arXiv subjects

Prim Plansangkate

Publications and source records attributed to Prim Plansangkate.

9 recordsLinked to original sources

arXiv2025

Conserved quantities of distinguished curves on conformal sphere

We give conserved quantities of two generalizations of conformal circles on the conformal sphere. One generalization concerns curves carrying a distinguished parallel tractor along them, which can be used to construct the conserved quantities. The other is the class of curves satisfying the conformal Mercator equation, the conserved quantities of which are computed using Lagrangian formalism. The two generalizations are not disjoint, and our main result is the relation between their conserved quantities. Explicit examples are also presented.

math.DG↗

arXiv2023

A quadric ansatz method for a certain class of second order PDEs

We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional Kähler-Einstein equation with symmetry and the one-sided type-D Einstein equation with nonzero scalar curvature. The procedure, which reduces the PDEs to ordinary differential equations (ODEs), involves imposing additional constraints to the form of the ansatz, depending on the exact form of the PDEs. Thus its applicability varies within the class. We identify applicable subclasses of a class of modified Toda equations, one of which includes the Kähler-Einstein equation with symmetry, and give the corresponding reduced ODEs. In addition, we obtain a quadric ansatz reduction for a family of Einstein-Weyl equations in arbitrary dimensions. Lastly, we suggest a particular form of equations for which our procedure appears to be most effective as the aforementioned constraints vanish, and also present some reduction results using another type of ansatz, namely the hyperplane ansatz.

nlin.SI↗

arXiv2021

The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces

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 Painlevé 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.

nlin.SI↗

arXiv2019

From the conformal self-duality equations to the Manakov-Santini system

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.

nlin.SI↗

arXiv2018

Skyrme fields, multi-instantons and the $SU(\infty)$-Toda equation

We construct Skyrme fields from holonomy of the spin connection of multi-Taub-NUT instantons with the centres positioned along a line in $\mathbb{R}^3.$ 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 $SU(2)$ 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 $SU(\infty)$-Toda equation.

hep-th↗

arXiv2015

Compactified Twistor Fibration and Topology of Ward Unitons

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.

hep-th↗

arXiv2011

Scalar--flat Kähler metrics with conformal Bianchi V symmetry

We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b}, and construct all four-dimensional Kahler metrics with vanishing scalar curvature which are invariant under the conformal action of Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem with two double poles. The resulting metrics are non-diagonal in the left-invariant basis and are explicitly given in terms of Bessel functions and their integrals. We also make a connection with the LeBrun ansatz, and characterise the associated solutions of the SU(\infty) Toda equation by the existence a non-abelian two-dimensional group of point symmetries.

gr-qc↗

arXiv2009

Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of $SL(3, \C)$ 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.

math.DG↗

arXiv2006

Topology and Energy of Time Dependent Unitons

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.

hep-th↗