Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painlevé analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painlevé test. An auto-B?cklund transformation is provided by use of the truncated Painlevé expansion and symbolic computation. Via the given auto-B?cklund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.     

Nonlinear propagation of ultra-low-frequency electromagnetic modes in a magnetized dusty plasma

A theoretical investigation has been made of nonlinear propagation of ultra-low-frequency electromagnetic waves in a magnetized two fluid (negatively charged dust and positively charged ion fluids) dusty plasma. These are modified Alfvén waves for small value of and are modified magnetosonic waves for large , where is the angle between the directions of the external magnetic field and the wave propagation. A nonlinear evolution equation for the wave magnetic field, which is known as Korteweg de Vries (K-dV) equation and which admits a stationary solitary wave solution, is derived by the reductive perturbation method. The effects of external magnetic field and dust characteristics on the amplitude and the width of these solitary structures are examined. The implications of these results to some space and astrophysical plasma systems, especially to planetary ring-systems, are briefly mentioned. Received 8 July 1999 and Received in final form 11 October 1999     

Planar and nonplanar ion-acoustic envelope solitary waves in a very dense electron-positron-ion plasma

Ion-acoustic envelope solitary waves in a very dense plasma comprised of the electrons, positrons and ions are investigated. For this purpose, the quantum hydrodynamic model and the Poisson equation are used. A modified nonlinear Schrödinger equation is derived by employing the reductive perturbation method. The effects of the quantum correction and of the positron density on the propagation and stability of the envelope solitary waves are examined. The nonplanar (cylindrical/spherical) geometry gives rise to an instability period. The latter cannot exist for planar case and it affected by the quantum parameters, as well as the positron density. The present investigation is relevant to white dwarfs.     

Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation

An investigation to deepen the connection between the family of nonlinear Schr?dinger equations and the one of Korteweg-de Vries equations is carried out within the context of the Madelung's fluid picture. In particular, under suitable hypothesis for the current velocity, it is proven that the cubic nonlinear Schr?dinger equation, whose solution is a complex wave function, can be put in correspondence with the standard Korteweg-de Vries equation, is such a way that the soliton solutions of the latter are the squared modulus of the envelope soliton solution of the former. Under suitable physical hypothesis for the current velocity, this correspondence allows us to find envelope soliton solutions of the cubic nonlinear Schr?dinger equation, starting from the soliton solutions of the associated Korteweg-de Vries equation. In particular, in the case of constant current velocities, the solitary waves have the amplitude independent of the envelope velocity (which coincides with the constant current velocity). They are bright or dark envelope solitons and have a phase linearly depending both on space and on time coordinates. In the case of an arbitrarily large stationary-profile perturbation of the current velocity, envelope solitons are grey or dark and they relate the velocity u₀ with the amplitude; in fact, they exist for a limited range of velocities and have a phase nonlinearly depending on the combined variable x-u_{0 s} (s being a time-like variable). This novel method in solving the nonlinear Schr?dinger equation starting from the Korteweg-de Vries equation give new insights and represents an alternative key of reading of the dark/grey envelope solitons based on the fluid language. Moreover, a comparison between the solutions found in the present paper and the ones already known in literature is also presented. Received 20 February 2002 and Received in final form 22 April 2002 Published online 6 June 2002     

Transformation from the nonautonomous to standard NLS equations

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevé analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.     

Impact of electron exchange-correlation on drift acoustic solitary waves

The impact of electron exchange-correlation term on the linear and nonlinear quantum ion (QIA) acoustic drift waves in a highly degenerate plasma is investigated. An analytical approach is employed to derive the differential equation which is later on turned into Sagdeev energy integral equation that can be utilized to get drift solitons under existence conditions. It is noted that phase speed/frequency of the linear drift quantum ion acoustic (QIA) waves increases with electron exchange-correlation effect, but the amplitude of the corresponding solitons decreases with inclusion of these effects. Present study is carried out with reference to highly dense plasma environments like fast ignition inertial confinement fusion and white dwarfs etc.     

On the steady motions of a flat domain wall in a ferromagnet

A new derivation is presented of Walker's exact solution to Gilbert equation, a solution which mimicks the travelling-wave motion of a flat domain wall at 180^°. It is shown that a process during which the working of the applied magnetic field exactly compensates dissipation (the Walker condition) exists both under the constitutive circumstances considered in the standard Gilbert equation and when either the internal free-energy or the dissipation, or both, are generalized by the introduction of higher-gradient terms; but that such a process cannot solve the generalized Gilbert equation. It is also shown that, when dry-friction dissipation is considered and a suitable magnetic field is applied, the associated Gilbert equation has a Walker-type solution mimicking a flat wall, at 90^° this time, which however does not satisfy the Walker condition. Received 16 November 2001     

