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It is a great pleasure to dedicate this special issue of Physica Scripta to Professor Lennart Stenflo, my mentor and close friend for more than a quarter of a century. Lennart Stenflo is one of the prominent figures in plasma physics. Considering his many outstanding contributions in nonlinear sciences, it was strongly felt by his friends that we honour Lennart by bringing out these proceedings when he turns sixty (November 27, 1999). Accordingly, we sent out invitations to distinguished physicists who collaborated with Lennart in physics and in other spheres of life, and asked them to write an article dedicated to Lennart's Proceedings. Many colleagues responded overwhelmingly to our proposal. Due to page limitations we could only select 36 contributions, however. They are compiled in this special issue of Physica Scripta. The executive Editor of Physica Scripta Roger Wäppling generously supported our endeavors to celebrate this festive event. The whole-hearted support from our colleagues clearly reflects our respect for the nice personality of Lennart, and it also demonstrates his international leadership.

I have the pleasure of writing a few words about Lennart and his achievements in science and in other parts of life. I have known Lennart as my mentor since, in 1971, he got the chair at the Plasma Physics Department at Umeå University (UmU). Lennart built up an excellence in plasma physics at UmU, and trained a large number of doctoral students including myself. Besides, the kind personality of Lennart has attracted many scientists from all over the world to visit Umei for collaborating with him in various areas of nonlinear sciences.

Lennart has worked in many diverse fields of physics. His scientific accomplishments are too numerous to be listed here. However, here are some highlights of his most distinguished contributions in physics:

Lennart Stenflo has derived many original results in the areas of parametric instabilities (wave–wave interactions), coherent nonlinear structures (solitons and vortices), and chaos. Specifically, his contributions in parametric theory is, for example, recognized by his discoveries of stimulated electromagnetic emissions from plasmas, where it is demonstrated how the wave–wave interactions can generate radiation. His ideas have been verified experimentally in the Earth's ionosphere and magnetosphere.

In the area of nonlinear structures, we know the Kaufman–Stenflo theory for the super-Alfvénic upper-hybrid envelope solitons. Lennart Stenflo and his team has opened up a new field of strongly nonlinear surface plasma waves, which is followed by many physicists around the globe. Furthermore, Lennart Stenflo presented a set of nonlinear equations for acoustic gravity waves, which are now refered to as the Lorenz–Stenflo equations; the latter admit chaotic trajectories. Recently, he has been working in modern astroparticle physics where our team has suggested that the neutrinos are the building block of the universe, backed by the idea of collective neutrino plasma interactions. The strength of Lennart Stenflo lies in the fact that he has very broad interests and feels at home with both microscopic and macroscopic nonlinear theories in plasmas, as well as masters the physics and mathematics of fluid turbulence. Thus, many important contributions of Lennart Stenflo have helped us in understanding the complex nonlinear phenomena in space and astrophysical plasmas, in fluids and optics, as well as in laser produced and magnetically confined laboratory plasmas. Such a high level of Lennart's perfection is truly admirable.

The present proceedings focus on topics which are central to Lennart's scientific interest. Thus, the authors have considered the physics of various types of waves and their generation mechanisms, the development of turbulence and the formation of coherent structures, particle and heat transport, neutrino-plasma coupling, as well as various aspects of the nonlinear Schrödinger equation in plasmas, fluids, and nonlinear optics. The articles have dealt with the general picture of the subject matter at hand and the underlying physics, as well as described the present state-of-art in the field. It was noted that despite the diversity of the physical problems, the mathematical equations governing the particular phenomena and their solutions remain somewhat similar. The diverse materials included in this topical issue of Physica Scripta simply reflect the fact that Lennart's scientific knowledge is so rich, which is worth sharing.

I have had the privilege to work very closely with Lennart for such a long period of time and, therefore, have been able to admire his fine personality and appreciate his deep dedication for science. We have also collaborated in organizing many international conferences. On this particular occasion I also appreciate his warm feelings towards his fellow physicists whom he always likes to support. We wish Lennart a very long and healthy life and hope that in the coming millennium he will continue enlightening us with new physics and also show us a right path for doing high quality physics and for appreciating each other as human beings as well.

