arXiv subjects

Ioannis Kourakis

Publications and source records attributed to Ioannis Kourakis.

18 recordsLinked to original sources

arXiv2025

Travelling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework

The Black-Scholes framework is crucial in pricing a vast number of financial instruments that permeate the complex dynamics of world markets. Associated with this framework, we consider a second-order differential operator $L(x, {\partial_x}) := v^2(x,t) (\partial_x^2 -\partial_x)$ that carries a variable volatility term $v(x,t)$ and which is dependent on the underlying log-price $x$ and a time parameter $t$ motivated by the celebrated Dupire local volatility model. In this context, we ask and answer the question of whether one can find a non-linear evolution equation derived from a zero-curvature condition for a time-dependent deformation of the operator $L$. The result is a variant of the Harry Dym equation for which we can then find a family of travelling wave solutions. This brings in extensive machinery from soliton theory and integrable systems. As a by-product, it opens up the way to the use of coherent structures in financial-market volatility studies.

math.NA↗

arXiv2024

Nonlinear Ion-Acoustic Waves with Landau Damping in Non-Maxwellian Space Plasmas

The dynamics of nonlinear ion-acoustic solitary waves in the presence of kinetic (Landau type) damping have been investigated in a collisionless, non-magnetized electron-ion plasma. A cold ion fluid model, coupled to a Vlasov-type kinetic equation for the electron dynamics, has been adopted as a starting point. The electron population was assumed to be in a kappa-distributed state, in account of the non-Maxwellian behavior of energetic (suprathermal) electrons often observed in Space. A multiscale perturbation technique has led to an evolution equation for the electrostatic potential, in the form of a modified Korteweg-de Vries (KdV) equation, incorporating a non-local term accounting for Landau damping (associated with the electron statistics). Exact analytical solutions have been obtained, representing solitary waves undergoing amplitude decay over time. The combined effect of Landau damping and non-Maxwellian electron statistics (via the kappa parameter) on the characteristics of IASWs has been examined. Numerical integration of the evolution equation has been undertaken, to elucidate the importance of kinetic Landau damping on a shock-shaped initial condition. The results of this investigation aim to improve our understanding of the dynamics of nonlinear electrostatic waves under the influence of Landau damping in various space plasma environments.

physics.plasm-ph↗

arXiv2024

Coupled Nonlinear Schrödinger (CNLS) Equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

The nonlinear dynamics of two co-propagating electrostatic wavepackets, characterized by different wavenumbers and amplitudes, in a 1D non-magnetized plasma fluid model is considered, from first principles. The original plasma model, consisting of κ-distributed electrons evolving against a cold ion background, is reduced, by means of a multiple-scale perturbation method to a pair of asymmetric coupled nonlinear Schrödinger (CNLS) equations for the dynamics of the wavepacket envelopes. Exact analytical expressions are derived for the dispersion, self-modulation, and cross-modulation coefficients involved in the CNLS equations, as functions of the wavenumbers and the spectral index κcharacterizing the electron profile. An analytical investigation of the modulational instability (MI) properties of this pair of wavepackets reveals that MI occurs in most parts of the parameter space. The instability windows and the corresponding growth rate are calculated in a number of case studies. Two-wave interaction favors MI by extending its range of occurrence and by enhancing its growth rate. Growth rate patterns obtained for different κsuggest that deviation from Maxwellian equilibrium, for low κvalues, leads to enhanced MI of the interacting wave pair. To the best of our knowledge, the dynamics of two co-propagating wavepackets in a plasma described by a fluid model with κ-distributed electrons is investigated thoroughly with respect to their MI properties as a function of κfor the first time, in the framework of an asymmetric CNLS system. Although we have focused on electrostatic wavepacket propagation in non-Maxwellian plasma, the results are generic and may be used as basis to model energy localization in nonlinear optics, in hydrodynamics or in dispersive media with Kerr-type nonlinearities where MI is relevant.

physics.plasm-ph↗

arXiv2024

Modulational electrostatic wave-wave interactions in plasma fluids modeled by asymmetric coupled nonlinear Schrödinger (CNLS) equations

The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid evolving against a thermalized (Maxwell-Boltzmann distributed) electron background. A multiple-scale perturbation method is employed to reduce the original model equations to a pair of coupled nonlinear Schrödinger (CNLS) equations governing the dynamics of the wavepacket amplitudes (envelopes). The CNLS equations are in general asymmetric for arbitrary carrier wabvenumbers. Similar CNLS systems have been derived in the past in various physical contexts, and were found to support soliton, breather, and rogue wave solutions, among others. A detailed stability analysis reveals that modulational instability (MI) is possible in a wide range of values in the parameter space. The instability window and the corresponding growth rate are determined, considering different case studies, and their dependence on the carrier and the perturbation wavenumber is investigated from first principles. Wave-wave coupling is shown to favor MI occurrence by extending its range of occurrence and by enhancing its growth rate. Our findings generalize previously known results usually associated with symmetric NLS equations in nonlinear optics, though taking into account the difference between the different envelope wavenumbers and thus group velocities.

