arXiv subjects

D. S. Agafontsev

Publications and source records attributed to D. S. Agafontsev.

At least 19 recordsLinked to original sources

arXiv2026

Rogue waves from noise-induced modulational instability of a plane wave

In the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type, the present paper studies numerically rogue waves (RWs) that emerge in the nonlinear stage of noise-induced modulational instability of a plane wave. For a large ensemble of simulations, all sufficiently large local maximums of the wavefield (i.e., RWs) are systematically collected and analyzed. It is shown that the frequency of occurrence of such maximums begins to increase from zero at the time, when the fourth-order moment of amplitude reaches its first (largest) local maximum, and then grows in an oscillatory manner approaching its asymptotic value at long time. Near the statistically stationary state, the 1D-NLSE generates a much larger number of RWs than a comparable linear system, but, in average, one RW affects a much smaller spatiotemporal area. The distribution of these RWs by maximum intensity represents an exponential-like function without noticeable changes in behavior, indicating a similar origin for RWs of significantly different amplitudes. The collected RWs are compared within a sufficiently large spatiotemporal window with nine exact solutions, of which two models reproduce RW dynamics much better than the others: a general collision of two solitons and a general collision of two Tajiri-Watanabe breathers; RWs are fitted with these collisions using pre-compiled databases of such solutions. The parameters of the two models turn out to be quite similar to the direct scattering transform spectrum of the whole wavefield, supporting a hypothesis that RWs may emerge due to synchronization of a few nonlinear modes (solitons or breathers) in presence of many other nonlinear modes of the wavefield.

nlin.PS↗

arXiv2026

Drift towards isotropization during the 3D hydrodynamic turbulence onset

The incompressible three-dimensional Euler equations develop very thin pancake-like regions of exponentially increasing vorticity. The characteristic thickness of such regions decreases exponentially with time, while the other two dimensions do not change considerably, making the flow near each pancake strongly anisotropic. The pancakes emerge in increasing number with time, which may enhance the anisotropy of the flow, especially if they orient similarly in space. In the present paper, we study numerically the anisotropy by analyzing the evolution of the so-called isotropy markers [Phys. Rev. Fluids 10, L022602 (2025)]. We show that these functions drift slowly towards unity, indicating the process of slow isotropization, which takes place without the viscous scales getting exited and despite the similar orientation of the emerging pancakes.

physics.flu-dyn↗

arXiv2024

Spontaneous modulational instability of elliptic periodic waves: the soliton condensate model

We use the spectral theory of soliton gas for the one-dimensional focusing nonlinear Schrödinger equation (fNLSE) to describe the statistically stationary and spatially homogeneous integrable turbulence emerging at large times from the evolution of the spontaneous (noise-induced) modulational instability of the elliptic ``dn'' fNLSE solutions. We show that a special, critically dense, soliton gas, namely the genus one bound-state soliton condensate, represents an accurate model of the asymptotic state of the ``elliptic'' integrable turbulence. This is done by first analytically evaluating the relevant spectral density of states which is then used for implementing the soliton condensate numerically via a random N-soliton ensemble with N large. A comparison of the statistical parameters, such as the Fourier spectrum, the probability density function of the wave intensity, and the autocorrelation function of the intensity, of the soliton condensate with the results of direct numerical fNLSE simulations with dn initial data augmented by a small statistically uniform random perturbation (a noise) shows a remarkable agreement. Additionally, we analytically compute the kurtosis of the elliptic integrable turbulence, which enables one to estimate the deviation from Gaussianity. The analytical predictions of the kurtosis values, including the frequency of its temporal oscillations at the intermediate stage of the modulational instability development, are also shown to be in excellent agreement with numerical simulations for the entire range of the elliptic parameter $m$ of the initial dn potential.

nlin.PS↗

arXiv2023

Multisoliton interactions approximating the dynamics of breather solutions

Breather solutions are considered to be generally accepted models of rogue waves. However, breathers are not localized, while wavefields in nature can generally be considered as localized due to the limited spatial dimensions. Hence, the theory of rogue waves needs to be supplemented with localized solutions which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multi-soliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact $N$-soliton solution converging asymptotically to the plane wave at large number of solitons $N$. On the example of the Peregrine, Akhmediev, Kuznetsov-Ma and Tajiri-Watanabe breathers, we show that the constructed with our method multi-soliton solutions, being localized in space with characteristic width proportional to $N$, are practically indistinguishable from the breathers in a wide region of space and time at large $N$. Our method makes it possible to build solitonic models with the same dynamical properties for the higher-order rational and super-regular breathers, and can be applied to general multi-breather solutions, breathers on a nontrivial background (e.g., cnoidal waves) and other integrable systems. The constructed multi-soliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronizations conditions represents a challenging problem for future studies.

nlin.PS↗

arXiv2022

Bound-state soliton gas as a limit of adiabatically growing integrable turbulence

