New solvable variants of the goldfish many-body problem (Q2816390)

The original ``goldfish'' many body model is characterized by the translation-invariant equation of motion NEWLINE\[NEWLINE \ddot{z} = \sum_{_{\substack{ \ell=1 \\ \ell\neq n}}}^N \frac{2\dot{z}_n \dot{z}_\ell}{z_n-z_\ell}.NEWLINE\]NEWLINE This system of \(N\) ordinary differential equations can be interpreted as the Newtonian equations of the motion of \(N\) unit-mass point-particles moving in the complex \(z\)-plane and interacting pairwise with forces equal to twice the product of the velocities of the two interacting particles divided by their mutual distances (see a. o. the author [``Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related ``solvable'' many body problems'', Nuovo Cimento B (11) 43, No. 2, 177--241 (1978)]).NEWLINENEWLINEIn the paper under review, the author introduces a technique which allows him to identify and investigate additional solvable models ''of goldfish type''. In Subsection 1.1, four representative selection of solvable many-body problems of goldfish type are introduced, whereas more general classes of such models are reported in Section 2. In order to avoid interruptions in the flow of the presentation of the examples, some technical developments are confined in two appendices.