physics.plasm-ph↗

arXiv2024

Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas

An asymmetric pair of coupled nonlinear Schr{ö}dinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct carrier wavenumbers and amplitudes are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a $κ-$distributed electron background in 1D. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the carrier wavenumbers. The system admits various types of vector solitons (VSs), physically representing nonlinear localized electrostatic plasma modes. The possibility for either bright (B) or dark (D) type excitations for either of the two waves provides four combinations for the envelope pair (BB, BD, DB, DD). Moreover, the soliton parameters are also calculated for each type of VS in its respective area of existence. The dependence of the VS characteristics on the carrier wavenumbers and the spectral index $κ$ has been explored. In certain cases, the amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among the wavepackets. As $κ$ decreases from large values, modulational instability (MI) occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of VSs on the parameter plane(s) also varies significantly with decreasing $κ$, and in fact dramatically for $κ$ between $3$ and $2$. Deviation from the Maxwell-Boltzmann picture therefore seems to favor MI as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

physics.plasm-ph↗

arXiv2023

On the stability of two-dimensional modulated electrostatic wavepackets in non-Maxwellian dusty plasma-application in Saturn's magnetosphere

Motivated by observations of localized electrostatic wavepackets by the Voyager 1 and 2 and Cassini missions in Saturn's magnetosphere, we have investigated the evolution of modulated electrostatic wavepackets in a dusty plasma environment. The well-known dust-ion acoustic (DIA) mode was selected to explore the dynamics of multi-dimensional structures, by means of a Davey-Stewartson (DS) model, by taking into account the presence of a highly energetic (suprathermal, kappa-distributed) electron population in combination with heavy (immobile) dust in the background. The modulational (in)stability profile of DIA wavepackets for both negative as well as positive dust charge is investigated. A set of explicit criteria for modulational instability (MI) to occur is obtained. Wavepacket modulation properties in 3D dusty plasmas are shown to differ from e.g. Maxwellian plasmas in 1D. Stronger negative dust concentration results in a narrower instability window in the $K$ (perturbation wavenumber) domain and a suppressed growth rate. In the opposite manner, the instability growth rate increases for higher positive dust concentrations and the instability window gets larger. In a nutshell, negative dust seems to suppress instability while positive dust appears to favor the amplitude modulation instability mechanism.

physics.plasm-ph↗

arXiv2023

Dust ion-acoustic dromions in Saturn's magnetosphere

Motivated by observations of localized electrostatic wavepackets by the Cassini and (earlier) by Voyager 1 and 2 mission(s) in Saturn's magnetosphere, we have investigated the existence conditions and the dynamical evolution of localized multi-dimensional structures in the Saturnian dusty plasma environment. To this effect, we have adopted a plasma-fluid model for dust-ion acoustic (DIA) excitations, taking into account the presence of a highly energetic (suprathermal, kappa-distributed) electron population in combination with massive dust particulates in the background. A multiple scales perturbation method was shown to lead to a Davey-Stewartson (DS) system of evolution equations, if one assumes perpendicular carrier wave propagation across the magnetic field (direction). The system is then shown to possess two regimes mainly, known in the literature as DS-I and DS-II. In the former case, if certain conditions are fulfilled, exponentially localized solutions are obtained, known as dromions. The combined effects of various physical parameters such as the electron spectral index, the ambient magnetic field (strength), and the dust concentration have been examined. A numerical investigation reveals that the dromion amplitude increases with higher dust concentration, while it decreases for lower $κ_e$ (i.e. with an increase in the suprathermal electron population component). A stronger magnetic field results in higher amplitude but narrower dromions. Our results provide a comprehensive framework for modeling modulated electrostatic wavepackets, in direct comparison with experimental data in planetary environments, in Saturn's magnetosphere, and elsewhere.

physics.plasm-ph↗

arXiv2011

Note on the single-shock solutions of the Korteweg-de Vries-Burgers equation

The well-known shock solutions of the Korteweg-de Vries-Burgers equation are revisited, together with their limitations in the context of plasma (astro)physical applications. Although available in the literature for a long time, it seems to have been forgotten in recent papers that such shocks are monotonic and unique, for a given plasma configuration, and cannot show oscillatory or bell-shaped features. This uniqueness is contrasted to solitary wave solutions of the two parent equations (Korteweg-de Vries and Burgers), which form a family of curves parameterized by the excess velocity over the linear phase speed.

