Topical issue on Geometry, Integrability and Nonlinearity in Condensed Matter Physics

Recent developments in the study of nonlinear phenomena have led to the realization that a combination of the concepts of integrability, geometry and topology provides a new powerful framework for describing a great variety of physical systems. It was therefore felt that the compilation of a special issue comprising articles on the interdiseiplinary topic of Geometry, Integrability and Nonlinearity in Condensed Matter Physics, would indeed be timely. The enthusiastic response and support that we received from the active researchers in this subject, when we organized an International Conference on the above topic from July 15 to July 20, 2001, in Bansko, Bulgaria, provided a further motivation for undertaking this task. As the topic is interdisciplinary in nature, the articles in this volume contain new results on a wide range of subjects. These include among others, integrable equations and the interplay between geometry and nonlinearity, the role of optical solitons in communication. (and, possibly, computation), common nonlinear and geometrical aspects of condensed matter, field theory, and so on. The increasingly important role played by geometry and topology in diverse areas such as the quantum Hall effect, localization, deformation and elasticity, quasiparticle kinetics and dynamics, spin systems, membranes, is highlighted in some of the articles. There are papers in which essential links of nonlinearity to differential geometry are identified and many elegant mathematical methods are presented. Some other articles focus on how the mathematical tools of geometry and nonlinear analysis can be applied to solve certain physical problems. Given the vast range of titles, it was difficult to strictly divide the contributions into distinct categories. Except for the pedagogical introductory article by Rajaraman titled "CP _N Solitons in Quantum Hall Systems", which essentially "sets the stage" for the various themes covered, we have grouped the articles broadly under the following headings: Geometry, integrability and mathematical physics; Solitons: Interaction phenomena, nonlinear optics; Condensed matter physics; Soft condensed matter physics; Quantum phenomena. We gratefully acknowledge the support from Los Alamos National Lab, USA; Université de Cergy-Pontoise, France; The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and the Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria, in putting this special volume together. We believe that the cross-fertilization and synergy of a host of ideas in seemingly disparate fields of physics would lead to the natural emergence of new paradigms, which in turn could pave the way for collaborative research to arrive at new solutions of complex nonlinear problems. It is our hope that this topical issue will be useful in providing an impetus for achieving this broad objective. Radha Balakrishnan, Chennai, India Rossen Dandoloff, Cergy-Pontoise, France Vladimir Gerdjikov, Sofia, Bulgaria Dimitar Pushkarov, Sofia, Bulgaria Avadh Saxena, Los Alamos, USA     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

Instabilities of an electron cloud in a Penning trap

We have measured the storage instabilities of electrons in a Penning trap at low magnetic fields. These measurements are carried out as a function of the trapping voltage, for different magnetic fields. It is seen that these instabilities occur at the same positions when the trapping voltage is expressed as a percentage of the maximum voltage, given by the stability limit. The characteristic frequencies at which these instabilities occur, obey a relation that is given by n _zω _z + n ₊ω ₊ + n _-ω _- = 0, where ω _z, ω ₊ and ω _- are the axial, perturbed cyclotron and the magnetron frequencies of the trapped electrons respectively, and the n's are integers. The reason for these instabilities are attributed to higher order static perturbations in the trapping potential. Received 5 August 2002 / Received in final form 14 October 2002 Published online 17 December 2002 RID="a" ID="a"Present address: Dept. of Physics, Rampurhat College, Rampurhat, Birbhum, West Bengal, India. RID="b" ID="b"e-mail: werth@mail.uni-mainz.de     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Intra-Channel Collision of Dual-Power Law Optical Solitons

The intra-channel collision of optical solitons, with dual-power law nonlinearity, is studied in this paper by the aid of quasi-particle theory. The perturbation terms that are considered in this paper are the nonlinear gain and saturable amplifiers along with filters. The suppression of soliton-soliton interaction, in presence of these perturbation terms, is acheived. The numerical simulations support the quasi-particle theory. OCIS codes: 060.2310; 060.4510; 060.5530; 190.3270; 190.4370.     

Study of gain flatness for multi-channel amplification in single stage EDFA for WDM applications

In this paper, gain flatness is studied for simultaneous 16-ITU-T channel amplifications at C-band (1,532–1,558 nm) in a single stage EDFA for WDM application at different average inversion levels. The inversion levels are varied due to the change of the input signal levels from the targeted operating point and also for dropping few numbers of channels. Specially designed gain flattening filter (GFF) is used in order to get the flat gain with gain variation ±0.5 dB for −20 dBm/ch input signal power (total input signal power is −8.0 dBm) at a fixed average inversion level which is maintained by proper selection of optimum fibre length and pump power. A specific loss spectrum of GFF is obtained by writing a chirped fibre Bragg grating of length 20 mm. Gain variations are studied by changing the total input signal levels from −8.0 dBm to −20.0 dBm and maintained within 20.0 ± 0.5 dB by using automatic gain control (AGC) circuit. About 15 out of 16 channels are dropped and observed ± 0.5 dB gain-variation which is an important parameter in optical network system.     

A numerical study of the target system of an ADSS with different flow guides

The mechanical design of the target module of an accelerator driven subcritical nuclear reactor system (ADSS) calls for an analysis of the related thermal-hydraulic issues because of large amount of heat deposition in the spallation region during the course of nuclear interactions with the molten lead bismuth eutectic (LBE) target. The LBE also should carry the entire heat generated as a consequence of the spallation reaction. The problem of heat removal by the LBE is a challenging thermal-hydraulic issue. For this, one has to examine the flows of low Prandtl number fluids (LBE) in a complex ADSS geometry. In this study, the equations governing the laminar flow and thermal energy are solved numerically using the streamline upwind Petrov-Galerkin (SUPG) finite element (FE) method. The target systems with a straight and a nozzle guide have been considered. The principal purpose of the analysis is to trace the flow and temperature distribution and thereby to check the suitability of the flow guide in avoiding the recirculation or stagnation zones in the flow space that may lead to hot spots.      

Calculation of CWKB envelope in boson and fermion productions

We present the calculation of envelope of boson and of both low-and high-mass fermion production at the end of inflation when the coherently oscillating inflatons decay into bosons and fermions. We consider three different models of inflation and use CWKB technique to calculate the envelope to understand the structure of resonance band formation. We observe that though low-mass fermion production is not effective in preheating because of Pauli blocking, it is quite probable for high-mass fermion to take part in pre-heating.