We study numerically the integrable turbulence in the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type using a new approach called the "growing of turbulence". In this approach, we add a small linear pumping term to the equation and start evolution from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and examine the resulting integrable turbulence. For sufficiently small initial noise and pumping coefficient, and also for not very wide simulation box (basin length), we observe that the turbulence grows in a universal adiabatic regime, moving successively through the statistically stationary states of the integrable 1D-NLSE, which do not depend on the pumping coefficient, amplitude of the initial noise or basing length. Waiting longer in the growth stage, we transit from weakly nonlinear states to strongly nonlinear ones, characterized by a high frequency of rogue waves. Using the inverse scattering transform (IST) method to monitor the evolution, we observe that the solitonic part of the wavefield becomes dominant even when the (linear) dispersion effects are still leading in the dynamics and with increasing average intensity the wavefield approaches a dense bound-state soliton gas, whose properties are defined by the Fourier spectrum of initial noise. Regimes deviating from the universal adiabatic growth also lead to solitonic states, but solitons in these states have noticeably different velocities and a significantly wider distribution by amplitude, while the statistics of wavefield indicates a much more frequent appearance of very large waves.

nlin.SI↗

arXiv2022

Compressible vortex structures and their role in the onset of hydrodynamic turbulence

We study formation of quasi two-dimensional (thin pancakes) vortex structures in three-dimensional flows, and quasi one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at high Reynolds numbers, when their evolution is described at the leading order by the Euler equations for an ideal incompressible fluid. We show numerically and analytically that the compression of these structures and, as a consequence, the increase in their amplitudes is related to the compressibility of the frozen-in-fluid fields: the field of continuously distributed vortex lines in the three-dimensional case and the field of vorticity rotor (divorticity) for two-dimensional flows. We find that the growth of vorticity and divorticity can be considered as a process of breaking of the corresponding fields. At high intensities, the process demonstrates a Kolmogorov-type scaling relating the maximum amplitude with the characteristic width of the structures. The possible role of these coherent structures is analyzed in the formation of the turbulent Kolmogorov spectrum, as well as the Kraichnan spectrum corresponding to a constant flux of enstrophy in the case of two-dimensional turbulence.

physics.flu-dyn↗

arXiv2021

Stability of tangential discontinuity for the vortex pancakes

Within the incompressible three-dimensional Euler equations, we study the pancake-like high vorticity regions, which arise during the onset of developed hydrodynamic turbulence. We show that these regions have an internal fine structure consisting of three vortex layers. Such a layered structure, together with the power law of self-similar evolution of the pancake, prevents development of the Kelvin-Helmholtz instability.

physics.flu-dyn↗

arXiv2020

Rogue waves with rational profiles in unstable condensate and its solitonic model

In this brief report we study numerically the spontaneous emergence of rogue waves in (i) modulationally unstable plane wave at its long-time statistically stationary state and (ii) bound-state multi-soliton solutions representing the solitonic model of this state [Gelash et al, PRL 123, 234102 (2019)]. Focusing our analysis on the cohort of the largest rogue waves, we find their practically identical dynamical and statistical properties for both systems, that strongly suggests that the main mechanism of rogue wave formation for the modulational instability case is multi-soliton interaction. Additionally, we demonstrate that most of the largest rogue waves are very well approximated -- simultaneously in space and in time -- by the amplitude-scaled rational breather solution of the second order.

nlin.PS↗

arXiv2020

Growing of integrable turbulence

We study numerically the integrable turbulence in the framework of the focusing one-dimensional nonlinear Schrodinger equation using a new method -- the "growing of turbulence". We add to the equation a weak controlled pumping term and start adiabatic evolution of turbulence from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and realize that the "grown up" turbulence is statistically stationary. We measure its Fourier spectrum, the probability density function (PDF) of intensity and the autocorrelation of intensity. Additionally, we show that, being adiabatic, our method produces stationary states of the integrable turbulence for the intermediate moments of pumping as well. Presently, we consider only the turbulence of relatively small level of nonlinearity; however, even this "moderate" turbulence is characterized by enhanced generation of rogue waves.

nlin.PS↗

arXiv2020

Integrable turbulence developing from strongly nonlinear partially coherent waves

We study numerically the integrable turbulence developing from strongly nonlinear partially coherent waves, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. We find that shortly after the beginning of motion the turbulence enters a state characterized by a very slow evolution of statistics (the quasi-stationary state - QSS), and we concentrate on the detailed examination of the basic statistical functions in this state depending on the shape and the width of the initial spectrum. In particular, we show that the probability density function (PDF) of wavefield intensity is nearly independent of the initial spectrum and is very well approximated by a certain Bessel function representing an integral of the product of two exponential distributions. The PDF corresponds to the value of the second-order moment of intensity equal to 4, indicating enhanced generation of rogue waves. All waves of large amplitude that we have studied are very well approximated - both in space and in time - by the rational breather solutions of either the first (the Peregrine breather), or the second orders.

nlin.PS↗

arXiv2019

Statistical properties of the velocity field for the 3D hydrodynamic turbulence onset

