Dept Math & Stat UiT, Forskningsparken B459
  12-19 Apr 2024 Fri 14:15-15:45

Prim Plansangkate (Prince of Songkla University, Thailand)

"Anti-self-dual Equations and Integrable Systems"

In the first lecture, I will introduce the anti-self-dual Yang-Mills (ASDYM) equation and discuss two aspects of its relation to integrable systems. Firstly, an equation can be identified as being integrable by twistor method if it can be realized as a symmetry reduction of the ASDYM equation, with an example of the affine sphere equation. Secondly, an equation which is a reduction of the ASDYM equation will inherit a Lax pair, which can be useful in a study of properties of solutions. A result about soliton solutions of the Ward integrable chiral model will be presented as an example in this case.

In the second lecture, I will present two results in connection with the anti-self-dual (ASD) conformal structures and Einstein-Weyl structures. Firstly, motivated by the fact that any 4-dimensional ASD structure with a non-null conformal Killing symmetry gives an Einstein-Weyl structure on the 3-dimensional space of orbits, I will show that the ASD conformal equation in neutral signature can be explicitly reduced, by a simple transformation, to the Manakov-Santini system which governs generic Lorentzian Einstein-Weyl structures. Secondly, a generalization of the dKP equation which determines a family of Einstein-Weyl structures in an arbitrary dimension will be discussed, and an extension of the quadric ansatz method presented as an attempt to find solutions of the equation.