Two particles on a star graph,I

We consider a two-particle system on a star graph with δ function interaction. A class of eigensolutions is described, which are constructed from appropriate one-particle solutions, and hence are parametrized by two momenta. These solutions include a family of solutions with discontinuous derivative on the diagonal. Dedicated to the memory of Vladimir Geyler     

Classical electrodynamics of a point particle

Even for the simplest physical situations the Lorentz-Dirac equation, solved as an initial value problem, gives unphysical ‘run-away’ solutions. Dirac’s method for avoiding these unphysical solutions generates solutions which exhibit unphysical acausal pre-acceleration. A careful examination of the application of the conservation of momentum in the derivation of the Abraham self-force reveals a fundamental error concerning the force acting on the particle. This error, originally made by Abraham (1903), has been repeated by subsequent investigators. When corrected, a new equation of motion results. A discussion of the general properties of the new equation of motion is given, and solutions for several important special cases are presented. The behaviour of these solutions is causal, physically reasonable, and easily understood.     

Globally regular solutions of a schwarzschild black hole and a Reissner-Nordström black hole

A new physical concept about globally regular solutions is suggested. The globally regular solutions corresponding to the Schwarzschild black hole and the Reissner-Nordström black hole are examined.     

Some non-perturbative semi-classical methods in quantum field theory (a pedagogical review)

A pedagogical introduction is given to non-perturbative semi-classical methods for finding solutions to quantum field theories. Both the weak coupling method based on a time-independent classical solution, and the WKB method based on all periodic orbits are developed in detail, proceeding ffrom elementary quantum mechanics to field theory in stages. Both methods are then illustrated in model field theories. The [λø⁴]₂ theory to which the weak coupling method is applied yields a new family of “kink” states whose properties are discussed.The WKB method is illustrated by quantizing “soliton” and “doublet” solutions of the two-dimensional sine-Gordon theory. The results are compared to consequences of Coleman's equivalence proof relating this system to the massive Thirring model. The speculation that solitons may be fermions is discussed, along with indications that the WKB method may ne yielding exact mass ratios for this theory.A final section is devoted to solutions of more realistic four-dimensional models containing fermions, internal symmetry etc. Some quark-confinement models and vortex type solutions come under this category. General remarks are made on this family of solutions, and illustrated using 't Hooft's monopole solution.     

Relationship Among Solutions of a Generalized Riccati Equation

In this paper, some solutions of a generalized Riccati equation are investigated, which are given in the recent articles [Chaos, Solitons ＆ Fractals 24 （2005） 257; Phys. Lett. A 336 （2005） 463], and the relationship among the solutions is revealed.     

Investigation of the Dynamics of a Soliton-Like Solution to the Nonlinear Schrödinger Equation with an External Field

The properties of asymptotic soliton-like solutions to the 1-D nonstationary nonlinear Schrödinger equation with the external-field potential of a special form are studied. A comparative analysis of the asymptotic solutions and simulation results is performed to show the range of parameter values where asymptotic and numerical soliton-like solutions are in agreement, the localization being preserved.     

Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation

In this paper, by virtue of symbolic computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrödinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Bäcklund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.     

Mean curvature flow of a hyperbolic surface

A four-parameter family of self-similar solutions is obtained to the mean curvature flow equation for a surface. This family is shown to be stable with respect to a small deformation of a hyperbolic surface. At time instant t*, a singular point is formed within a finite time interval, that is accompanied by a change in the topology of the surface. The solution is continued beyond the singular point. A relationship between the parameters describing the hyperbolic surface before and after the change in the surface topology is obtained. A particular case is analyzed when the unperturbed surface is a cylinder. A cylindrical surface is weakly unstable with respect to a perturbation in the form of a “wide neck.” At the final stage of the development of the neck when its transverse size becomes much less than the cylinder radius at large distances from the neck, the surface flow in a wide region in the neighborhood of the neck is described by a universal two-parameter family of self-similar solutions. These solutions are stable with respect to small perturbations of the surface.     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Gravitational fields with groups of motions on two-dimensional transitivity hypersurfaces in a model with matter and a magnetic field

For gravitational fields with metrics which admit of groups of motions multiply — transitive on 2-dimensional space-like invariant varieties, the exact solutions of the Einstein gravitational equations are given for the case when the sources of the gravitational field are dust-like matter and a magnetic field. A magnetic field is orientated along a direction orthogonal to transitivity hypersurface. The solutions contain arbitrary functions. In the case of transitivity hypersurface of positive curvature and in the absence of a magnetic field, the solution is reduced to the Tolman spherically symmetric solution for dust-like matter. The conditions are studied under which the solutions with a magnetic field become asymptotically isotropic and approach the flat and the open Friedmann models. The case of transitivity hypersurfaces with signature (+ –) is also considered.     

