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Efim Pelinovsky

Publications and source records attributed to Efim Pelinovsky.

At least 19 recordsLinked to original sources
arXiv2026

Reconstruction of the non-linear wave at a buoy from shoreline data and applications to the tsunami inverse problem for piece-wise sloping bathymetry

We discuss the following inverse problem: given the run-up data of a tsunami wave, can we recover its initial shape? We study this problem within the framework of the non-linear shallow water equations, a model widely used to study tsunami propagation and inundation. Previously, it has been demonstrated that in the case of infinite sloping bathymetry, it is possible to recover the initial water displacement and velocity from shoreline readings \cite{Rybkin23,Rybkin24,Rybkin25}. We consider a finite sloping bathymerty. We show that it is possible to recover boundary conditions (water displacement and velocity) on a virtual buoy from the shoreline data. Further, we discuss stitching together the shallow water equations and the Boussinesq equation in a more complex piece-wise sloping bathymetry in order to recover the initial conditions, while incorporating the dispersion to our model.

math.AP↗
arXiv2025

Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory

We investigate the inverse tsunami wave problem within the framework of the 1D nonlinear shallow water equations (SWE). Specifically, we focus on determining the initial displacement $η_0(x)$ and velocity $u_0(x)$ of the wave, given the known motion of the shoreline $R(t)$ (the wet/dry free boundary). We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any $η_0$ and $u_0$, provided the wave does not break. In particular, we show that the knowledge of $R(t)$ enables the unique recovery of both $η_0(x$) and $u_0(x)$ in terms of the Abel transform. It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R(t)$ can be computed exactly only for $u_0(x)=0$, our algorithm can recover $η_0$ and $u_0$ exactly for any non-zero $u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented in \cite{Rybkin23,Rybkin24}, where the inverse problem was solved for $u_0(x)=0$. As in previous work, our approach utilizes the Carrier-Greenspan transformation, which linearizes the SWE for inclined bathymetries. Extensive numerical experiments confirm the efficiency of our algorithms.

math.AP↗
arXiv2025

Spring-mass behavior of solitons under the influence of an external force field within the modified Korteweg-de Vries equation

We investigate the interaction of solitons with an external periodic field within the framework of the modified Korteweg-de Vries (mKdV) equation. In the case of small perturbation a simple dynamical system is used to describe the soliton behaviour. Equilibrium points of this dynamical system are computed when the external force travels at a constant speed. Assuming that the external force moves with sinusoidal speed, we demonstrate that the soliton behavior is qualitatively similar to the constant-speed case. Besides, a resonant frequency is derived from the asymptotic theory without using the classical broad force approximation. The results obtained from the dynamical system are compared with fully direct numerical simulations, which reveal that the soliton solution exhibits spiral-like behavior in the soliton amplitude versus soliton phase space. Moreover, when the external force oscillates at the resonant frequency, the trajectories in the soliton phase versus soliton amplitude exhibit chaotic behavior.

nlin.PS↗
arXiv2024

Inverse non-linear problem of the long wave run-up on coast

The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erdélyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach.

physics.flu-dyn↗
arXiv2024

On the linearity and stability of electrostatic structures based on the Schamel equation

This paper contributes in the first part to the correct understanding of the linear limit in the Schamel equation (S-equation) from the perspective of structure formation in collisionless plasmas. The corresponding modes near equilibrium turn out to be nonlinear modes of the underlying microscopic Vlasov-Poisson (VP) system for which particle trapping is responsible. A simple shift of the electrostatic potential to a new pedestal leads to non-negativity and thus mitigates the positivity problem of the S-equation. The stability of a solitary electron hole (bright soliton), based on both the S-equation and an earlier transverse but limited VP instability analysis, exhibits marginal stability and linear perturbations in the form of the asymmetric shift eigenmode of a solvable Schrödinger problem. This finding of a dominant shift mode perturbation also seems to have been observed in a numerical PIC simulation, thus providing some confirmation for both theories.

physics.plasm-ph↗
arXiv2024

Wave evolution within the Cubic Vortical Whitham equation

In this work, we study the evolution of disturbances within the framework of the Cubic Vortical Whitham (CV-Whitham) equation, considering both positive and negative cubic nonlinearities. This equation plays important role for description of the wave processes in the presence of shear flows. We find well-formed breather-type structures arising from the evolution of depression disturbances with positive cubic nonlinearity. For elevation disturbances, the results are two-fold. When the cubic nonlinearity is negative, we show that the CV-Whitham equation and the Gardner equation are qualitatively similar, differing only by a small phase lag due to differences in the dispersion term. However, with positive cubic nonlinearity, the differences between the solutions become more pronounced, with the CV-Whitham equation producing sharper waves that suggest the onset of wave breaking.

