The Volterra lattice admits two non-Abelian generalizations that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to Painlev\'e-type equations. In the case of symmetries of low order, including the scaling and master-symmetry, this constraint can be reduced to second order equations. This gives rise to two non-Abelian generalizations for the discrete Painlev\'e equations dP$_1$ and dP$_{34}$ and for the continuous Painlev\'e equations P$_3$, P$_4$ and P$_5$.

We will consider one of the first finite amplitude, meaning inherently nonlinear, solutions for waves on the surface of the fluid, namely, Stokes waves. Linear stability of waves, which were obtained with high precision in our previous works, is investigated. We linearize Dyachenko equations with respect to small perturbations of Stokes waves for incompressible fluid of infinite depth. Resulting equations are reformulated as an eigenvalue problem for a nonlocal linear operator, which is solved on inhomogeneous grid using Arnoldi algorithm with shift-and-invert preconditioner. The first three branches of instability are demonstrated and predictions of the system behavior for higher nonlinearity Stokes waves are proposed.

Suppression of the critical temperature in homogeneously disordered superconducting films is a consequence of the disorder-induced enhancement of Coulomb repulsion. We demonstrate that for the majority of thin films studied now this effect cannot be completely explained in the assumption of two-dimensional diffusive nature of electrons motion. The main contribution to the $T_c$ suppression arises from the correction to the electron-electron interaction constant coming from small scales of the order of the Fermi wavelength that leads to the critical temperature shift $\delta T_c/T_{c0} \sim - 1/k_Fl$, where $k_F$ is the Fermi momentum and $l$ is the mean free path. Thus almost for all superconducting films that follow the fermionic scenario of $T_c$ suppression with decreasing the film thickness, this effect is caused by the proximity to the three-dimensional Anderson localization threshold and is controlled by the parameter $k_Fl$ rather than the sheet resistance of the film.
D.S. Antonenko, M.A Skvortsov,. JETP Lett. (2020). https://doi.org/10.1134/S0021364020190017

We obtain a dispersionless integrable system describing a local form of a general three-dimensional Einstein–Weyl geometry with a Euclidean (positive) signature, construct its matrix extension, and show that it leads to the Bogomolny equations for a non-Abelian monopole on an Einstein–Weyl background. We also consider the corresponding dispersionless integrable hierarchy, its matrix extension, and the dressing scheme.