Energy density of Abrikosov vortices           for t = -1/2, 0, +1/2.

Solitons play an important role in a wide variety of phenomena. They appear in benign form as solitary waves in canals, or threaten the Japanese west coast as tsunamis, giant waves which can propagate over great distances. As optical solitons, they increase the capacity of fibres for telecommunications and, as sine-Gordon solitons, they occur in condensed matter. In all its manifestations, a soliton’s hall-mark is its stability.

Whereas the solitons mentioned so far travel in one space dimension, the solitons studied extensively in field theory during the last 25 years live in two or three space dimensions. Magnetic flux tubes in superconductors are one example of solitons in two space dimensions; magnetic bubbles in ferromagnets and vortices in He3 and He4 are others. These extended objects are very important for the performance of the corresponding materials. Not of practical, but of fundamental importance for our understanding of the universe, are cosmic strings and monopoles.

The stability of these extended objects is due to their nontrivial topology far away from the centre. Although the bulk of their energy is concentrated near the centre, the shape of the fields at infinity is very important. Their nontrivial shape can be explained in terms of a strip twisted by 180° and then glued together end-to-end. Such a strip cannot be deformed into an untwisted strip without tearing it. In a similar fashion, the fields of our extended objects are twisted at infinity. To unravel them would require an infinite amount of energy.

Whereas topology explains the existence of solitons in field theory, nonlinear partial differential equations provide the proper mathematical framework for their description. The mathematical problem is to find solutions to the relevant equations which describe the properties and the behaviour of the extended objects. The Differential Equations Group of the School of Mathematical Sciences in DCU is engaged in this project as part of its wider research efforts.

In some models, by a suitable choice of coupling constants, the forces can be made to balance out. For magnetic flux tubes (Abrikosov vortices), this occurs at the borderline between type-I and type-II superconductivity. This makes it possible to look for static vortex solutions, which simplifies the mathematical problem considerably. Even so, no explicit solutions of the equations have been found. Recently we constructed these solutions in terms of a series. Using the terms of lower order, the energy density of two Abrikosov vortices can be calculated. It is plotted in the figure for three different values of the separation parameter.

Slowly moving vortices are at all times close to static configurations such as those depicted in the figure. This idea led to a mathematical technique, called the slowmotion approximation, for certain models with wave-like propagation. Within this approximation, the dynamics of solitons can be studied. The surprising result is 90° scattering, also depicted in the figure if we read it left-to-right as an evolution in time. In this scattering process, the extended objects merge in a head-on collision and then emerge on the sides, to the utter bewilderment of snooker players.