He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients

In this Letter, He's homotopy perturbation method is applied to heat-like and wave-like equations with variable coefficients. The solutions are introduced in this Letter are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.     

Application of Exp-function method to the Whitham–Broer–Kaup shallow water model using symbolic computation

In this letter, the Exp-function method is applied to the Whitham–Broer–Kaup shallow water model. With the help of symbolic computation, several kinds of new solitary wave solutions are formally derived.     

Computational study of collisional energy transfer and quasi-continuous population inversions in laser-pumped Ca vapor

A computational study of population inversion between several pairs of excited states viz 3d4p ³ F-4s3d ³ D, 4s5s ³ S-4s4p ³ P and 4s3d ³ D-4s4p ³ P in Ca vapor pumped on the 4s ^{2 1} S ₀-4s4p ³ P ₁ transition is presented. The main aim is to investigate the influence of various atomic processes in creating and sustaining the population inversion for long times after the excitation pulse. The delicate interplay between superelastic energy transfer to free electrons, energy pooling collisions and cascaded recombination is particulary examined. It is noted that quasi-continuous population inversion can be readily excited on the 4s3d ³ D-4s4p ³ P transitions; and under some conditions, also on the 4s5s ³ S-4s4p ³ P transitions. Furthermore, inversion on the 3d4p ³ F-4s3d ³ D transitions can also be excited for a considerable length of time. The results may be useful in designing and developing quasi-cw metal vapor lasers.     

Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem

The hierarchy of integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of linearization of these equations and their conservation law in the terms of solutions of corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and conservation laws explicitly expressed through the variables of the nonlinear equations are derived.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

Painleve Property and New Analytic Solutions for a Variable-Coefficient Kadomtsev--Petviashvili Equation with Symbolic Computation

A variable-coemcient Kadomtsev-Petviashvili equation is investigated. The Painlev6 analysis leads to its explicit Painlevd-integrable conditions. An auto-Backlund transformation and the bilinear form are presented via the truncated Painlev6 expansion and symbolic computation. Several families of new analytic solutions arepresented, including the soliton-like and periodic solutions.     

Cross Soliton-Like Waves for the （2＋1）-Dimensional Breaking Soliton Equation

We construct a two-soliton-like solution for the （2＋1）-dimensionai breaking soliton equation. The obtained solution contains two arbitrary functions and hence can model various cross soliton-like waves including the two-solitary waves. We show the evolution of some special cross soliton-like waves diagrammatically.     

Thermal Diffusion Process Estimation for Fabrication of Graded Plastic Optical Fibre

Thermal diffusion of dopants is investigated in the process of generating the graded-index profile of plastic optical fibres. Because the diffusion coefficient in high polymers has been shown to depend strongly on dopant concentration, it is allowed in this work to vary with the radial coordinate of the multistep-core fibre. A novel multi-layer model is presented for solving the diffusion equation with the variable diffusion coefficient. It is solved by the finite difference method. The solution determines the dopant diffusion profile in the fibre. It is verified against a solution from the literature and two cases of fibres with diffused profiles. The application is demonstrated on two examples of graded-index plastic optical fibres, one originally with a two-step and the other with four-step core. The results indicate that closer to the core-cladding interface, the computed diffused profile with variable diffusion coefficient D is closer to target profile than the profile obtained with constant D for the same time of thermal process.     

Method of group foliation and non-invariant solutions of partial differential equations

Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, i.e. the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: sheftel@gursey.gov.tr     

Some geometric aspects of maps

In this talk we introduce a Weierstrass-like system of equations corresponding to CP N -1 fields in two dimensions. Then using this representation we introduce a vector in R N 2-1 and treating this vector as the radius vector of a surface immersed in R N 2-1 we discuss to what extent the associated metric describes the geometry of the CP N -1 maps. We show that for the holomorphic maps - the correspondence is exact; while for the more general fields we have to go beyond the Weierstrass system and add extra terms. Received 1st August 2001 / Received in final form 18 October 2001 Published online 2 October 2002 RID="a" ID="a"Work done in collaboration with M. Grundland e-mail: w.j.zakrzewski@durham.ac.uk