A New Approach to Energy Integral for Investigation of Dust-Ion Acoustic(DIA)Waves in Multi-Component Plasmas with Quantum Effects in Inertia Less Electrons

Dust-ion acoustic waves are investigated in this model of plasma consisting of negatively charged dusts,cold ions and inertia less quantum effected electrons with the help of a typical energy integral.In this case,a new technique is applied formulating a differential equation to establish the energy integral in case of multi-component plasmas which is not possible in general.Dust-ion acoustic(DIA) compressive and rarefactive,supersonic and subsonic solitons of various amplitudes are established.The consideration of smaller order nonlinearity in support of the newly established quantum plasma model is observed to generate small amplitude solitons at the decrease of Mach number.The growths of soliton amplitudes and potential depths are found more sensitive to the density of quantum electrons.The small density ratio r(=1-f) with a little quantized electrons supplemented by the dust charges Z_d and the in-deterministic new quantum parameter C_2 are found responsible to finally support the generation of small amplitude solitons admissible for the model.     

Intrinsic optical bistability in Kerr ferroelectric materials

In this paper we investigate the intrinsic optical bistability (IOB) in a ferroelectric (FE) single layer using an alternative analysis. The FE material is considered to have an intensity-dependent refractive index where the third order nonlinear susceptibility χ⁽³⁾ acts like Kerr coefficient. The nonlinear response of the FE medium is modeled using the Landau-Khalatnikov (LK) dynamical equation with the nonlinear anharmonic potential obtained from the Landau-Devonshire free energy expressed in terms of polarization. Within a single frequency approximation, the electromagnetic wave equation is written in terms of the polarization P rather than the electric field E as the dependent variable. With the application of the nonlinear boundary conditions we have derived expressions for both reflectance and transmittance as a function of the electric field incident amplitude, polarization and other material parameters. The formalism proves to be more suitable for FE materials since most of these materials have highly linear and nonlinear coefficients especially when the operating frequency is in the resonance region. The effects of thickness, operating frequency and temperature on BaTiO₃ single film are investigated theoretically. The results presented here agree in principle with the recent experimental observations of intrinsic OB in BaTiO₃ monocrystal and other FE photorefractive materials.     

Rarefactive ion-acoustic electrostatic solitary structures in nonthermal plasmas

An investigation has been made of ion-acoustic solitary waves in an unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. The properties of stationary solitary structures are briefly studied by the pseudo-potential approach, which is valid for arbitrary amplitude waves, and by the reductive perturbation method which is valid for small but finite amplitude limit. The time evolution of both compressive and rarefactive solitary waves, which are found to coexist in this nonthermal plasma model, is also examined by solving numerically the full set of fluid equations. The temporal behaviour of positive (compressive) solitary waves is found to be typical, i.e., the positive initial disturbance breaks up into a series of solitary waves with the largest in front. However, the behaviour of negative (rarefactive) solitary waves is quite different. These waves appear to be unstable and produce positive solitary waves at a later time. The relevancy of this investigation to observations in the magnetosphere of density depressions is briefly pointed out. Received 12 October 1999     

Multistability and hysteresis of non-linear electron oscillations induced in a rippled plasma

Summary In the framework of the hydrodynamic approximation, we study the Langmuir oscillations of an ion density ripple background plasma under the influence of a single external harmonic r.f. field. An analytic frequency response curve is derived for the late-time behaviour exhibiting multistability and hysteresis.     

On dissipation mechanisms in micromagnetics

Within the framework of a dynamic version of micromagnetics [1,2], the space-time evolution of magnetization in a rigid, saturated ferromagnet is governed by the following equation: γ^-1 = ×( + + div ), where the interaction couple × and the couple stress are to be constitutively specified. Under constitutive assumptions for , , and the free energy ψ, that allow for equilibrium response and viscosity out of equilibrium and agree with the dissipation principle - ^. + ^. ∇ - ≥ 0, the above evolution equation yields a broad generalization of the standard Gilbert equation. In particular, while the standard Gilbert equation only incorporates relativistic dissipation, it is shown that the dissipation mechanisms compatible with the generalized Gilbert equation include exchange dissipation [2], dry-friction dissipation [3], and others. It is also shown that the additional term proposed in [4] to account for exchange dissipation, rather than having a genuine dissipative nature, modifies instead the nature of possible equilibria; and that such a modification is an automatic side effect when dry-friction dissipation is incorporated in the manner of [3]. Received 31 October 2000     

Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases

A two-dimensional axisymmetric model of the propagation of intense femtosecond laser pulses through dispersion-free transparent media is described. The effects of diffraction, nonlinear Kerr effect (instantaneous and retarded) and multiphoton ionisation are included. Numerical results concerning air and other gases are discussed. In particular, time self-compression of femtosecond pulses is predicted. Stable self-guided pulses are simulated, in agreement with recent experimental observations. Received: 19 June 1998 / Received in final form: 14 January 1999     

