Vector systems of Painlevé type
V. E. Adler, V. V. Sokolov
TL;DR
The paper develops vector (multicomponent) Painlevé-type reductions of the NLS, mKdV, and KdV equations via similarity reductions, producing isomonodromic Lax representations for the resulting ODE systems. It identifies two main deformation families of Garnier-type systems: one Liouville-integrable nonautonomous deformation and a second linked to the Veselov–Shabat quasiperiodic dressing chain, with a detailed connection to the scalar Painlevé equations, including $P_4$ in the $n=1$ case. The Kulish–Sklyanin system and vector mKdV/KdV reductions are analyzed, yielding autonomous and nonautonomous deformations on symmetric spaces and their corresponding Lax pairs, traveling-wave solutions, and isotropic reductions that reveal fixed-order dynamics independent of the vector dimension. A unifying theme is the construction of isomonodromic Lax representations across the vector reductions, demonstrating a rich interplay between Liouville integrability, Painlevé-type dynamics, and classical dressing chains, and highlighting open problems such as vector Bäcklund transformations and even-period dressing chains. Overall, the work extends scalar Painlevé reductions to multicomponent settings, offering a framework with potential applications in integrable systems, spectral theory, and mathematical physics.
Abstract
The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlevé equations P$_1$, P$_2$, P$_{34}$, and P$_4$. Some of them can be interpreted as nonautonomous deformations of well-known systems integrable in the Liouville sense, in particular, the Garnier and Hénon--Heiles systems. In one case, an unexpected connection with the equations of quasiperiodic dressing chain for the Schrödinger operator is established.