I am most grateful to the Editor of this issue, Professor Padma K Shukla, for his generous initiative to celebrate my birthday in this way. It is a great honour for me that so many outstanding scientists have contributed to this Topical Issue of Physica Scripta.

Much has happened in the field of plasma physics since I started my work on nonlinear science in 1964 in Trieste. We are all products of the previous generation of scientists, to which I owe a debt of gratitude, and we can consider ourselves successful if also we can contribute in a similar way to forthcoming generations. During more than three decades I have been fortunate in meeting and working with some very talented students, who are now internationally leading scientists.

I am convinced that the future prospects of plasma physics are very bright, in particular with regard to its significant influence on the development of space physics, astrophysics, condensed matter physics and computational physics. This future however depends on our ability to go on recruiting and educating new generations of students from all over the world.

During my years in nonlinear plasma science I have got many close friends from a variety of physics institutes in a large number of countries. I treasure this friendship and all that I have learnt from you. Once more I wish to express my sincere gratitude to Padma and all the other contributors to the present issue. I look forward to having the privilege to work with you for many more years.

The Kaufman-Stenflo equations for nonlinearity coupled upper-hybrid (UH) and magnetosonic waves are generalized for the multi-dimensional wave propagation case in collisional magnetoplasmas. Thus, a two-dimensional envelope equation for the UH waves, and three-coupled equations for magnetosonic waves are derived, by taking into account the combined effects of the UH ponderomotive force and the UH wave Joule heating of the electrons. The newly derived mode coupling equations are then used to derive a general dispersion that is appropriate for studying the parametric instabilities in collisional magnetoplasmas. The relevance of our investigation to low-temperature laboratory and space plasmas is pointed out.

A review of the bifurcation and chaos characteristics of the Lorenz-Stenflo equations for nonlinear acoustic gravity waves in a rotational system is presented.

The evolution of weakly nonlinear electrostatic drift waves in an externally imposed strong homogeneous magnetic field is investigated numerically in three spatial dimensions. The analysis is based on a set of coupled, nonlinear equations, which are solved for an initial condition which is perturbed by a small amplitude incoherent wave–field. The initial evolution is exponential, following the growth of perturbations predicted by linear stability theory. The fluctuations saturate at relatively high amplitudes, by forming a pair of magnetic field aligned vortex–like structures of opposite polarity, i.e. a pair of electrostatic convective cells.

Nonlinear drift–Alfvén waves in an inhomogeneous, relativistic low-pressure electron–positron plasma are studied. A set of coupled nonlinear equations describing these waves are deduced. It is shown that the nonlinear equations for stationary drift–Alfvén waves are similar to those for shear Alfvén and drift–Alfvén waves in an electron–ion plasma or shear Alfvén waves in an electron–positron plasma. Dipolar drift–Alfvén vortices can thus exist in relativistic electron–positron plasmas.

A set of coupled nonlinear differential equations governing the dynamics of low-frequency electromagnetic waves in the presence of equilibrium sheared flow for inhomogeneous collisional magnetized plasma has been derived. In the linear limit of a uniform density plasma, it is shown that equilibrium sheared plasma flows can cause an instability of resistive Alfvén-like waves. Furthermore, a quasi-stationary solution of the mode coupling equations can be represented in the form of localized vortex structures. On the other hand, the temporal behavior of the mode coupling equations is governed by six coupled equations, which are a generalization of the Lorenz-Stenflo equations and which admit chaotic trajectories.

An investigation of electron acceleration in the auroral precipitation region of the magnetosphere is made. In particular a determination of the mechanisms that could produce the observed power law distribution in energy and the positive slope in the distribution function are described.

The dynamics of vortical structures in magnetized plasmas with nonuniform density is investigated numerically. In particular the dynamics of monopolar vortices is considered and the results are discussed in terms of the conservation of potential vorticity. It is found that individual vortex structures tend to transport density up the gradient in contrast to structures embedded in developed turbulence, which on the average will lead to a flattening of the density gradient. The role of vortical structures in connection with electrostatic plasma turbulence and the associated cross-field plasma transport is also addressed.

