The return current induced in a plasma by a relativisitc electron beam generates a new electron-ion two-stream instability (return current instability). Although the effect of these currents on the beam-plasma e-e instability is negligible, there exists a range of wave numbers which is unstable only to return current (RC) instability and not to e-e instability. The electromagnetic waves propagating along the direction of the external magnetic field, in which the plasma is immersed, are stabilized by these currents but the e.m. waves with frequencies,ω²≪Ω_e²≪ω_pe² (Ω_e andω_pe being cyclotron and plasma frequency for the electrons of the plasma respectively) propagating transverse to the magnetic field get destabilized. Heuristic estimates of plasma heating, due to RC instability and due to decay of ion-acoustic turbulence generated by the return current, are made. The fastest time scale on which the return current delivers energy to the plasma due to the scattering of ion-sound waves by the electrons can be ∼ω_pi⁻¹ (ω_pi being the plasma frequency for the ions).

The wave-particle interactions in a beam-plasma system in the presence of finite but small thermal motions of the particles are investigated in the linear as well as nonlinear regime. During the linear growth, the thermal motions are found to have a stabilizing effect. The nonlinear evolution is studied by using the Perturbed Orbit Formalism. Due to the thermal motions the nonlinear saturation of growth is found to take place at a level lower than that of the cold case. A detailed study of the energy balance shows that nonresonant particles pick up a bigger fraction of the energy lost by the streaming motion of the beam, thus leading to more efficient ‘heating’.

The nonlinear Schrödinger equation describing the evolution of the plane wave solutions of the Hirota equation and of the Boussinesq equation are obtained. The conditions for modulational instability and the localised stationary solutions are derived.

The envelope properties of ion-acoustic waves in a two-electron-temperature plasma are studied. The nonlinear Schrödinger equation describing the envelope of these waves is obtained from the plasma fluid equations by employing the Krylov-Bogoliubov-Mitropolsky perturbation method. It is shown that the ion-acoustic waves can be modulationally unstable or stable depending on the ratios of the densities and the temperatures of the hot and the cold electron components. Even a small fraction of the cold electron component can drastically affect the stability of the system.

The radiation, resulting from the nonlinear interaction of whistler solitons, which act as localized antennae, with the upperhybrid waves in the coronal loop, is shown to give rise to fine structures in solar radio bursts. All the observed features of microwave spikes in radio flares,e.g. their frequency, polarization and short duration can be explained by the presence of about 10⁶ solitons occupying a volume of ∼ 10⁸ m³, provided this interaction takes place at low altitudes. However, if this interaction takes place close to the top of the coronal loop, it gives rise to the isolated tadpole ‘eyes’ features in the dynamic spectra. About 10⁹ solitons are needed to account for the observed flux of these ‘eyes’.

The evolution of nonlinear Langmuir waves in the interplanetary medium is investigated by appropriately accounting for the random density irregularities of the medium. A pair of modified Zakharov equations, which describe these waves, is solved numerically as an initial value problem for large scale (≫ 10² km) initial pertubations. For an ion acoustic-Langmuir solitary wave, the random irregularities damp the Langmuir wave by way of scattering and let the ion density perturbation radiate away in a few days. However an initial solitary or shock-like Langmuir wave excites the ion density perturbations within a fraction of a second, and then itself gets damped. These effects will strongly decelerate the collapse of large scale Langmuir waves. The possibility of detecting these processes, by means of interplanetary scintillation, is discussed.

Some interesting features of a class of two-dimensional Hamiltonians with indefinite kinetic energy are considered. It is shown that such Hamiltonians cannot be reduced, in general, to an equivalent dynamical Hamiltonian with positive definite kinetic energy quadratic in velocities. Complex nonlinear evolution equations like the K-dV, the MK-dV and the sine-Gordon equations possess such Hamiltonians. The case of complex K-dV equation has been considered in detail to demonstrate the generic features. The two-dimensional real systems obtained by analytic continuation to complex plane of one-dimensional dynamical systems are also discussed. The evolution equations for nonlinear, amplitude-modulated Langmuir waves as well as circularly polarized electromagnetic waves in plasmas, are considered as illustrative examples.

Starting with the very definition of chaos, we demonstrate that the study of chaos is not an abstract one but can lead to some useful practical applications. With the advent of some powerful mathematical techniques and with the availability of fast computers, it is now possible to study the fascinating phenomena of chaos — the subject which is truly interdisciplinary. The essential role played by fractals, strange attractors, Poincare maps, etc., in the study of chaotic dynamics, is briefly discussed. Phenomena of self-organization, coherence in chaos and control of chaos in plasmas is highlighted.