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2. electrostatic streaming instabilities Califano, Galeotti, and Briand ( 2007 ); Silin, Sydora, and Sauer ( 2007 ); Califano and Lontano ( 2005 |
3. nonlinear Landau damping Manfredi ( 1997 ); Brunetti, Califano, and Pegoraro ( 2000 ); Abbasi et al. 2007 4. chirp-driving (external driving) mechanism Bertsche, Fajans, and Friedland ( 2003 ); Trivedi and Ganesh ( 2016 ); Peinetti et al. 2005 |
5. non-periodic plasmas Fijalkow and Nocera ( 2003 6. random phase approach Pezzi, Valentini, and Veltri ( 2014 7. transient approach Zhou and Hutchinson ( 2016 |
These approaches develop the solitary waves and its corresponding electron hole naturally and during the temporal evolution of the simulation. They basically start from an unstable initial condition and reach a steady state which is a subclass of Bernstein-Greene-Kruskal (BGK) modes Bernstein, Greene, and Kruskal ( 195... |
However, one should note that solitary waves and their accompanying holes coming from these mechanisms are not necessarily equal to one another. In other words, they can belong to different subclasses of BGK modes. The above list refers to just the recent papers on these mechanisms and obviously there are many other wo... |
The family of solutions generated through the Sagdeev method does not generally correspond to asymptotic solutions in the dynamic evolution. Hence electron holes depending on the parameters. Furthermore, these is no guarantee that the BGK modes created in the different generation mechanisms can be reproduced by Sagdeev... |
To give a few example of other related topics, it is reported by Saeki and Genma Saeki and Genma ( 1998 that ion dynamics cause an electron hole to split into to oppositely-propagating holes if its velocity is in order or slower than ion-acoustic speed. We have adopted these well-known phenomena as the benchmark of our... |
It is also can be generally assumed that any large hole can break up into smaller holes in a process called “chain formation”. It is shown that the pulses produced by this process are truly solitary waves via long-time simulation Hosseini Jenab and Spanier ( 2016 |
and solitons by focusing on their mutual collisions Hosseini Jenab and Spanier ( 2017c in the fully kinetic regime. There has been a strong line of research on the standing electron holes and their dynamics and interaction with ions and each other by focusing on their merging and break-up process Eliasson and Shukla ( ... |
I.3 The scope of this investigation The question of our study is: Are “Sagdeev solutions” equal to “solitary waves”? We will provide an answer to this question in electrostatic fully-kinetic ion-acoustic regime. To do so, we need to study the temporal behavior of Sagdeev solutions and their corresponding electron holes... |
Here, we are reporting the results in the ion-acoustic regime, i.e. considering electron-ion plasmas. Fully-kinetic ion-acoustic regime refers to the fact that the dynamics of both electrons and ions are followed by the Vlasov equation. Two types of background distribution functions have been considered; the Maxwellian... |
We proceed as follows: Firstly, the pulse profile is determined numerically based on the Sagdeev pseudo-potential approach while accounting for trapped and reflected particles (see Sec. II.1 ). Secondly, using this pulse profile and the corresponding distribution function as initial conditions for the Vlasov code, toge... |
II Basic Equations and the Numerical Schemes Equations and variables are normalized to ionic parameters such as the ion plasma frequency, the ion Debye length and the ion thermal velocity. The details are presented in table |
II.1 Fully kinetic Sagdeev pseudo-potential approach II.1.1 Constructing Sagdeev pseudo-potential The Sagdeev approach starts with manipulation of Poisson’s equation |
[EQUATION] in which [MATH] stands for the density of each species, namely electrons [MATH] ) and ions ( [MATH] ). This equation is multiplied by [MATH] and then integrated over [MATH] |
to produce the Sagdeev pseudo-potential: [EQUATION] After constructing [MATH] , one can easily integrate once more over it and construct the electric potential [MATH] and the electric field [MATH] |
In order to derive a solution, the species’ densities should be computed as a function of electrical potential ( [MATH] ). Analytically, there are two approaches for such computation, i.e. fluid and kinetic. In the first approach, fluid equations (including equation of momentum and continuity equation) are used in a co... |
[EQUATION] in which, [MATH] (unperturbed number density multiplied by normalized charge) stands for unperturbed charge density. Due to the normalization used, unperturbed number density of each species is equal to one [MATH] and [MATH] |
Here, we have adopted the kinetic approach in our numerical model, and hence a valid distribution function is needed as the starting point of our numerical approach. We start by recalling a well-known property of the Vlasov equation; a distribution function that depends in any way on the particle constants of motion is... |
In this study, we are focusing on two different types of distribution functions, namely the Maxwellian distribution function: [EQUATION] |