A variational method is used to derive stationary solutions of the generalized nonlinear Schrödinger equation with complex coefficients that describes growth and damping which has been introduced and solved in the 1 + 1 dimensional case by Pereira and Stenflo (1977 Phys. Fluids20 1733). In the 1+1 dimensional case, the exact classical Pereira-Stenflo soliton solution is reproduced. Application to the 1 + 2 dimensional case with cylindrical symmetry, provides an explicit approximate single soliton solution which is named the cylindrical Pereira-Stenflo soliton.

Evolution of a solitary pulse in the cubic complex Ginzburg-Landau (CGL) equation, including third-order dispersion (TOD) as a small perturbation, is studied in detail. Starting from the exact Pereira-Stenflo soliton solution, we develop analytical approximations which yield an effective velocity c of the pulse induced by TOD. The analytical predictions are compared to direct numerical simulations, showing acceptable agreement at small values of the TOD parameter, provided that the second-order dispersion coefficient D takes values D > -3/2 or D < -30 (very different analytical approximations are used in these two cases). Between these regions, the numerically found dependence c(D) shows a very steep jump at D ≅ -3/2, and a less steep jump in the opposite direction at -30 < D < -20, each jump changing the sign of the velocity. The simulations also demonstrate that there is a maximum of the laminar propagation distance (before the onset of the ultimate turbulent stage) attained at D ≅ -18. The action of the sliding-frequency filtering on the soliton dynamics is also investigated numerically, and it is found that it slightly increases the laminar propagation distance.

It is shown that the nonlinear Schrödinger equation describing pulse propagation in optical fibers in the presence of a properly space-tailored damping or amplification is exactly integrable. A simple transformation of variables is given which transforms the inhomogeneous nonlinear Schrödinger equation into the standard form with constant coefficients, thus generating new explicit bright and dark soliton solutions in the cases of anomalous and normal dispersion, respectively.

The resonant radiation of a soliton described by the nonlinear Schrödinger equation with third and fourth order derivative terms is investigated by different methods. The results are in good agreement between themselves.

Some breather type solutions of the NLS equation have been suggested by Henderson et al (to appear in Wave Motion) as models for a class of 'freak' wave events seen in 2D-simulations on surface gravity waves. In this paper we first take a closer look on these simple solutions and compare them with some of the simulation data (Henderson et al to appear in Wave Motion). Our findings tend to strengthen the idea of Henderson et al. Especially the Ma breather and the so called Peregrine solution may provide useful and simple analytical models for 'freak' wave events.

Nonlinear propagation of coupled upper-hybrid and magnetoacoustic modes in a magnetized plasma is investigated within the framework of generalized Schrödinger–Boussinesq (or, –KDV) system of equations. The latter includes nonlinearities in the low-frequency response of the plasma up to cubic order in the magnetoacoustic wave amplitude. For stationary propagation, a new type of upper-hybrid soliton consisting of triple-hump structured field intensity accompanied by density trough having single-dip profile is obtained. This solution may be considered as the higher-order nonlinear stationary eigenstate of the upper-hybrid field trapped in the self-created density cavity. The existence of single-hump and double-hump solitons as well as parameter regimes for the integrable coupled mode propagation are also discussed.

The nonlinear coupling between broadband-upper-hybrid (BBUH) waves and electrostatic ion-cyclotron perturbations in a uniform collisional magnetoplasma is considered, taking into account the combined effects of the UH ponderomotive and thermal nonlinear forces. This coupling is governed by a pair of equations which is employed to derive the nonlinear dispersion relation, which admits modulational instabilities of BBUH waves in some interesting cases. The relevance of our investigation to the nonlinear propagation of random phase upper-hybrid waves in a weakly collisional Earth's ionosphere as well as in low-temperature laboratory discharges is pointed out.

The nonlinear motions of plasma electrons are investigated for a semi-infinite cold plasma for the case when the one-dimensional limit is applicable. Nonlinear oscillations, where the frequency is a function of the amplitude, are found. The general relations between the magnitudes of the density perturbations and the values of the corresponding harmonic frequencies are determined.

Nonlocality in the mechanism of the ionization nonlinearity in stationary surface–wave sustained discharges is considered. It is shown that axial diffusion combined with charged particle flux from the region of the wave launcher governs the maintenance of the discharge in the vicinity of the launcher and may be responsible for the complicated behaviour of the discharge characteristics in the beginning of a plasma column.

