Surface water waves constitute an important example of so-called dynamic turbulence. Natural evolution of wave spectra results in development of the Kolmogorov cascade that transfer wave energy from low to high frequencies (Zakharov--Filonenko solution). This solution may be obtained from the basic
principles of resonant interaction of waves.

The natural wave spectra are anisotropic. This begs a question: how this anisotropy affects the Zakharov--Filonenko (ZF) spectrum. An anisotropic solution, also obtained from the principles of wave resonance, was found by Kats and Kontorovich. But a general study of anisotropy requires the detailed examination of the kinetic equation for waves. This is the main topic of the present paper.

The evolution of the generic powerlike-cosine anisotropic additions to ZF spectrum was determined analytically via the quadruplet integrals. These integrals were calculated numerically (the example graph here shows the exponent coefficient for the first cosine mode). The calculation has shown the existence of some new solutions.

The pure cosine mode has proved to be special in that it decays much slower than other ones. As a result, general wave spectra in the inertial interval usually includes ZF spectrum with the addition of the pure cosine and virtually no other harmonics.

V.Geogjaev
JETP Letters 124, issue 5 (2026)