and the Kappa distribution function Hellberg et al. 2009 [EQUATION] in which [MATH] and [MATH] stands for the usual gamma function. The Kappa distribution function was proposed for the first time by V. M. Vasyliunas Vasyliunas ( 1968 |
when studying “low-energy electrons in the evening sector of the magnetosphere” to explain the long tail appearing in the velocity distribution at high (velocity) values. It is designed to address the case of superthermal particle populations. This type of distribution function has been used for modeling non-Maxwellian... |
Pierrard and Lazar ( 2010 ); Sarri et al. 2010 ); Maksimovic, Pierrard, and Riley ( 1997 ); Hellberg and Mace ( 2002 The Kappa distribution function depends on a real parameter ( [MATH] ), which measures the magnitude of the superthermal tail in the distribution function, representing the excess of highly energetic par... |
II.1.2 Constructing the potential dependency The aim of this subsection is to present a simple and easy way to derive the Schamel distribution function from the energy point of view Schamel ( 1971 This approach can be easily implemented into the Vlasov-solver. Furthermore it can be easily extend to different types of d... |
Assuming a potential pulse moving with a velocity ( [MATH] ) in the laboratory frame, the following steps should be taken in order to consider its effect on the distribution fucntion: |
1. transforming the kinetic energy into the co-moving frame: [MATH] in which [MATH] 2. finding the shifted velocity in the co-moving frame: |
[MATH] 3. transforming the kinetic energy back to the laboratory frame: [MATH] in which [MATH] The shifted velocity equation indicates that the particle with positive potential energy [MATH] has lost some portion of its kinetic energy due to the interaction with the potential and stored it as potential energy (in the c... |
In order to add the effect of trapped particles, one can use a shifted Maxwellian distribution function for modeling them, as suggested by Schamel Schamel ( 1971 The trapped population can be achieved by following the steps below: |
1. the velocity is transformed into the moving frame: [MATH] 2. the shifted velocity [MATH] and the shifted kinetic energy in the moving frame |
[MATH] are calculated. 3. kinetic energy of the moving frame [MATH] ) is added to the kinetic energy of the particles in the moving frame ( [MATH] with a coefficient [MATH] |
[MATH] Based on [MATH] , the distribution function of trapped particles can take three different types of shapes, namely hole [MATH] ), plateau [MATH] ) and hump [MATH] ). Note that since the above method is implemented for both positive and negative potential energy, the two cases of reflected and trapped particles ar... |
Hence, the total form of the distribution function can be written as follows: [EQUATION] in which: [EQUATION] The above distribution function (written in form of kinetic energy) can be shown to be equivalent to the Schamel distribution function Schamel ( 1971 |
II.1.3 Summary In summary, the path to find the Sagdeev solutions [MATH] consists of four steps: 1. [MATH] : by using the above recipe (Sec. II.1.2 ), the distribution function’s dependency on the potential is calculated (Eq. |
2. [MATH] : by integration over velocity, the density can be obtained (Eq. ). 3. [MATH] : by following the same routine for all the species, the Sagdeev pseudo-potential can be constructed by integration over [MATH] (Eq. ). |
4. [MATH] : the potential profile [MATH] can found from the Sagdeev pseudo-potential [MATH] by integrating (Eq. ). Three variables affect the Sagdeev pseudo-potential ( [MATH] ) and pulse potential profile ( [MATH] ) , namely the shape of the distribution function, the velocity of the solitary wave ( [MATH] and the tra... |
While the above procedure does generate steady state solutions, there is no guarantee these solutions are stable. One should note that through out the derivation of the Sagdeev solutions, there is no temporal evolution in the model. In the Sec. III.2 we will study the long-time behavior (i.e. stability ) of Sagdeev sol... |
II.2 Fully kinetic Vlasov-Poisson method Here, the study is restricted to electron-ion plasmas, and both species’ dynamics are followed by the Vlasov equations: |
[EQUATION] while Poisson’s equation provides the force (electric) field: [EQUATION] where [MATH] represents the corresponding species. Vlasov and Poisson equations are coupled by density integrations for each species to form a closed set of equations: |
[EQUATION] [EQUATION] In which [MATH] stands for the number density. [MATH] is the (normalized) unperturbed value of the charge density. The quasi-neutrality condition stays true: |
[EQUATION] The kinetic simulation approach utilized here is based on the Vlasov-Hybrid Simulation (VHS) method in which a distribution function is modeled by phase points |
Nunn ( 1993 ); Kazeminezhad, Kuhn, and Tavakoli ( 2003 ); Abbasi, Jenab, and Pajouh ( 2011 The arrangement of phase points in the phase space at each time step provides the distribution function, and hence all the kinetic momentums such as density, entropy, etc. The initial value of distribution function associated to ... |
Each temporal update of the simulation consists of three steps which are summarized below. 1. Integration of the distribution function over velocity direction to achieve the number density (Eq. ). |
