Prim Plansangkate's research while affiliated with Prince of Songkla University and other places

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev–Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painlevé I or II for the dKP equation, but fails to pass the Painlevé test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein–Weyl (EW) structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of EW spaces belonging to this class, and depending on one arbitrary function of one variable.

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev-Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painleve I or II for the dKP equation, but fails to pass the Painlev\'e test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein-Weyl structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of Einstein-Weyl spaces belonging to this class, and depending on one arbitrary function of one variable.

Under two separate symmetry assumptions, we demonstrate explicitly how the equations governing a general anti-self-dual conformal structure in four dimensions can be reduced to the Manakov–Santini system, which determines the three-dimensional Einstein–Weyl structure on the space of orbits of symmetry. The two symmetries investigated are a non-null translation and a homothety, which are previously known to reduce the second heavenly equation to the Laplace’s equation and the hyper-CR system, respectively. Reductions on the anti-self-dual null-Kähler condition are also explored in both cases.

Under two separate symmetry assumptions, we demonstrate explicitly how the equations governing a general anti-self-dual conformal structure in four dimensions can be reduced to the Manakov-Santini system, which determines the three-dimensional Einstein-Weyl structure on the space of orbits of symmetry. The two symmetries investigated are a non-null translation and a homothety, which are previously known to reduce the second heavenly equation to the Laplace's equation and the hyper-CR system, respectively. Reductions on the anti-self-dual null-K\"ahler condition are also explored in both cases.

We construct Skyrme fields from holonomy of the spin connection of multi-Taub–NUT instantons with the centres positioned along a line in Our family of Skyrme fields includes the Taub–NUT Skyrme field previously constructed by Dunajski. However, we demonstrate that different gauges of the spin connection can result in Skyrme fields with different topological degrees. As a by-product, we present a method to compute the degrees of the Taub–NUT and Atiyah–Hitchin Skyrme fields analytically; these degrees are well defined as a preferred gauge is fixed by the symmetry of the two metrics. Regardless of the gauge, the domain of our Skyrme fields is the space of orbits of the axial symmetry of the multi-Taub–NUT instantons. We obtain an expression for the induced Einstein–Weyl metric on the space and its associated solution to the -Toda equation.

We use the compactified twistor correspondence for the (2+1)-dimensional integrable chiral model to prove a conjecture of Ward. In particular, we construct the correspondence space of a compactified twistor fibration and use it to prove that the second Chern numbers of the holomorphic vector bundles, corresponding to the uniton solutions of the integrable chiral model, equal the third homotopy classes of the restricted extended solutions of the unitons. Therefore we deduce that the total energy of a time-dependent uniton is proportional to the second Chern number.

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of SL(3, \mathbbC){SL(3, \mathbb{C})} anti–self–dual Yang–Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin–Yau–Zaslow construction to give an explicit expression for a six–real dimensional semi–flat Calabi–Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3rd Painlevé transcendents.

We consider a class of time dependent finite energy multi-soliton solutions of the U(N) integrable chiral model in $(2+1)$ dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these extended solutions to any spacelike plane in $\R^{2,1}$ have trivial monodromy and give rise to maps from a three sphere to U(N). We demonstrate that the total energy of each multi-soliton is quantised at the classical level and given by the third homotopy class of the extended solution. This is the first example of a topological mechanism explaining classical energy quantisation of moving solitons.