On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy

A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations.Comment: 16 pages, 1 figur

Lagrangian Approach to Dispersionless KdV Hierarchy

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and pplications) at http://www.emis.de/journals/SIGMA

Dynamical systems theory for nonlinear evolution equations

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the $K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the $K(3,\,2)$ equation for which the parameter can take only negative values. The $K(2,\,3)$ equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.Comment: 5 pages, 4 figure

Polar Network Index as a magnetic proxy for the solar cycle studies

The Sun has a polar magnetic field which oscillates with the 11 year sunspot cycle. This polar magnetic field is an important component of the dynamo process which is operating in the solar convection zone and produces the sunspot cycle. We have systematic direct measurements of the Sun's polar magnetic field only from about mid 1970s. There are, however, indirect proxies which give us information about this field at earlier times. The Ca K spectroheliograms taken in Kodaikanal Solar Observatory during 1904 - 2007 have now been digitized with the 4k x 4k CCD and have higher resolution (0.86 arcsec) than the other available historical datasets. From these Ca-K spectroheliograms, we have developed a completely new proxy (Polar Network Index, PNI) for the Sun's polar magnetic field. We calculate the PNI from the digitized images using an automated algorithm and calibrate our measured PNI against the polar field as measured by the Wilcox Solar Observatory for the period of 1976 - 1990. This calibration allows us to estimate polar fields for the earlier period up to 1904. The dynamo calculations done with this proxy as input data reproduce the Sun's magnetic behavior for the past century reasonably well.Comment: 19 pages, 5 figures Accepted for publication in APJ

Research and development work on substitute electrical resistance alloys for heating elements

From the start of the Second Five- Year Plan great emphasis has been laid on production and utilisation of electric power in various industrial and domestic appliances. Electric heating is thus gradually repla- cing solid-fuels, gas and oil heating . Increasing application of electric heat with all its attendant advantages will fail to register full impact unless suitable electrical heating elements , having long high temperature service life are indigenously avai- lable at reasonable cost