2. Calculation of the electric potential and field by solving Poisson’s equation (Eq. ). 3. Determination of the new arrangement of phase points in the phase space for the next step by solving the Vlasov equation for each species based on the characteristics method utilizing a leap-frog scheme (Eq. ). |
As for the initial condition of our Vlasov simulations, we need to have the distribution function of each of the species. In order to implement the calculated pulse profile [MATH] self-consistently, we use the distribution function described in Sec. II.1.2 |
to produce the initial distribution functions, which have the trapped and reflected population inside them. Furthermore, the initial conditions include some parameters and variables which need to set before the simulation starts. These are explained in details in the next section. |
II.3 variables and parameters The constant parameters which remain fixed through all of our simulations include: the mass ratio [MATH] the temperature ratio [MATH] |
and [MATH] where [MATH] is the grid size on the spatial direction. The length of the simulation box is [MATH] or [MATH] which is specified in each section. Periodic boundary conditions are adopted on the spatial direction. Furthermore, the input parameters for each set of the simulation set consists of three variables: |
the shape of the distribution function, either Maxwellian or Kappa the trapping parameter ( [MATH] the velocity of the pulse ( [MATH] |
Note that by ionic normalization (table ), the ion sound velocity, the electron plasma frequency and the electron thermal velocity are |
[MATH] [MATH] and [MATH] respectively. In order to follow the form of many theoretical papers. In what follows, we will express the velocity of solitary waves using the Mach number [MATH] |
In the case of [MATH] the simulation grid consists of [MATH] cells for electrons phase space and [MATH] for ions. The difference between the species phase space is in the velocity direction, the electron velocity cut-off is [MATH] |
while for ions it is [MATH] Initially there are 16 phase points per cell in the phase space. Hence the total number of phase points for electrons (ions) is [MATH] [MATH] For the case of [MATH] , grid size is [MATH] |
and hence [MATH] and [MATH] Since the dynamical part of the simulation method utilizes the Vlasov equation, which is the collision-less Boltzman equation, simulations must conserve entropy and other forms of Casimir invariants Elskens, Escande, and Doveil ( 2014 as well as the total energy. It is vital for any (collisi... |
[EQUATION] for all the simulations reported in this study in order to be fully transparent about the research presented here. [MATH] represents the total energy ( [MATH] ) and entropy ( [MATH] ). |
[MATH] implies kinetic energy and [MATH] indicates potential energy of the system. III Results and Discussion III.1 Benchmark: the effect of ion dynamics on standing electron holes |
In this section, the well-known example Saeki and Genma ( 1998 ); Saeki and Juul Rasmussen ( 1991 of the effect of ion dynamics on an initially standing electron hole is taken as a benchmark of our simulation code. It is considered a well-established fact that an electron hole with zero velocity will stand fixed if the... |
and theoretical studies Saeki and Juul Rasmussen ( 1991 However, when the ion dynamics is considered (the ions can move and react to the field) the result is totally different. In the electron phase space, the standing hole breaks up into two oppositely moving holes, which are completely symmetric to one another since ... |
by considering electron following the Maxwellian distribution function, and ions as motionless species providing the background positive charge. |
Fig. presents the results of a simulation without the dynamics of ions. The results show that the electron hole will stand still as predicted. When the dynamics of ions is added to the simulation (see Fig. ), the electron hole breaks up into two oppositely propagating holes. The break-up happens in the following steps.... |
Three things should be noted which can establish the correctness and precision of the simulation approach in these two test simulations. In case of the first simulation, the trapped population has to be confined in the trapped region for the entire time of the simulation. Fig. shows that despite the rapid dynamics insi... |
In case of the second simulation, the symmetry of the dynamics in the positive and negative side of phase space should be observed. The simulation code can follow the symmetry between the two moving holes with high precision indicating that the numerical code is capable of modeling nonlinear processes. Furthermore, the... |
III.2 Stability of the Sagdeev solutions Initially, we are presenting the results of simulations carried out employing the Maxwellian distribution functions for both electrons and ions. The first simulation (case I) includes the electron hole with [MATH] and [MATH] The flat-topped area in the electron distribution func... |
[MATH] Long running simulation ( [MATH] ) verifies the stability of the solution. The temporal evolution of the system in presented in three kinds of plots to analyze this stability. |
