A study of Langmuir waves in plasmas

This paper talks about Langmuir waves in plasmas. The integration of the governing nonlinear Schrödinger’s equation with perturbation terms is carried out. Both Kerr law as well as the power law nonlinearity are considered.     

Stochastic perturbation of solitons for Alfven waves in plasmas

The stochastic perturbation of solitons due to Alfven waves in plasmas, is studied in this paper, in addition to the deterministic perturbation terms. The Langevin equations are derived and it is proved that the soliton travels through the plasma with a fixed mean velocity.     

Envelop solitons in dusty plasmas for warm dust

A nonlinear Schrödinger equation is obtained for the warm dusty plasmas. The modulational instability of envelop soliton is investigated in this paper. Both the temperature of the dust grains and the charge variations of dust grains affect the instability regions of the dusty plasmas. It also affect the amplitude and the width of the envelop soliton.     

Steady bifurcation and solitons in relativistic laser plasmas interaction

In this paper, steady bifurcation and solitons in relativistic laser plasmas interaction are investigated. At first, a new coupled equation for wake wave and the circularly polarized transversal electromagnetic wave is derived. It is a Hamiltonian system with two degrees of freedom. Then, a steady bifurcation analysis based on the coexistence of three different equilibrium states is given. Finally, a condition for predicting the existence of solitons is obtained in terms of the bifurcation control parameter and Hamiltonian function value. The soliton solutions are found numerically. It is shown that the solitons can exist in appropriate regime of vector potential frequency.     

Dynamics of solitons in plasmas for the complex KdV equation with power law nonlinearity

This paper obtains the 1-soliton solution of the complex KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The soliton perturbation theory for this equation is developed and the soliton cooling is observed for bright solitons. Finally, the dark soliton solution is also obtained for this equation.     

On the stability of spatially uniform Langmuir oscillations of electronic plasmas

We investigate the stability of spatially uniform, time-periodic solutions of the one-dimensional Vlasov–Maxwell system describing the longitudinal oscillations of an electronic plasma in an uniform neutralizing ion background. We show that such a stability problem can be trivially solved since the zero wave number mode of the electric field, i.e. its space average, performs pure Langmuir oscillations independently of the other modes. We however point out that such oscillations do affect on time average the evolution of the velocity distribution function in the frame at rest.     

Obliquely propagating dust-acoustic solitary waves in cosmic dust-laden plasmas

Obliquely dust-acoustic solitary waves in a collisional, magnetized dusty plasmas having cold dust grains, isothermal electrons, two temperature isothermal ions and stationary neutrals are studied via a reductive perturbation method. It is found that the effects of two temperature ions, collisions, magnetic field and directional cosine of the waves vector k along the x-axis have vital roles in the behavior of the dust acoustic solitary waves. The present investigation can be relevance to the electrostatic solitary structures observed in various cosmic dust-laden plasmas, such as Saturn’s E-ring, noctilucent clouds, Halley’s comet and interstellar molecular clouds.     

Quasi-stationary approximation for reaction-diffusion systems

Let S denote an idempotent semigroup, let W denote a Banach space. The space BV (S, W), which is the space of functions of bounded variation from S into W, is considered. It is shown that if f is in BV (S, W) and if W⁷ contains no copy of l_∞ then the value of f at every point is ∫_Γγ(s) dμ_f(γ), where Γ is the structure space of S and μ_f is an appropriate W valued measure. The hypothesis that $\text{W}{}^7$ has no copy of l_∞ is then dropped and necessary and sufficient conditions are given for μ_f to still have values in W. An application is made to Lipschitz functions and conditions are derived for μ_ff to be a Gelfand or a Pettis indefinite integral. Another application is made to product measures.     

Dynamical analysis of turbulence in fusion plasmas and nonlinear waves

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos.     

A route to chaos in a conservative system of three interacting Langmuir waves

This paper presents the results of numerical calculations of a route to chaos in a conservative Hamiltonian system of three Langmuir waves interacting with each other through three-wave couplings. The route is investigated by studying time series, power spectra, phase space portraits and Lyapnov exponents of wave variables for several combinations of wave vectors. The results show that the system follows a route which is very similar to the Ruelle–Takens–Newhouse scenario observed in dissipative systems, and widths and shifts of peaks in power spectra appeared due to the three moderate strength wave interactions. The breaks of tori in the system are also numerically investigated by studying the dependency of Maximum Lyapnov exponents for wave-variables on a parameter which represents the nonlinearity of the system.     

Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains

The stationary conditional, doubly limiting conditional and limiting conditional mean ratio quasi-stationary distributions are given for continuous-time Markov chains with denumerable state space both in terms of the transition matrixP(t) and the infinitesimal, generatorQ.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

Exact and approximate nonlinear waves generated by the periodic superposition of solitons

Toda [1], Boyd [2], Zaitsev [3], Korpel & Banerjee [4], and Whitham [5] have proved that many species of solitons may be cloned and superposed with even spacing to generateexact nonlinear, spatially periodic solutions (“cnoidal waves”). The equations solved by such “imbricate” series of solitary waves include the Korteweg-deVries, Cubic Schroedinger, Benjamin-Ono, and resonant triad equations. However, all existing theorems apply only when the solitons arerational ormeromorphic functions and the cnoidal waves areelliptic functions. In this note, we ask: does the exact soliton-superposition apply to non-elliptic solitons and cnoidal waves? Although a complete answer to this (very broad!) question eludes us, it is possible to offer a revealing counterexample. The quartic Korteweg-deVries equation has solutions which arehyperelliptic, and thus very special. Nevertheless, its periodic solutions are not the exact superposition of the infinite number of copies of a soliton. This is highly suggestive that non-elliptic extensions of the Toda theorem are rare or non-existent. It is intriguing, however, that the soliton-superposition generates a very goodapproximation to the hypercnoidal wave even when the solitons strongly overlap.     

Isoinertial family of operators and convergence of KdV cnoidal waves to solitons

In this paper we show the convergence of Korteweg-de Vries cnoidal waves to the limit soliton. It is proved that the convergence is uniform and in H²-norm, as the period of the solutions tends to infinity. Families of Hill operators are also studied. We obtain a condition under which families of operators are isoinertial. This condition is satisfied for classes of Hill operators that are obtained by linearization. Our application is to the family of linearized operators at the KdV cnoidal waves. It is proved that this family is isoinertial and also the value of the inertial index is calculated.     

Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution—actually an infinite family of quasi-stationary distributions— if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step (killing) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth–death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.      

Quasi-stationary distributions for structured birth and death processes with mutations

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.