The conversion of energy from one wave type to another is spatially localized by nonuniformity of the medium. This process can be analyzed in terms of the local linear interaction of wave fields on rays that propagate in phase space. The methodology presented results in the replacement of partial differential equations for the wave fields by ordinary differential equations for rays, wave amplitudes, and wave phases.

The mode coupling equations for Rossby waves in a bounded periodic channel are derived by means of a hamiltonian method. The streamfunction is used as field variable and we need not introduce Clebsch potentials or a lagrangian description. The Manley-Rowe relations follow from general hamiltonian theory.

Relative diffusion between ions and electrons in electrostatic turbulence in strongly magnetized plasmas is considered, with particular attention to the effects of finite ion Larmor radii. Analytical expressions are presented which account for the observations. Numerical results are obtained from a simulation in two spatial dimensions, based on a model where the finite Larmor radius corrections are introduced by a filtering operation.

Turbulent transport may drive a plasma towards a state of Turbulent EquiPartition (TEP). It can be regarded as an attractor with a uniform distribution of Lagrangian invariants respected by the turbulent perturbations in the medium, notably a tokamak plasma. An attractor (the canonical profiles) has been described by invariants, while adequate hydrodynamics has not been derived. In the present paper the model of two dimensional hydrodynamics is suggested assuming low-frequency electrostatic perturbations and the hypothesis that the poloidal, but not toroidal component of the frozen-in low holds in tokamaks. It is shown that instability which drives plasma to the state of TEP is drift-flute type instability and the driving force of the instability is the pressure gradient combining with the shear of the magnetic field. The analysis of the condition for the instability to be developed leads to the important conclusion that the negative magnetic shear stabilizes the instability and thus can suppress the source of drift turbulence in large scale tokamaks, as has already been seen in many recent experiments and in gyro-particle simulations.

The propagation of Langmuir waves in the vicinity of the upper hybrid resonance in an inhomogeneously heated magnetoplasma is studied by means of exact analytical solutions. These are obtained through special transformations of the dispersive equations for the waves. These exact solutions are independent of WKB approximations for the wide classes of monotonic and non-monotonic spatial profiles of electron temperature. Thermal acceleration and retardation of Langmuir waves which depends on the shapes of these profiles is studied. The possibility of formation of a waveguide for Langmuir waves, caused by an inhomogeneous temperature distribution is demonstrated and the spectrum of trapped modes in such a thermal waveguide is presented.

The relaxation in systems with several sources of free energy is discussed. Examples from widely different systems are considered both for homogeneous systems with particle streams and for inhomogeneous systems with configuration space instabilities.

A review is given on an extended form of Maxwell's equations which is based on Lorentz invariance in combination with a nonzero divergence of the electric field Ein vacuo. In addition to the displacement current, this form includes a "space-charge current" ε₀ (div E)C where the modulus of C is equal to the velocity c of light and the direction of C depends on the particular geometry. This form predicts new states to exist, such as steady equilibria and additional types of wave phenomena.

Among the equilibria there are axisymmetric "particle-shaped" and "string-shaped" states. The former result in one class of solutions with nonzero integrated charge q₀, magnetic moment M₀, mass m₀, and angular momentum s₀, and in another class with vanishing q₀ and M₀ but nonzero m₀ and s₀. These solutions could contribute to the understanding of charged and neutral leptons. The latter states can be of interest to the string model of the hadron structure.

Among the new wave types there are plane and axisymmetric modes. The former provide a possible solution to the problem of total reflection at the interface between a dissipative medium and vacuum. The latter types provide a model of the individual photon as a boson wave packet with an angular momentum, a nonzero but small rest mass. and a radius being in agreement with microwave experiments.

We have considered the generation of a low-frequency nonlinear current by high-frequency waves, using the relativistic Vlasov equation. A general formula for the nonlinear current has been derived, inlcuding the ponderomotive part as well as the magnetization current. We have found that the magnetization current can be significantly changed by the relativistic effects, even for a weakly relativistic temperature.

Parallel propagation of nonlinear electrostatic waves in multispecies plasmas is described via a Sagdeev-type pseudo-potential or by Korteweg-de Vries and modified Korteweg-de Vries equations for weak nonlinearities, almost regardless of the assumptions about plasma compositions or pressure closures. We explore the reasons for that in some detail and show why this should be the case.