Firstly, profiles of physical quantities in both phase space and real space such as distribution function, charge density, electric potential and field are depicted in Fig. It shows the state of the system in the beginning of the simulation (initial condition) and at the last time step ( [MATH] ). Since the time step i... |
comes after [MATH] computational time steps. This long time simulation can easily show any instability or deformation in the profile of the solitary waves. As can be observed, the profile stands unaltered during the simulation. Fig. also displays the temporal evolution of energy and entropy, which shows robust conserva... |
In order to provide a clear-cut image of the phase space evolution, the initial phase points are marked with extra markers based on their energy. These markers are just used for diagnosis purposes and has no effect or involvement in the dynamical process. By depicting these markers in the phase space, it is possible to... |
shows the stability of these trajectories for electron phase space in case I of the simulations. The closed rings among these trajectories refers to the trapped electrons, which are marked with negative values, reflecting that their energy is smaller than the potential energy keeping them trapped. |
The trajectories of energy are presented for ions as well, which reflects the dynamics of reflected ions (see Fig. ). Depicting the distribution function of these particles can not represent the dynamics. Since the value of the distribution function for this area of the phase space is close to zero, at least ten orders... |
[MATH] Therefore, the reflected ions population are presented using the markers’ values in stead of the distribution function. Furthermore, we have presented the temporal evolution of the three major characteristics of the solitary waves, e.g. amplitude, width and velocity. Their stability during propagation reflects t... |
shows the similarity between the figures and confirm our interpretation of these oscillations. The oscillation originating from the propagation of ion-acoustic waves can be witnessed on the larger time scale and later in the simulation. Fig. shows these oscillations happening clearly for [MATH] in the two features of s... |
Changing the value of [MATH] from zero results in the modification of the amplitude, width and shape of the Sagdeev solutions. We have carried out simulations with different values to examine the stability in the parameter space of ( [MATH] ). The simulation results are presented in Fig for case II ( [MATH] ). The para... |
as well which can be found as multimedia alongside this paper. Case II presents a much faster solitary wave than case I with a deeper hole in the electron phase space. The simulation results prove the stability of this fast moving solitary waves during the long time propagation. Based on the analytical fluid approach t... |
[EQUATION] Our simulations reveals that by removing these constrains and considering the kinetic effects, solitary waves can move much faster, at least as high as [MATH] based on case II of our simulations. |
Next, we have employed our fully kinetic simulation approach for the Kappa distribution function with [MATH] Fig. 11 shows the stability of the ion-acoustic solitary waves in this regime. |
III.3 Instability of the Sagdeev solutions Instability of Sagdeev solutions has been reported just once (to the best of our knowledge), recently in a PIC simulation study by Zhou and Hutchinson Zhou and Hutchinson ( 2017 They have employed a transient approach to seed the electron hole and when it reached its steady-st... |
The process of instability observed in our simulation appears as an energy exchange between electrons and ions. Fig. 14 demonstrates the growth of kinetic energy of ions in the expense of electrons’ kinetic energy, while the change in electric energy is negligible. This process has been predicted to exist by the theory... |
Our simulations clearly show that the trapping parameter play a crucial role in the stability of the Sagdeev solution. For any specific velocity (Mach number), there exists a threshold in the trapping parameter ( [MATH] ) which separates the stable solutions from unstable ones. In cases reported here, it is [MATH] for ... |
A full study of this phenomena will be communicated in the future reports. However, we are able to claim that the fully kinetic Sagdeev solutions in the ion-acoustic regime are not guaranteed to be stable. This can affect the implication of the Sagdeev method in more complex scenarios such as multi-species plasmas. Wha... |
IV Conclusions The nonlinear solutions of the Sagdeev pseudo-potential approach to the ion-electron plasmas are studied in the fully-kinetic ion-acoustic regime. This research stands as the first attempt to verify Sagdeev solutions as solitary waves in the kinetic regime (to the best of our knowledge). Our simulation r... |
Moreover, our simulations show the instability of Sagdeev solutions for both slow and fast moving ones. Instability grows from small numerical noises and causes the nonlinear pulse to lose its trapped population by emitting ion-acoustic waves packets on its propagation trail. We have found that there exists a clear thr... |
