Holm shallow water equation**
and the
**Degasperis–Procesi equation**.
Although peakons were introduced in the context of water wave theory, the
methods we use to find and to analyze peakon solutions come from fields which might seem like
"pure" mathematics, such as Stieltjes continued fractions,
Padé approximation, (bi-)orthogonal polynomials, and even the
Chaundy–Karlin–McGregor–Lindström–Gessel–Viennot path counting
lemma from combinatorics.

Another interesting integrable peakon equation,
with cubic nonlinearities instead of quadratic,
was found by
Vladimir Novikov:
*m*_{t}+(*m*_{x}*u*+3*m**u*_{x})*u* = 0, where *m* = *u*−*u*_{xx}.
This equation is in a curious way "dual" to the Degasperis–Procesi equation
on the level of the associated spectral problems;
see my joint paper with Jacek Szmigielski and
Andy Hone.

My research activity has mostly been concentrated
on questions related to peakons.
During the period 2008–2010, I was funded by a grant from
the Swedish Research Council (Vetenskapsrådet)
for the project
*Integrable systems, in particular the Degasperis–Procesi equation*
(details),
and 2011–2013 I had a grant for the project
*Mathematical questions related to peaked solitons*
(details).

I also have a little project "on the side" together with Alain Albouy at the IMCCE (a division of the Paris Observatory), where we are trying to understand cofactor systems from the point of view of "projective dynamics". The results are quite interesting, but nothing is written down yet. Stay tuned!

(*Projective dynamics* is to Newtonian dynamics what projective
geometry is to Euclidean geometry; one studies dynamics of lines
through the origin under the influence of a force field in an
(*n*+1)-dimensional space, and the Newtonian dynamics is
recovered as the dynamics of the point of intersection between the
moving line and a fixed *n*-dimensional affine hyperplane. It
turns out, for example, that the structure of constants of motion
depending polynomially on the velocity components becomes much clearer
when written in terms of homogeneous coordinates on the bigger space.)

And last, but not least, I have been blessed with some wonderful Ph.D. students!

I was the main advisor of Budor Shuaib (بدور شعيب),
who defended her Ph.D. thesis *Ghostpeakons*
in January 2019, with Krzysztof Marciniak as co-supervisor.

I was the main advisor of Marcus Kardell
who defended his Ph.D. thesis *New Phenomena in the World of Peaked Solitons*
in February 2016, with Stefan Rauch and Joakim Arnlind as co-supervisors.

I was also the co-supervisor of Nils Rutstam, who defended his Ph.D. thesis about the dynamics of the Tippe Top in February 2013, with Stefan Rauch as his main advisor.

Last modified 2024-05-20. Hans Lundmark (hans.lundmark@liu.se)