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Summary of Hans Lundmark's research

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 a couple of years ago by Vladimir Novikov: mt+(mxu+3mux)u = 0, where m = uuxx. 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 during the last few years 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'm the main advisor of Budor Shuaib (بدور شعيب), who works on peakon-related topics. Krzysztof Marciniak is her 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 2016-05-11. Hans Lundmark (hans.lundmark@liu.se)