physics.space-ph↗

arXiv2011

Electron beam-plasma interaction in a dusty plasma with excess suprathermal electrons

The existence of large-amplitude electron-acoustic solitary structures is investigated in an unmagnetized and collisionless two-temperature dusty plasma penetrated by an electron beam. A nonlinear pseudopotential technique is used to investigate the occurrence of stationary-profile solitary waves, and their parametric dependence on the electron beam and dust perturbation is discussed.

astro-ph.HE↗

arXiv2011

Electron-acoustic solitary waves in the presence of a suprathermal electron component

The nonlinear dynamics of electron-acoustic localized structures in a collisionless and unmagnetized plasma consisting of "cool" inertial electrons, "hot" electrons having a kappa distribution, and stationary ions is studied. The inertialess hot electron distribution thus has a long-tailed suprathermal (non-Maxwellian) form. A dispersion relation is derived for linear electron-acoustic waves. They show a strong dependence of the charge screening mechanism on excess suprathermality (through κ). A nonlinear pseudopotential technique is employed to investigate the occurrence of stationary-profile solitary waves, focusing on how their characteristics depend on the spectral index κ, and the hot-to-cool electron temperature and density ratios. Only negative polarity solitary waves are found to exist, in a parameter region which becomes narrower as deviation from the Maxwellian (suprathermality) increases, while the soliton amplitude at fixed soliton speed increases. However, for a constant value of the true Mach number, the amplitude decreases for decreasing κ.

astro-ph.SR↗

arXiv2004

Modulated dust-acoustic wave packets in a plasma with non-isothermal electrons and ions

Nonlinear self-modulation of the dust acoustic waves is studied, in the presence of non-thermal (non-Maxwellian) ion and electron populations. By employing a multiple scale technique, a nonlinear Schrodinger-type equation (NLSE) is derived for the wave amplitude. The influence of non-thermality, in addition to obliqueness (between the propagation and modulation directions), on the conditions for modulational instability to occur is discussed. Different types of localized solutions (envelope excitations) which may possibly occur are discussed, and the dependence of their characteristics on physical parameters is traced. The ion deviation from a Maxwellian distribution comes out to be more important than the electron analogous deviation alone. Both yield a de-stabilizing effect on (the amplitude of) DAWs propagating in a dusty plasma with negative dust grains. The opposite effect, namely a tendency towards amplitude stabilization, is found for the case of positively charged dust presence in the plasma.

physics.plasm-ph↗

arXiv2004

Nonlinear theory of dust lattice mode coupling in dust crystals

Quasi-crystals formed by charged mesoscopic dust grains (dust lattices), observed since hardly a decade ago, are an exciting paradigm of a nonlinear chain. In laboratory discharge experiments, these quasi-lattices are formed spontaneously in the sheath region near a negative electrode, usually at a levitated horizontal equilibrium configuration where gravity is balanced by an electric field. It is long known (and experimentally confirmed) that dust-lattices support linear oscillations, in the longitudinal (acoustic mode) as well as in the transverse, in plane (acoustic-) or off-plane (optic-like mode) directions. Either due to the (typically Yukawa type) electrostatic inter-grain interaction forces or to the (intrinsically nonlinear) sheath environment, nonlinearity is expected to play an important role in the dynamics of these lattices. Furthermore, the coupling between the different modes may induce coupled nonlinear modes. Despite this evidence, the elucidation of the nonlinear mechanisms governing dust crystals is in a rather preliminary stage. In this study, we derive a set of (coupled) discrete equations of motion for longitudinal and transverse (out-of-plane) motion in a one dimensional model chain of charged dust grains. In a continuum approximation, i.e. assuming a variation scale which is larger than the lattice constant, one obtains a set of coupled modified Boussinesq-like equations. Different nonlinear solutions of the coupled system are discussed, based on localized travelling wave ansatze and on coupled equations for the envelopes of co-propagating quasi-linear waves.

physics.plasm-ph↗

arXiv2004

Modulated envelope localized wavepackets associated with electrostatic plasma waves

The nonlinear amplitude modulation of known electrostatic plasma modes is examined in a generic manner, by applying a collisionless fluid model. Both cold (zero-temperature) and warm fluid descriptions are discussed and the results are compared. The moderately nonlinear oscillation regime is investigated by applying a multiple scale technique. The calculation leads to a Nonlinear Schrodinger-type Equation (NLSE), which describes the evolution of the slowly varying wave amplitude in time and space. The NLSE admits localized envelope (solitary wave) solutions of bright- (pulses) or dark- (holes, voids) type, whose characteristics (maximum amplitude, width) depend on intrinsic plasma parameters. Effects like amplitude perturbation obliqueness, finite temperature and defect (dust) concetration are explicitly considered. The relevance with similar highly localized modulated wave structures observed during recent satellite missions is discussed.

