Peter Vingaard Larsen
Ph.D. in Applied Mathematics
 Valdemarsgade 83, 5.th
1665 Copenhagen V
Denmark      
Telephone (Direct): (+45) 6171 0927
E-mail:  peter.vingaard(a)gmail.com


PhD Research Areas
 

RESEARCH

My general interest is the field of Nonlinear Science, a very broad, versatile, and interesting field of study. As a general rule, linear systems are easy to handle mathematically, but the true physics is often found in nonlinear models. Nonlinearity underlies the much celebrated chaos theory, where the emphasis is on the non-predictable nature of complex systems, such as e.g. weather systems. The main issue is that in linear systems, a small change to initial conditions will only result in a small change of the outcome (e.g., a slightly harder kick to a ball results in a slightly larger distance of the kick), but for nonlinear systems a small change in initial conditions may (or will!) result in unpredicted outcomes (in the ball kick analogy: the ball might roll backwards!).

Above, the uncertain nature of nonlinearity is presented. My main interest, however, is the case where nonlinearity acts to stabilize the system! In linear dispersive systems, an initial excitation will eventually die out. With a water example: a wave generated from a tossed stone will decrease in height and eventually disappear due to this dispersion. However, when nonlinearity comes into play (and conditions are otherwise suitable), the wave may travel undistorted for miles and miles. This was first discovered by John Scott Russell in 1834 in a channel and led to the term soliton. Another tragic example is the Boxing Day tsunami in South East Asia in 2004.

Solitons occur in a number of fields, including hydrodynamics, optics, plasma physics and biophysics. Find more information about solitons on these pages (and their links):

In the modeling of physical systems the main issue is to find a description that is both sufficiently accurate as well as computational feasible. Before modern computer power, this put a big limitation on the accuracy of the models, but this is no longer the case. It should be clear that nonlinear systems are harder to solve than linear, but a special class of nonlinear systems are even harder: the nonlocal models.

The term nonlocality refers to that the state of a physical system may not be described locally, i.e., that its features at a certain point (of space and/or time) cannot be correctly reproduced by merely taking into account the system parameters (e.g. mass, velocity, spin, etc.) at that very point, but one needs to consider the variable values at all points! This greatly increases the complexity of the calculation, but it also allows for a more physically sound model. Solitons can also be found in nonlocal models, along with a plethora of other interesting effects!

In particular, I apply the notion of nonlocality to

To carry out the investigation of nonlocal, nonlinear phenomena, I have studied partial differential equations and their solutions - both analytically and numerically (in particular Runge-Kutta solvers and the Fourier Split Step method, implemented in C++ and Fortran).

General research interests

Apart from the above mentioned two instances of nonlinear partial differential equations I studied in my PhD, I am also interested in applied mathematics in a broader sense, particularly with emphasis on solitons. I have some affinity towards the biological aspects of mathematics; not only DNA modeling, but I have also investigated nerve signal transmission (the Hodgkin-Huxley model). Furthermore, I have also fully enjoyed studying atmosphere--ocean dynamics.