Ultracold bosons with 3-body attractive interactions in an optical lattice

We study the effect of an optical lattice (OL) on the ground-state properties of one-dimensional ultracold bosons with three-body attractive interactions and two-body repulsive interactions, which are described by a cubic-quintic Gross-Pitaevskii equation with a periodic potential. Without the optical lattice and with a vanishing two-body interaction term, normalizable soliton solutions of the Townes type are possible only at a critical value of the interaction strength, at which an infinite degeneracy of the ground state occurs; a repulsive two-body interaction makes such localized solutions unstable. We show that the OL opens a stability window around the critical point when the strength of the periodic potential is above a critical threshold. We also consider the effect of an external parabolic trap, studying how the stability properties depend on the matching between minima of the periodic potential and the minimum of the parabolic trap.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Envelope solitons induced by high-order effects of light-plasma interaction

The nonlinear coupling between light beams and non-resonant ion density perturbations in a plasma is considered, taking into account the relativistic particle mass increase and the light beam ponderomotive force. A pair of equations comprising a nonlinear Schr?dinger equation for light beams and a driven (by the light beam pressure) ion-acoustic wave response is derived. It is shown that the stationary solutions of our nonlinear equations can be represented in the form of a bright and dark/gray soliton for the one-dimensional problem. We also present numerical results which exhibit that our bright soliton solutions are stable exclusively for the values of the parameters compatible with our theory. Received 24 July 2002 Published online 31 October 2002 RID="a" ID="a"Permanent address: Dipartimento di Scienze Fisiche, Universitá Federico II and INFN, Complesso Universitario di M.S. Angelo, Via Cintia, 80126 Napoli, Italy e-mail: renato.fedele@na.infn.it RID="b" ID="b"Permanent address: Dipartimento di Fisica Generale, Universitá di Torino, Via Pietro Giuria 1, 10125 Torino, Italy RID="c" ID="c"Permanent address: Institute of Physics, Georgian Academy of Sciences, Tbilisi 380077, Georgia     

Improved Speed and Shape of Ion-Acoustic Waves in a Warm Plasma

The basic set of fluid equations can be reduced to the nonlinear Kortewege-de Vries (KdV) and nonlinear Schrödinger (NLS) equations. The rational solutions for the two equations has been obtained. The exact amplitude of the nonlinear ion-acoustic solitary wave can be obtained directly without resorting to any successive approximation techniques by a direct analysis of the given field equations. The Sagdeev's potential is obtained in terms of ion acoustic velocity by simply solving an algebraic equation. The soliton and double layer solutions are obtained as a small amplitude approximation. A comparison between the exact soliton solution and that obtained from the reductive perturbation theory are also discussed.     

Modelling of Lévy walk kinetics of charged particles in edge electrostatic turbulence in tokamaks

We model and discuss the possible types of motion that charged particles may undergo in a stationary and spatially periodic electrostatic potential and a homogeneous magnetic field. The model is considered to be the simplest approximation of more complex phenomena of plasma edge turbulence in tokamaks. Therein, low frequency turbulence appears in the plasma edge, resulting in a fluctuation of the electron density, and also in the generation of a turbulent electrostatic field. Typical parameters of this turbulent electrostatic field are an electrical potential amplitude of 10–100 V and wave numbers k≈10³ m^-1. In our model, we consider these regimes, together with a homogeneous magnetic field with a magnitude of 1 T. We investigate the dynamics of singly-ionized carbon ions – a typical plasma impurity – with kinetic energies on the order of 10 eV. Besides the obvious Larmor and drift motions, a motion of random-walk and of Lévy walk character appear therein. All of these types of motion can play an important role in the modelling of the anomalous diffusion of particles from the plasma edge turbulence region. The dynamics mentioned will cause an inevitable escape of energetic particles and thus of power loss from the thermonuclear reactor. Moreover, Lévy walk kinetics represents a very interesting kind of kinetics, currently of great interest, which was previously not so often discussed.     

Nonlinear interactions between drift waves and zonal flows

Phase coherent interactions between drift waves and zonal flows are considered. For this purpose, mode coupling equations are derived by using a two-fluid model and the guiding center drifts. The equations are then Fourier analyzed to deduce the nonlinear dispersion relations. The latter depict the excitation of zonal flows due to the ponderomotive forces of drift waves. The flute-like zonal flows with insignificant density fluctuations have faster growth rates than those which have a finite wavelength along the magnetic field direction. The relevance of our investigation to drift wave driven zonal flows in computer simulations and laboratory plasmas is discussed. Received 5 April 2002 Published online 28 June 2002