The Robinson-Trautman Type III Prolongation Structure Contains K$_2$
J. D. Finley, III (Univ. of New Mexico)
TL;DR
This paper demonstrates that the minimal prolongation structure for Robinson-Trautman type III equations always contains the infinite-dimensional algebra K$_2$, which could facilitate finding new solutions through Bäcklund transformations.
Contribution
It reveals that the prolongation structure inherently includes the algebra K$_2$, providing a new perspective on solution generation for these equations.
Findings
The prolongation structure always contains K$_2$.
K$_2$ is of infinite growth.
Faithful representations could lead to new solutions.
Abstract
The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K, which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of B\"acklund transformations to evolve new solutions.
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