Christopher K. Anand

Ward's solitons

Using the ‘Riemann Problem with zeros’ method, Ward has constructed exact solutions to a (2 + 1)–dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact surface.

WARD'S SOLITONS II: EXACT SOLUTIONS

In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

Unitons and their moduli

We sketch the proof that unitons (harmonic spheres in U(N)) correspond to holomorphic ‘uniton bundles’, and that these admit monad representations analogous to Donaldson’s representation of instanton bundles. We also give a closed-form expression for the unitons involving only matrix operations, a finite-gap result (two-unitons have energy ≥ 4), computations of fundamental groups of energy ≤ 4 components, new methods of proving discreteness of the energy spectrum and of Wood’s Rationality Conjecture, a discussion of the maps into complex Grassmannians and some open problems.