This paper presents a critical discussion of the formation of ion-acoustic shocks in a collisionless plasma with negative ions.

It is shown that intense ion-cyclotron waves propagating at large angles to the magnetic field lines can generate second harmonic waves. The anisotropy of this effect, which is illustrated by the angular dependence of the frequencies of the interacting waves, is examined. The high efficiency of such frequency doubling for kHz waves in the upper ionosphere is demonstrated.

The nonlinear scattering of electromagnetic waves in an inhomogeneous magnetoplasma is considered. The differential cross-sections for the scattering of electromagnetic waves near the lower- and upper-hybrid resonance frequencies are obtained. For typical laboratory parameters, the differential cross sections in the presence of electromagnetic waves are much higher than the corresponding cross section when thermal fluctuations are present. The results could also be relevant for understanding the anomalous scattering of electromagnetic waves from the ionospheric plasma.

In this review we consider new type of nonlinear interactions of particles and waves in plasmas which allows us to convert wave energy with large frequency difference without inverted particle population and special matching conditions between wave frequencies and wave numbers. The main emphasis is given to recent concepts clarifying the physics of the plasma–maser phenomena.

It is shown that the ion drag force acting on the charged dust grains can cause an instability of electrostatic waves in dusty plasmas. The instability can be responsible for the great void, which is observed in low-temperature dusty plasma discharges.

We present analytical evidence of a new plasma wave interaction mechanism which reveals caviton-like states of hf Langmuir waves trapped in the density trough provided by an electron hole. Localized structures of this type move with typical electron thermal velocity and are hence outside the realm of strong Langmuir turbulence described by Zakharov's equations.

It is shown that photons and neutrinos with nearly constant velocity can radiate a large spectrum of electromagnetic waves when they cross the boundary between a plasma and vacuum. This is a direct consequence of the existence of an equivalent electric charge of photons or neutrinos in a plasma. The general expression for the radiation field in the vacuum region far away from the boundary is established, and the particular case of a sharp plasma boundary is discussed.

The magnetic properties of neutrinos, as a relativistic gas, are described. Thermodynamics of the neutrino gas in a strong magnetic field is considered, including the contribution of the interaction between the magnetic moment of neutrinos and the external magnetic field. The magnetic susceptibility is found for some special cases.

The nonlinear propagation of neutrinos and gravitons in a plasma has been considered within the framework of the general relativity. A set of governing equations for plasmas as well as for neutrinos and gravitational waves has been developed. The results should be useful for formulating the nonlinear coupling scenarios between high-energy neutrinos, intense radiation/plasmons, and gravitons, which should help to understand the combined influence of the weak nuclear, strong electromagnetic, and gravitational forces in astrophysical settings as well as in the early universe.

Self-organised criticality (SOC) has been proposed as a potentially powerful unifying paradigm for interpreting non-diffusive avalanche-type transport in laboratory, space and astrophysical plasmas. After reviewing the most promising astrophysical sites where SOC might be observable, we consider the theoretical arguments for supposing that SOC can occur in accretion discs. Perhaps the most rigorous evidence is provided by numerical modelling of energy dissipation due to magnetohydrodynamic turbulence in accretion discs by G Geertsema and A Achterberg (1992 Astron. Astrophys.255 427); we investigate how "sandpile"-type dynamics arise in this model. It is concluded that the potential sites for SOC in accretion systems are numerous and observationally accessible, and that theoretical support for the possible occurrence of SOC can be derived from first principles.

It is shown that non-parallel density and temperature gradients can produce magnetic fields which can account for the asteroid Gaspra surface magnetic field of about one Gauss.

The Sun's visible spectrum is filled with a wealth of polarized spectral structures due to coherent scattering via bound-bound atomic transitions. Here we outline the theory for these coherence phenomena, including the effects of the external magnetic field (the Hanle effect), and demonstrate the correspondence between classical coherence and quantum interference between excited atomic states. Observational examples of the Hanle effect and of spectral features that can only be understood in terms of rather exotic quantum physics are presented. The signatures of coherent scattering have revealed the presence of a small scale, turbulent magnetic field in the Sun's photosphere, which has not been detectable by other techniques.