nlin.PS↗
arXiv2024

Soliton dynamics in random fields: The Benjamin-Ono equation framework

Algebraic soliton interactions with a periodic or quasi-periodic random force are investigated using the Benjamin-Ono equation. The random force is modeled as a Fourier series with a finite number of modes and random phases uniformly distributed, while its frequency spectrum has a Gaussian shape centered at a peak frequency. The expected value of the averaged soliton wave field is computed asymptotically and compared with numerical results, showing strong agreement. We identify parameter regimes where the averaged soliton field splits into two steady pulses and a regime where the soliton field splits into two solitons traveling in opposite directions. In the latter case, the averaged soliton speeds are variable. In both scenarios, the soliton field is damped by the external force. Additionally, we identify a regime where the averaged soliton exhibits the following behavior: it splits into two distinct solitons and then recombines to form a single soliton. This motion is periodic over time.

nlin.PS↗
arXiv2024

Dynamics of irregular wave fields in the Schamel equation framework

The present article is devoted to the study of the dynamics of narrowband wave fields within the non-integrable Schamel equation, which plays an important role in plasma physics, wave dynamics in metamaterials, and electrical circuits. A Monte Carlo approach is used to obtain a large number of random independent realizations of the wave fields, allowing for an investigation of the evolution of the following statistical characteristics: spectra, moments, and distribution functions. The simulations are conducted for different values of the Ursell number (the ratio of nonlinearity to dispersion) to study the impact of nonlinearity and dispersion on the processes under consideration.

physics.flu-dyn↗
arXiv2023

Bipolar solitary wave interactions within the Schamel equation

Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we investigate the features and properties of bipolar soliton interactions within the framework of the non-integrable Schamel equation, contrasting them with the integrable modified Korteweg-de Vries equation. We examine variations in moments and extrema of the wave fields. Additionally, we identify scenarios in which, in the bipolar solitary wave interaction, the smaller solitary wave transfers a portion of its energy to the larger one, causing an increase in the amplitude of the larger solitary wave and a decrease in the amplitude of the smaller one, returning them to their pre-interaction state. Notably, we observe that non-integrability can be considered a factor that triggers the formation of rogue waves.

physics.flu-dyn↗
arXiv2023

Non-integrable soliton gas: The Schamel equation framework

Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of almost solitons may exist. In the present paper, the complex dynamics of ensembles of solitary waves is studied within the Schamel equation using direct numerical simulations. Some important statistical characteristics (distribution functions, moments) are calculated numerically for unipolar and bipolar soliton gases. Comparison of results with integrable Korteweg-de Vries (KdV) and modified KdV (mKdV) models are given qualitatively. Our results agree well with the predictions of the KdV equation in the case of unipolar solitons. However, in the bipolar case, we observed a notable departure from the mKdV model, particularly in the behavior of kurtosis. The observed increase in kurtosis signifies the amplification of distribution function tails, which, in turn, corresponds to the presence of high-amplitude waves.

physics.flu-dyn↗
arXiv2023

Evolution and statistical analysis of random wave fields within the Benjamin-Ono equation

This study investigates the numerical evolution of an initially internal random wave field characterized by a Gaussian spectrum shape using the Benjamin-Ono (BO) equation. The research focuses on analyzing various properties associated with the BO random wave field, including the transition to a steady state of the spectra, statistical moments, and the distribution functions of wave amplitudes. Numerical simulations are conducted across different Ursell parameters, revealing intriguing findings. Notably, it is observed that the spectra of the wave field converges to a stationary state in a statistical sense, while exhibiting statistical characteristics that deviate from a Gaussian distribution. Moreover, as the Ursell parameter increases, the positive skewness of the wave field intensifies, and the kurtosis increases. The investigation also involves the computation of the probability of rogue wave formation, revealing deviations from the Rayleigh distribution. Notably, the study uncovers distinct types of rogue waves, specifically referred to as "two sisters" and "three sisters" phenomena.

physics.flu-dyn↗
arXiv2023

Interaction of interfacial waves with an external force: The Benjamin-Ono equation framework