In case of stable solutions, verification has been presented in the three types of plots reflecting different aspects of the dynamics in both phase space and real space. In the real space, the temporal evolution of three quantities, namely charge density, electric field and potential are analyzed for any instability. F... |
Furthermore, high-speed ion-acoustic solitary waves ( [MATH] ) are presented which demonstrates the existence of solitary waves out of existence regime proposed by fluid model. This shows the great impact of the kinetic effects on the existence regime, ignored in the fluid model. Kinetic effects such as trapping and re... |
In order to validate our simulation code, we have chosen the nonlinear problem of the effect of ion dynamics on standing electron holes as a benchmarking test. The results confirms the theoretical predictions and shows the robustness of the VHS (Vlasov-Hybrid Simulation) method in following the temporal evolution of th... |
Furthermore, we have presented the conservation of both energy and entropy for all the simulations discussed in this study. Deviation from the initial value for all the conserved quantities reported here, stays below [MATH] |
Supplementary Material Temporal evolution of ion acoustic solitary waves for a movie for [MATH] and [MATH] is presented for different variables including: electron distribution function (first row), electron number density(second row), ion distribution function (third row), ion number density (fourth row), charge densi... |
VI Outlook This self-consistent kinetic approach provides an accurate and strong method to study the solitary waves in different environments such as multi-species plasmas with different types of distribution functions. It is being developed and adopted to study structures such as super-solitons Verheest, Hellberg, and... |
Sheridan and Lonngren ( 1999 and other types of multi-species plasmas and has revealed some interesting phenomenon, which will be discussed in forthcoming publications. |
The work on examining the relationship between stability of Sagdeev solutions and other parameters such as mass and temperature ratio is underway. We are also examining the dependency of the critical trapping parameter ( [MATH] ) on the velocity ( [MATH] ). |
Acknowledgements. G. Brodin and S. M. Hosseini Jenab would like to acknowledgment financial support by the Swedish Research Council, grant number 2016-03806. This work is based upon research supported by the National Research Foundation (NRF) and Department of Science and Technology (DST) from Republic of South Africa.... |
# Source: arxiv 1806.07194 # Title: A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations # Sections: all # Downloaded: 2026-03-02T08:56:17.671200+00:00 |
A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations Abstract. |
We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs... |
1. Introduction Stochastic partial differential equations (SPDEs) are a standard part of a scientist’s toolbox to study the effect of fluctuations and to include mesoscopic effects in natural and engineered systems. Applications range from genetics, epidemic outbreaks, and population dynamics to chemical engineering |
. The inclusion of noise can have profound impact on the dynamics of the propagation of coherent structures We treat here SPDEs where the deterministic part exhibits symmetry; in particular, we study SPDEs with translational symmetry supporting travelling waves. We set out here to reduce the complexity of such infinite... |
. Rather than employing perturbative expansions to study small noise perturbations or judiciously chosen moment closure schemes , we adopt here the perspective of decomposing the dynamics into the dynamics on the symmetry group and the dynamics orthogonal to it, established for deterministic partial differential equati... |
). This symmetry perspective of spatially extended systems has proved successful in studying and classifying pattern formation , constructing efficient numerical methods for equivariant systems |
, and in studying Hamiltonian systems such as planetary dynamics and observed spectra of [MATH] molecules Within the symmetry perspective PDEs are cast into a skew product of ordinary differential equations whereby the dynamics on the symmetry group is driven by the so-called shape dynamics. In the case of a travelling... |
[EQUATION] where [MATH] represents the translation variables. If the shape dynamics ( ) consists of an equilibrium [MATH] we obtain [MATH] with [MATH] . This includes the case of a travelling wave moving witorganizh constant speed [MATH] mentioned above. |
This paper is concerned with employing this framework to SPDEs with weak noise. The main idea of this work is that the effect of the noise will be controlled in the shape dynamics which is dominated by strong contraction to the travelling wave solution; along the neutral direction on the group orbit, however, the noise... |
subjected to multiplicative noise as well as spatially localized additive noise. We show how the collective coordinate approach quantitatively describes the speed of the front propagation and the shape of the front. We find that the addition of noise leads to a slowing down and a steepening of the front in the case of ... |
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