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.     

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.     

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.     

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.     

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.     

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.      

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.     

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.     

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.     

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.     

Vector acoustic solitons from the coupling of long and short waves in a paramagnetic crystal

We investigate the propagation of a longitudinal-transverse elastic pulse in a statically deformed crystal containing paramagnetic impurities and placed in an external magnetic field. We derive a system of three nonlinear wave equations describing the interaction of the pulse with the paramagnetic impurities in the quasiresonance approximation in the Faraday geometry. We assume that the transverse components of the pulse, which cause quantum transitions, have carrier frequencies and are short-wave (acoustic), while the longitudinal component has no carrier frequency and is long-wave. We show that in the case of an equilibrium initial distribution of populations of quantum levels of paramagnetic impurities, the coupling between the longitudinal and transverse components is weak, the pulse is therefore strictly transverse, and its dynamics are described by the Manakov system. With a nonequilibrium initial distribution of populations, conditions of effective interaction between all components of the elastic pulse can be reached, and their nonlinear dynamics are described by a vector generalization of the Zakharov equations. In the case of a unidirectional propagation of the pulse, these equations reduce to the Yajima-Oikawa vector system. We show that the obtained system of equations and its version with an arbitrary number of short-wave components can be integrated using the inverse scattering transform. We construct infinite hierarchies of solutions of the Yajima-Oikawa vector system (including a solution on a nontrivial background). We consider stationary (complex-valued Garnier system) and self-similar reductions of that system, also admitting a representation in the form of compatibility conditions.     

Stability criterion for solitons

Institute of Chemical Physics, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 1, pp. 28–33, January, 1990.     

Approximating the Quasi-stationary Distribution of the SIS Model for Endemic Infection

Probably the simplest model for endemic infection is the susceptible-infected-susceptible (SIS) logistic model. Long-term behaviour of this model prior to disease extinction is described by the quasi-stationary distribution. This quasi-stationary distribution has been the subject of much previous work, including derivation of a variety of approximations, using both standard distributional forms and specialized approximating formulae. The aim of this paper is to carry out a systematic comparison between approximations. As well as comparing previously available approximations, we derive several new variants. Taking into account both accuracy (measured using total variation distance) and simplicity, and denoting by R ₀ the basic reproduction number, our main findings are: (a) in the subcritical region R ₀ < 1 a geometric distribution approximation is preferred; (b) in the supercritical region R ₀ ≫ 1 a beta-binomial distribution is preferred. Both of these preferred approximations are new.     

A theoretical description for solitons in polyacetylene

The bond-alternation domain walls or the solitons in the dimerized polyacetylene are analyzed theoretically. The width of the soliton is many times the period of the chain, so that the soliton can be reasonably well described by a continuum model. Because of the existence of the bond-alternation domain walls, the electron density is different definitely. Thus the electron density can be used to describe the formation of the domain walls, and a self-trapped potential is discussed and introduced in the Hamiltonian. It is shown that the envelope of the wave functions of the chain is governed by the nonlinear Schr?dinger equation which has soliton solutions. Then the shape of the soliton is determined analytically which is in accordance with the numerical calculations by Su, Schrieffer and Heeger. This implies that the bond-alternation domain wall or the soliton is observed as the envelope of the wave function.