A simple and very efficient gas jet levitation technique for levitating inertial confinement fusion (ICF) targets has been developed. A low velocity gas jet through diverging nozzle generates precisely controlled low Reynolds number flow pattern, capable of levitating polymer microballoons up to 2500 µm diameter. Different shaped diverging nozzle are investigated, satisfactory levitation is achieved with simple conical shapes. With this setup microballoon can be levitated for hours with excellent stability, continuous rotation and at the desired height (reproducible with in less than 100 µm). The height of stabilization depends upon cone angle of diverging nozzle and velocity of levitating gas. This technique is very robust and highly insensitive to external disturbances like nonuniform temperature fields and vibrations.

This setup is very economical to fabricate, easy to operate and can be used efficiently in various spray coating application involving plastic and metallic layers on microballoons.

In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with determination of symmetry algebra, the one- and two-dimensional optimal systems are constructed, and optimality is also established using various invariant functions of full adjoint action. Apart from classification and construction of several conservation laws, the Painlevé analysis is also performed in a symbolic manner which describes the non-integrability of equations.

The degenerate coupled multi-Korteweg–de Vries equations for coupled multiplicity $l = 3$ are studied. The equations, also known as three-field Kaup–Boussinesq equations, are considered for invariant analysis and conservation laws. The classical Lie’s symmetry method is used to analyse the symmetries of equations. Based on the Killing’s form, which is invariant of adjoint action, the full classification for Lie algebra is presented. Further, one-dimensional optimal group classification is used to obtain invariant solutions. Besides this, using general theorem proved by Ibragimov, we find several non-local conservation laws for these equations. The conserved currents obtained in this work can be useful for the better understanding of some physical phenomena modelled by the underlying equations.

This paper investigates the new coupled (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) system with time-dependent coefficients for multiple types of exact solutions by using the Lie symmetry transformation method.Similarity transformation reduces the system of equations into ordinary differential equations and further, these are solved for solutions having bright, dark and singular solitons as well as periodic waves. Also, the solutions appeared in terms of time-dependent coefficient $\beta(t)$ and analysed graphically to show the effect of this arbitrary function. It is proved that the given system is nonlinear self-adjoint, and some conservation laws are obtained by applying the new conservation theorem.

Some new infinite-dimensional generalised Lie symmetries of Einstein’s field equations for perfect fluid distribution are found by using the Lie symmetry analysis. The reduced ordinary differential equations are solved to obtain new non-trivial exact solutions. The software MAPLE is used for computation and MAPLE code is given to facilitate the research in this field.