This study aims to explore the complex interactions between an internal solitary wave and an external force using the Benjamin-Ono equation as the theoretical framework. The investigation encompasses both asymptotic and numerical approaches. By assuming a small amplitude for the external force, we derive a dynamical system that describes the behavior of the solitary wave amplitude and the position of its crest. Our findings reveal three distinct scenarios: (i) resonance between the solitary wave and the external force, (ii) oscillatory motion with closed orbits, and (iii) displacement from the initial position while maintaining the wave direction. However, through numerical simulations, we observe a different relationship between the amplitude of the solitary wave and its crest position. Specifically, for external forces of small amplitude, the simulations indicate the presence of an unstable spiral pattern. Conversely, when subjected to external forces of larger amplitudes, the solitary wave exhibits a stable spiral trajectory which resembles the classical damped mass-spring system.

physics.flu-dyn↗
arXiv2023

Investigating overtaking collisions of solitary waves in the Schamel equation

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context of the Schamel equation. Our study reveals different regimes characterized by the behavior of the wave interactions. In certain regimes, the collisions maintain two well-separated crests consistently over time, while in other regimes, the number of local maxima undergoes variations following the patterns of $2\rightarrow 1\rightarrow 2\rightarrow 1\rightarrow 2$ or $2\rightarrow 1\rightarrow 2$. These findings demonstrate that the geometric Lax-categorization observed in the Korteweg-de Vries equation (KdV) for two-soliton collisions remains applicable to the Schamel equation. However, in contrast to the KdV, we demonstrate that an algebraic Lax-categorization based on the ratio of the initial solitary wave amplitudes is not feasible for the Schamel equation. Additionally, we show that the statistical moments for two-solitary wave collisions are qualitatively similar to the KdV equation and the phase shifts after soliton interactions are close to ones in integrable KdV and modified KdV models.

physics.plasm-ph↗
arXiv2023

Interactions of solitons with an external force field: Exploring the Schamel equation framework

This study aims to investigate the interactions of solitons with an external force within the framework of the Schamel equation, both asymptotically and numerically. By utilizing asymptotic expansions, we demonstrate that the soliton interaction can be approximated by a dynamical system that involves the soliton amplitude and its crest position. To solve the Schamel equation, we employ a pseudospectral method and compare the obtained results with those predicted by the asymptotic theory. Remarkably, our findings reveal a qualitatively agreement between the predictions and the numerical simulations at early times. Specifically, we classify the soliton interaction into three categories: (i) steady interaction occurs when the crest of the soliton and the crest of the external force are in phase, (ii) oscillatory behavior arises when the soliton's speed and the external force speed are close to resonance, causing the soliton to bounce back and forth near its initial position, and (iii) non-reversible motion occurs when the soliton moves away from its initial position without changing its direction.

physics.flu-dyn↗
arXiv2023

Nonlinear waves, modulations and rogue waves in the modular Korteweg-de Vries equation

Effects of nonlinear dynamics of solitary waves and wave modulations within the modular (also known as quadratically cubic) Korteweg - de Vries equation are studied analytically and numerically. Large wave events can occur in the course of interaction between solitons of different signs. Stable and unstable (finite-time-lived) breathers can be generated in inelastic collisions of solitons and from perturbations of two polarities. A nonlinear evolution equation on long modulations of quasi-sinusoidal waves is derived, which is the modular or quadratically cubic nonlinear Schrödinger equation. Its solutions in the form of envelope solitons describe breathers of the modular Korteweg - de Vries equation. The instability conditions are obtained from the linear stability analysis of periodic wave perturbations. Rogue-wave-type solutions emerging due to the modulational instability in the modular Korteweg - de Vries equation are simulated numerically. They exhibit similar wave amplification, but develop faster than in the Benjamin - Feir instability described by the cubic nonlinear Schrödinger equation.

nlin.PS↗
arXiv2022

Soliton interactions with an external forcing: the modified Korteweg-de Vries framework

The aim of this work is to study asymptotically and numerically the interaction of solitons with an external forcing with variable speed using the forced modified Korteweg-de Vries equation (mKdV). We show that the asymptotic predictions agree well with numerical solutions for forcing with constant speed and linear variable speed. Regarding forcing with linear variable speed, we find regimes in which the solitons are trapped at the external forcing and its amplitude increases or decreases in time depending on whether the forcing accelerates or decelerates.

physics.flu-dyn↗
arXiv2022

Solitary wave interactions with a periodic forcing: the extended Korteweg-de Vries framework

The aim of this work is to study numerically the interaction of large amplitude solitary waves with an external periodic forcing using the forced extended Korteweg-de Vries equation (feKdV). Regarding these interactions, we find that a solitary wave can bounce back and forth remaining close to its initial position when the forcing and the solitary wave are near resonant or it can move away from its initial position without reversing their direction. Additionally, we verify that the numerical results agree well within the asymptotic approximation for broad the forcings.

physics.flu-dyn↗