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Marcelo V. Flamarion

Publications and source records attributed to Marcelo V. Flamarion.

At least 19 recordsLinked to original sources
arXiv2026

The Kadomtsev-Petviashvili equation in conformal variables for waves over topography

The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by means of asymptotic expansions, obtain a Kadomtsev-Petviashvili type equation formulated in conformal variables as a model for weakly transversal surface waves propagating over topography. A key advantage of this formulation is that the topography, defined in the physical domain, does not need to be a smooth function, or even a function in the classical sense because, our asymptotic analysis relies on the effective depth, which comes through the Jacobian of the conformal map which is assumed to be a slowly varying function. The resulting equation provides a consistent extension of several well known weakly nonlinear dispersive wave models previously reported in the literature. Numerical simulations are performed to illustrate the newly derived equation.

math.AP↗
arXiv2025

The effect of normal electric fields on the Stokes drift

In periodic wave motion, particles beneath the wave undergo a drift in the direction of wave propagation, a phenomenon known as Stokes drift. While extensive research has been conducted on Stokes drift in water wave flows, its counterpart in electrohydrodynamic flows remains relatively unexplored. Addressing this gap, we investigate Stokes drift beneath periodic traveling irrotational waves on a dielectric fluid under the effect of normal electric fields. Through numerical simulations utilizing conformal mapping, we compute particle trajectories and analyze the resultant Stokes drift behaviors beneath periodic traveling waves. Our findings indicate that variations in the electric field impact particle velocities while maintaining trajectory shapes. Moreover, the kinetic energy associated with a particle depends on its depth location and is a non-decreasing convex function in a laboratory frame and a constant in a moving frame, as observed in water wave flows.

physics.flu-dyn↗
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

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

Full Euler equations for waves generated by vertical seabed displacements

We present a novel numerical method for simulating the generation and propagation of surface gravity waves by vertical seabed displacements. The cornerstone of our method is the computation of a time dependent conformal map which incorporates the time dependent geometry of the seabed and the wave profile along the free surface. This enables us to handle general geometric configurations of the seabed and the wave. As benchmark we reproduce the results of Hammack on tsunami generation and propagation. Our results show that Hammack's linear theory accurately predicts wave generation. However, as the velocity of the sea bed displacement increases, nonlinear effects become increasingly noticeable. Notably, when the seabed uplift occurs rapidly, the following nonlinear dynamics of the wave differ significantly from the linear dynamics usually associated with tsunami propagation.

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

Flow patterns induced by a moving disturbance in rotational flows within the forced Korteweg-de Vries equation

Flow structures beneath a moving disturbance along a water free surface in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. We compute the exact two branches of steady solutions in the disturbance moving frame. The velocity field in the bulk fluid is approximated which allows us to compute the flow structures beneath the free surface including stagnation points and Kelvin cat-eyes structures. We show that stagnation points exist only in one branch of solutions. The bifurcation of the flow is analyzed according to the intensity of the vorticity and the speed of the moving disturbance. Differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast.

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

Pressure anomalies beneath solitary waves with constant vorticity

While some works have investigated the particle trajectories and stagnation points beneath solitary waves with constant vorticity, little is known about the pressure beneath such waves. To address this gap, we investigate numerically the pressure beneath solitary waves in flows with constant vorticity. Through a conformal mapping that flats the physical domain, we develop a numerical approach that allows to compute the pressure and the velocity field in the fluid domain. Our experiments indicate that there exists a threshold vorticity such that pressure anomalies and stagnation points occur when the intensity of the vorticity is greater than this threshold. Above this threshold the pressure on the bottom boundary has two points of local maxima and there are three stagnation points in the flow, and below it the pressure has one local maximum and there is no stagnation point.

physics.flu-dyn↗
arXiv2023

An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields

The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked the attention of researchers from different areas. While there is an abundance of studies investigating the free surface wave properties, very few works have examined the associated velocity field within the bulk of the fluid. Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric fields in the framework of a weakly nonlinear theory. Elevation and depression solitary waves with constant vorticity are computed by a pseudo-spectral method and a parameter sweep on the intensity of the electric field is carried out in order to study its role in the appearance of stagnation points. We find that for elevation solitary waves the location of stagnation points does not change significantly with variations of the electric field. For depression solitary waves, on the other hand, the electric field acts as a catalyser that makes possible the appearance of stagnation points - in the sense that in its absence there is no stagnation point.

physics.flu-dyn↗
arXiv2022

Flow structure beneath periodic waves with constant vorticity under normal electric fields

Waves with constant vorticity and electrohydrodynamics flows are two topics in fluid dynamics that have attracted much attention from scientists for both the mathematical challenge and their industrial applications. The coupling of electric fields and vorticity is of significant research interest. In this paper, we study the flow structure of steady periodic travelling waves with constant vorticity on a dielectric fluid under the effect of normal electric fields. Through the conformal mapping technique combined with pseudo-spectral numerical methods, we develop an approach that allows us to conclude that the flow can have zero, two or three stagnation points according to variations in the voltage potential. We describe in detail the recirculation zones that emerge together with the stagnation points. Besides, we show that the number of local maxima of the pressure on the bottom boundary is intrinsically connected to the saddle points.

physics.flu-dyn↗
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↗