arXiv subjects

Sergey Dyachenko

Publications and source records attributed to Sergey Dyachenko.

4 recordsLinked to original sources

arXiv2026

Modulation theory for lumps and interactions between lumps and a mean field in the Kadomtsev-Petviashvili equation

A (2+1)-dimensional hyperbolic system of four quasi-linear partial differential equations is derived that describes the modulations of lump solutions of the Kadomtsev-Petviashvili I (KPI) equation in the presence of a mean field. The system is then shown to satisfy the necessary conditions for integrability of hydrodynamic chains. Moreover, a suitable reduction of the resulting modulation system is applied to study the interactions between lumps and a rarefaction wave for the mean field. Precise conditions are derived that describe how the lump parameters change as a result of the interaction, and which in particular determine whether the lump is transmitted through or trapped inside the rarefaction wave. The theoretical predictions are compared to direct numerical simulations of the KPI equation, showing excellent agreement.

nlin.SI↗

arXiv2026

Recurrent bifurcations of stability spectra for steep Stokes waves in a deep fluid

We study the modulational stability problem for the traveling periodic waves (called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in conformal variables. We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation, and (d) reconnection of figure-$\infty$ bands at each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously for Babenko's equation. The novelty of our work is the analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter. The derivation of the normal form uses some structural assumptions which are known to be true for the Stokes waves. For the first and second bifurcation cycles, we compute numerically with a higher-order accuracy the actual values of wave steepness for which the structural assumptions are satisfied and the numerical coefficients of the normal forms to show the excellent agreement between the normal form theory and the numerical approximations of the spectral bands.

physics.flu-dyn↗

arXiv2025

Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy

We address Euler's equations for irrotational gravity waves in an infinitely deep fluid rewritten in conformal variables. Stokes waves are traveling waves with the smooth periodic profile. In agreement with the previous numerical results, we give a rigorous proof that the zero eigenvalue bifurcation in the linearized equations of motion for co-periodic perturbations occurs at each extremal point of the energy function versus the steepness parameter, provided that the wave speed is not extremal at the same steepness. We derive the normal form for the unstable eigenvalues and, assisted with numerical approximation of its coefficients, we show that the new unstable eigenvalues emerge only in the direction of increasing steepness.

math.AP↗

arXiv2022

Dynamics of $2$D fluid in bounded domain via conformal variables

In the present work we compute numerical solutions of an integro-differential equation for traveling waves on the boundary of a $2$D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudo--differential equation by means of the Newton-Conjugate Residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to $N = 65536$.

physics.flu-dyn↗