Lattice equations and \(\tau\)-functions for a coupled Painlevé system. (English) Zbl 1013.34085
Summary: The author considers a pair of coupled Painlevé equations arising as a scaling similarity reduction of the Hirota-Satsuma system of partial differential equations. Bäcklund transformations constructed in a previous work are presented explicitly as discrete shifts in a two-dimensional parameter space. \(\tau\)-functions derived from a Hamiltonian description are also presented, which satisfy multilinear lattice equations built from Hirota bilinear operators, and these are used to calculate polynomial \(\tau\)-functions for rational solutions.
MSC:
34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
33E17 | Painlevé-type functions |
39A99 | Difference equations |