physics.plasm-ph↗

arXiv2004

Theory of solitary waves in complex plasma lattices

A comprehensive analytical theory for nonlinear excitations related to horizontal (longitudinal, acoustic mode) as well as vertical (transverse, optical mode) motion of charged dust grains in a dust crystal is presented. Different types of localized excitations, similar to those well known in solid state physics, are reviewed and conditions for their occurrence and characteristics in dusty plasma crystals are discussed. By employing a continuum approximation (i.e. assuming a long variation scale, with respect to the inter-particle distance) a dust crystal is shown to support nonlinear kink-shaped supersonic solitary excitations, associated with longitudinal dust grain displacement, as well as modulated envelope localized modes associated with either longitudinal or transverse oscillations. Although a one-dimensional crystal is considered for simplicity, the results in principle apply to a two-dimensional lattice if certain conditions are satisfied. The effect of mode-coupling is also briefly considered. The relation to previous results on atomic chains, and also to experimental results on strongly-coupled dust layers in gas discharge plasmas, is briefly discussed.

physics.plasm-ph↗

arXiv2004

Lagrangean formulation of ion- and dust-ion-acoustic waves

Ion-acoustic modes propagating in unmagnetized dusty plasmas are studied by applying a generic collisionless fluid model. An Eulerian-to-Lagrangean variable transformation leads to a new system of evolution equations, which may be combined into a single (rather complex) equation for the mean ion density. The linear and weakly nonlinear oscillations regimes are studied and their modulational stability is investigated. This study extends known previous results on the Lagrangian description of electron- or ion-acoustic electrostatic plasma modes. The relevance with the description of dust-ion acoustic waves propagating in a dusty plasma is also discussed.

physics.plasm-ph↗

arXiv2004

Intrinsic localized modes in dust lattices

Intrinsic Localized Modes (ILM) (or Discrete Breathers, DB) are localized oscillatory modes known to occur in atomic or molecular chains characterized by coupling and/or on-site potential nonlinearity. Quasi-crystals of charged mesoscopic dust grains (dust lattices), which have been observed since hardly a decade ago, are an exciting paradigm of such a nonlinear chain. In gas-discharge experiments, these crystals are subject to forces due to an externally imposed electric and/or magnetic field(s), which balance(s) gravity at the levitated equilibrium position, as well as to electrostatic inter-grain interaction forces. Despite the profound role of nonlinearity, which may be due to inter-grain coupling, mode- coupling and to the sheath environment, the elucidation of the nonlinear mechanisms governing dust crystals is still in a preliminary stage. This study is devoted to an investigation, from very first principles, of the existence of discrete localized modes in dust layers. Relying on a set of evolution equation for transverse charged grain displacements, we examine the conditions for the existence and sustainance of discrete localized modes and discuss the dependence of their characteristics on intrinsic plasma parameters. In addition, the possibility of DB stabilisation via an external force is discussed.

physics.plasm-ph↗

arXiv2004

Nonlinear whitlerons

Recently, observations from laboratory experiments have revealed amplitude modulation of whistlers by low-frequency perturbations. We here present theoretical and simulation studies of amplitude modulated whistler solitary waves (whistlerons) and their interaction with background low-frequency density perturbations created by the whistler ponderomotive force. We derive a nonlinear a nonlinear Schrodinger equation which governs the evolution of whistlers in the presence of finite-amplitude density perturbations, and a set of equations for arbitrary large amplitude density perturbations in the presence of the whistler ponderomotive force. The governing equations studied analytically in the small amplitude limit, and are solved numerically to show the existence of large scale density perturbations that are self-consistently created by localized whistlerons. Our numerical results are in good agreement with recent experimental results where the the formation of modulated whistlers and solitary whister waves were formed.

physics.plasm-ph↗

arXiv2004

Weakly nonlinear vertical dust grain oscillations in dusty plasma crystals in the presence of a magnetic field

The weakly nonlinear regime of transverse paramagnetic dust grain oscillations in dusty (complex) plasma crystals is discussed. The nonlinearity, which is related to the sheath electric/magnetic field(s) and to the inter--grain (electrostatic/magnetic dipole) interactions, is shown to lead to the generation of phase harmonics and, in the case of propagating transverse dust-lattice modes, to the modulational instability of the carrier wave due to self--interaction. The stability profile depends explicitly on the form of the electric and magnetic fields in the plasma sheath. The long term evolution of the modulated wave packet, which is described by a nonlinear Schrödinger--type equation (NLSE), may lead to propagating localized envelope structures whose exact forms are presented and discussed. Explicit suggestions for experimental investigations are put forward.

physics.plasm-ph↗