We study the statistical correlation functions for the three-dimensional hydrodynamic turbulence onset when the dynamics is dominated by the pancake-like high-vorticity structures. With extensive numerical simulations, we systematically examine the two-points structure functions (moments) of velocity. We observe formation of the power-law scaling for both the longitudinal and the transversal moments in the same interval of scales as for the energy spectrum. The scaling exponents for the velocity structure functions demonstrate the same key properties as for the stationary turbulence case. In particular, the exponents depend on the order of the moment non-trivially, indicating the intermittency and the anomalous scaling, and the longitudinal exponents turn out to be slightly larger than the transversal ones. When the energy spectrum has power-law scaling close to the Kolmogorov's one, the longitudinal third-order moment shows close to linear scaling with the distance, in line with the Kolmogorov's 4/5-law despite the strong anisotropy.

physics.flu-dyn↗

arXiv2018

Development of high vorticity structures and geometrical properties of the vortex line representation

The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $ω_{max}\sim\ell^{-2/3}$ between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D.S. Agafontsev et al, Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes' development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with respect to conservation of the Cauchy invariants and describes compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to $1536^3$ nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling $ω_{max}\sim\ell^{-2/3}$.

physics.flu-dyn↗

arXiv2018

Strongly interacting soliton gas and formation of rogue waves

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schr{ö}dinger (NLS) equation. In order to model such gas we use N-soliton solutions (N-SS) with $N\sim 100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wave intensity. We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multi-soliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

nlin.PS↗

arXiv2017

Development of high vorticity in incompressible 3D Euler equations: influence of initial conditions

The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $ω_{\max}(t)\propto\ell(t)^{-2/3}$ between the vorticity maximum and pancake thickness, and provide the leading contribution to the energy spectrum, where the gradual formation of the Kolmogorov interval $E_{k}\propto k^{-5/3}$ is observed for some initial flows [Agafontsev et. al, Phys. Fluids 27, 085102 (2015)]. With the massive numerical simulations, in the present paper we study the influence of initial conditions on the processes of pancake formation and the Kolmogorov energy spectrum development.

physics.flu-dyn↗

arXiv2016

Asymptotic solution for high vorticity regions in incompressible 3D Euler equations

Incompressible 3D Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudo-spectral method in anisotropic grids of up to 972 x 2048 x 4096.

physics.flu-dyn↗

arXiv2016

Integrable turbulence generated from modulational instability of cnoidal waves

We study numerically the nonlinear stage of modulational instability (MI) of cnoidal waves, in the framework of the focusing one-dimensional Nonlinear Schrodinger (NLS) equation. Cnoidal waves are the exact periodic solutions of the NLS equation and can be represented as a lattice of overlapping solitons. MI of these lattices lead to development of "integrable turbulence" [Zakharov V.E., Stud. Appl. Math. 122, 219-234 (2009)]. We study the major characteristics of the turbulence for dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of "overlapping" between the solitons within the cnoidal wave. Integrable turbulence, that develops from MI of dn-branch of cnoidal waves, asymptotically approaches to it's stationary state in oscillatory way. During this process kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as t^{-a}, 1<a<1.5, the phases contain nonlinear phase shift decaying as t^{-1/2}, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, s=2g_{max}. In the asymptotic stationary state the ratio of potential to kinetic energy is equal to -2. The asymptotic PDF of wave amplitudes is close to Rayleigh distribution for cnoidal waves with strong overlapping, and is significantly non-Rayleigh one for cnoidal waves with weak overlapping of solitons. In the latter case the dynamics of the system reduces to two-soliton collisions, which occur with exponentially small rate and provide up to two-fold increase in amplitude compared with the original cnoidal wave.

nlin.PS↗

arXiv2015

Intermittency in generalized NLS equation with focusing six-wave interactions

We study numerically the statistics of waves for generalized one-dimensional Nonlinear Schrodinger (NLS) equation that takes into account focusing six-wave interactions, dumping and pumping terms. We demonstrate the universal behavior of this system for the region of parameters when six-wave interactions term affects significantly only the largest waves. In particular, in the statistically steady state of this system the probability density function (PDF) of wave amplitudes turns out to be strongly non-Rayleigh one for large waves, with characteristic "fat tail" decaying with amplitude $|Ψ|$ close to $\propto\exp(-γ|Ψ|)$, where $γ>0$ is constant. The corresponding non-Rayleigh addition to the PDF indicates strong intermittency, vanishes in the absence of six-wave interactions, and increases with six-wave coupling coefficient.

physics.optics↗

arXiv2015

Integrable turbulence and formation of rogue waves

In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of "integrable turbulence" [V.E. Zakharov, Turbulence in integrable systems, Stud. in Appl. Math. 122, no. 3, 219-234, (2009)]. We study the time evolution of it's major characteristics averaged across realizations of initial data - the condensate solution seeded by small random noise with fixed statistical properties. The measured quantities are: (1) wave-action spectrum and spatial correlation function, (2) the probability density function (PDF) of wave amplitudes and their momenta, and (3) kinetic and potential energies.

nlin.SI↗