Solution electronic spectra of brassinosteroid and a synthesized conjugate of a steroid and a fluorescent label

Electronic absorption spectra of 24-epicastasterone and 24-epibrassinolide solutions have been recorded and analyzed. The long-wavelength band in solution spectra of the former steroid with maxima in the region 292–286 nm is assigned to the electronic nπ*-transition of a carbonyl chromophore. The latter steroid contains an O-C=O group, for which the extinction coefficient in this spectral region is approximately one order of magnitude less. These differences can be used to identify the studied steroids. A method to synthesize the conjugate of 24-epicastasterone with a fluorescent label, dansylhydrazine, which shows intense fluorescence and absorption, is described. Absorption spectra of the label and conjugate in diethylether, tetrahydrofuran, ethanol, and acetonitrile solutions are practically independent of the solvent properties. The exceptions are solutions of the conjugate in hexane and water. All investigated solutions of label and conjugate show intense fluorescence. As the dielectric constant of the solvent increases, the fluorescence spectra shift to long-wavelength. The fluorescence energetic and time characteristics of both the label and conjugate are similar, which enables the conjugate to be used in developing a method for immunofluorescent analysis of the brassinosteroid 24-epicastasterone. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 1, pp. 74–78, January–February, 2008.     

Classical pion fields in the presence of a source

A classical pion field that is similar to a disoriented chiral condensate is considered in the presence of an external source. The field is similar to the condensate in that the isotopic orientation of the field in the whole space is determined by a single vector. Within the nonlinear sigma model, classical solutions are considered both in the chiral limit, where the pions are massless, and in the case of a finite pion mass. In either case, the classical filed is similar to the Coulomb field of a charged particle; however, the nonlinear pion interaction results in the existence of several solutions. In the massless case and in the case where the source is sufficiently small, there are a great number of classical solutions characterized by finite discrete energies. In the more realistic case of heavy ions, there are no stable solutions of this type; however, long-lived quasistationary states, which slowly decay, emitting very soft pions, can be formed. The structure and the energies of these solutions are studied numerically.     

“Fine-tuning” renormalization and associated constants in a nonrelativistic four-fermion model

Exact solutions of a two-particle eigenvalue problem are obtained for a four-fermion contact interaction. Composite Goldstone modes are found. A relationship, is found between the renormalization procedure and the construction of self-adjoint solutions. Irkutsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 62–66, July, 1997.     

A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction

In this paper, new class of solutions of the Tzitzéica equation are derived by using the classical Lie symmetry analysis. The important aspect of this paper however is the fact that the analysis results in a new class of solitary-like solutions, the so-called cusp-solitary solutions.

This special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold. The studied equation was originally found in the field of geometry, otherwise the Tzitzéica equation takes place in many branches of non-linear sciences. Therefore, explicit class of solutions connected by a physical meaning are of great importance. The analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation. A complete characterization of the group properties is given. The classical Lie point symmetries are derived and algebraic properties are determined. Similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of Jacobian elliptic functions. It is worth to mention that the application of known powerful algebraic methods (e.g. special function transform methods) are not appropriate to study the solution manifold. Hence, the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture.     

Exact solution for the gravitational field of a charged,magnetized, spinning mass

A family of solutions of the Einstein-Maxwell field equations is presented, corresponding to the exterior of stationaryaxisymmetric sources with charge, mass, angular momentum, and magnetic dipole moment. The Riemann tensor vanishes asymptotically for each member of the family; some solutions are asymptotically flat and some have NUT-like behavior asymptotically. For the asymptotically flat solutions, the gyromagnetic ratio may vary from zero to one. The corresponding value for the Kerr-Newman solution is one. A method for generating infinite chains of families of solutions of the Einstein-Maxwell equations is described.     

Properties of exact solutions for a massive classical Yang-Mills field

Exact solutions for a massive Yang-Mills field are found and solutions of classical Wong equations and quantum Dirac equations are discussed for the field configurations obtained. A procedure for constructing constant fields is given and transition to solutions of the Yang-Mills equations in the case of a massless field is discussed.Translated from Izvestriya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 96–100, May, 1986.In conclusion, the authors express their gratitude to V. Ch. Zhukovskii and V. R. Khalilov for valuable remarks and discussions.     

Spherically symmetric solutions of Einstein-Maxwell theory with a Gauss-Bonnet term

The low energy expansion of supersymmetric string theory suggests that the leading correction to the Einstein action is given by the Gauss-Bonnet invariant. A generalisation of Birkhoff's theorem in the case of Einstein-Maxwell theory modified by a Gauss-Bonnet term is proved. The only spherically symmetric solutions of the theory are shown to be generalisations of the Reissner-Nordstrom and Robinson-Bertotti solutions. The “Reissner-Nordstrom” solutions have asymptotically flat and asymptotically anti-de Sitter branches, however, the latter are unstable.     

Discontinuity walls terminated by twist disclinations in smectic a liquid crystals

A discontinuity wall in smectic A liquid crystals was discussed in M. Kléman:Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media (J. Wiley, Chichester, 1983). Possible solutions describing a singularity — twist disclination which terminates the discontinuity wall in an infinite model smectic A liquid crystal are discussed. One of solutions proposed can be used to describe the so-called zigzag defect line in chiral smectic C liquid crystals on which opposite oriented chevrons are terminated. This work was supported by Grant No.: 202/96/1687 of the Grant Agency of the Czech Republic.     

一类相对论性非球谐振子系统的束缚态

给出了具有形式为12r²+A2r²的非球谐振子型标量势和矢量势 的相对论系 统在两种势相等的条件下三维Klein-Gordon方程，二维和三维Dirac方程的s波束缚态解. 关键词： 三维非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态