Boltzmann Equations For Mixtures of Maxwell Gases: Exact Solutions and Power Like Tails

We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.     

On the Problem of Electromagnetic-Field Quantization

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg’s equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.     

Generalized Knizhnik–Zamolodchikov equations,off-shell Bethe ansatz and non-skew-symmetric classical r-matrices

Using non-skew-symmetric sl(2)$\mathit{sl}(2)$-valued classical r-matrices with spectral parameters we construct a generalization of the Knizhnik–Zamolodchikov (KZ) equations. We obtain integral solutions of the constructed KZ-type equations using the “off-shell” Bethe ansatz technique. We consider several examples of the obtained generalized KZ equations and their integral solutions that correspond to the “K-twisted” non-skew-symmetric classical r-matrices parametrized by arbitrary complex parameter ξ.     

The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlevé property of the solutions.     

Reaction diffusion equation in the ultra cold coexisting atomic and molecular condensates and the importance of Feshbach interaction

We consider a gas of bosonic atoms near a Feshbach resonance. The dynamics of the atomic and molecular condensates can be described in the limit of small fluctuations by a set of coupled nonlinear Gross Piteaviskii equations. In the case of a strong atom-molecule conversion, the system has an integral motion for the spatially uniform solutions which exhibits temporal oscillation. The possible consequence of this oscillation may be the existence of Josephson like current in the condensates which has been investigated.     

Diffusive and dynamical radiating stars with realistic equations of state

We model the dynamics of a spherically symmetric radiating dynamical star with three spacetime regions. The local internal atmosphere is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the Schwarzschild exterior. A large family of solutions to the field equations are presented for various realistic equations of state. We demonstrate that it is possible to obtain solutions via a direct integration of the second order equations resulting from the assumption of an equation of state. A comparison of our solutions with earlier well known results is undertaken and we show that all these solutions, including those of Husain, are contained in our family. We then generalise our class of solutions to higher dimensions. Finally we consider the effects of diffusive transport and transparently derive the specific equations of state for which this diffusive behaviour is possible.     

Charged Tomimatsu-Sato metrics

We show that the solutions of Einstein-Maxwell field equations can be constructed from the Tomimatsu-Sato solutions of Einstein's field equations for empty space simply by inspection when δ has integral values.     

From Reactive Boltzmann Equations to Reaction–Diffusion Systems

We consider the reactive Boltzmann equations for a mixture of different species of molecules, including a fixed background. We propose a scaling in which the collisions involving this background are predominant, while the inelastic (reactive) binary collisions are very rare. We show that, at the formal level, the solutions of the Boltzmann equations converge toward the solutions of a reaction-diffusion system. The coefficients of this system can be expressed in terms of the cross sections of the Boltzmann kernels. We discuss various possible physical settings (gases having internal energy, presence of a boundary, etc.), and present one rigorous mathematical proof in a simplified situation (for which the existence of strong solutions to the Boltzmann equation is known).     

Solutions for ion-electron-radiation coupling with radiation and electron diffusion

We present semi-analytic solutions to the equations for radiation-electron-ion coupling, the so-called 3-T equations. Our solutions use a linearization based on Pomraning's form for the heat capacity given by with additional stipulations for the electron heat conduction and ion-electron coupling coefficients. To solve the linearized equations we use integral transform techniques and compute a Fourier integral numerically. We give solutions for a 3-T version and a 2-T with heat conduction version of the Su-Olson problem as well as solutions for spherical and spherical shell sources. We use the xRage radiation hydrodynamics code to demonstrate that our solutions are useful for code verification in multiple dimensions and axisymmetric geometries.     

Method for Solving Nonstationary Electrodynamic Problems in Dielectric Media with Time Dispersion

Radiophysics and Quantum Electronics - We consider three-dimensional singular integral equations with time delay that describe three-dimensional problems of interaction of a nonstationary...     

Partial diagonalization of Bethe-Salpeter type equations

We consider an equation of the Bethe-Salpeter type, with arbitrary potential and kernel, respectively, for space-like momentum transfer. The invariance group of the equation is then the Lorentz-group in three dimensions, the O(1, 2) group. The standard procedure for the diagonalization of such equations (valid for square integrable solutions only) is generalized to include the case of power bounded solutions, by means of a generalized O(1, 2) expansion formalism. The result is a two-dimensional integral equation for the O(1, 2) expansion coefficients. The right-most l-plane singularities of these determine the asymptotic behaviour of the amplitudes as in ordinary Regge theory. The formalism can be applied to other dynamical equations possessing O(1, 2) symmetry.     

Operation of the Faddeev Kernel in Configuration Space

We present a practical method to solve Faddeev three-body equations at energies above the three-body breakup threshold as integral equations in coordinate space. This is an extension of a previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.     

Integral Equations for Three-Body Coulomb Resonances

 We propose a novel method for calculating resonances in three-body systems with repulsive Coulomb interactions. The method is based on the solution of a set of Faddeev and Lippmann-Schwinger integral equations. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order to get the resonances. As a numerical illustration a model for the three-α system is solved. Received October 1, 1999; revised February 25, 2000; accepted for publication June 30, 2000     

On the logarithmic solutions of the WDVV equations

We consider restrictions and subsystems in the ∨-systems corresponding to the logarithmic solutions of the WDVV equations. We present certain solutions through restrictions of the Coxeter systems.     

Applications of the first integral method to nonlinear evolution equations

<正>In this paper,we establish travelling wave solutions for some nonlinear evolution equations.The first integral method is used to construct the travelling wave solutions of the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations.The obtained results include periodic and solitary wave solutions.The first integral method presents a wider applicability to handling nonlinear wave equations.     

The fermi-particles model spectrum

We consider the system of Hamilton’s equations and the system of variation equations whose solutions determine the excitation spectrum of the model fermion system.     

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.