Classical works, dedicated to the finite-gap intrgration, admits the following interpretation. One has g finite circles on the spectral curve, and each oval contains exactly one zero of the wave function. A set of g circles is exactly the g-dimensional torus, which is exactly the Liouville torus. The Abel transform linearizes the motion of this tours. If one consideres degenerate curves, corresponding to the multisoliton solutions, the defintion of divisors requires a refinement, including the resolution of singularities. We show that for the so-called MM-curves and Dubrovin-Natanzon divisors, generating real regular multisoliton KP-2 solutions, it is sufficient to apply only the two simplest types of resolutions.

Recently, a big number of papers dedicated to the study of PT-symmetric operators and integrable equations has been published. In the finite-gaps approach to the 1-d Schrodinger operators two alternative approaches can be used: 1) Matvees-Its approach based on the theory of Riemann theta-functions; 2) Dubrovin's approach based on the divisor's dynamics. The characterization of PT-symmetric Schrodinger operators in the first approach was recentky obtained by Taimanov. In our parer we describe the spectral curves and divisor trajectories for such operators in the first-order approximation.

Through a massive computation we reached the fourth superharmonic instability branch of the Stokes' wave. Using the obtained results we checked phenomenological formulae for the dependence of the instability growth rates corresponding to different branches of instability on the nonlinearity parameter (steepness, defined as the wave swing to wavelength ratio) in the vicinity of the new instability branch appearance and far from it. It is demonstrated, that the formulae, the one obtained as a least squares fit using the information from the first three branches of instability and a phenomenological asymptotics, work for the fourth branch and previously reported branches as well. Range of applicability of the relations was corrected. Growth rates for all four instability branches are reported.

We study the spatial dependence of pair correlation functions of velocity field components in a rotating turbulent fluid on a background of a coherent geostrophic vortex. The statistics of the turbulent pulsations are determined by their dynamics, which is the dynamics of inertial waves affected by the differential rotation in the vortex and a weak viscous damping. We are interested in distances which are larger than the scale of the wave forcing but smaller than the radius of the coherent vortex. We establish the anisotropy of the velocity field correlation function at the distances. All the diagonal elements of the correlation function decay logarithmically in the streamwise direction and power-like in radial direction and the direction along the rotation axis. This laws are independent of the details of the forcing correlation function that indicate “coherency” of the flow. On the contrary, the cross-correlation function of the radial-azimuth velocity components, which turns into the Reynolds stress for zero distance, demonstrates strong dependence on the forcing correlation function and decays quickly at distances larger than the forcing scale.

Leon L. Ogorodnikov, Sergey S. Vergeles. “Long-range spatial velocity statistics in a rotating coherent turbulent vortex”, Physical Review Fluids, vol. 10, p. 124702 (2025)