The Boltzmann Equation for Bose-Einstein Particles: Condensation in Finite Time

The paper considers the problem of the Bose-Einstein condensation in finite time for isotropic distributional solutions of the spatially homogeneous Boltzmann equation for Bose-Einstein particles with the hard sphere model. We prove that if the initial datum of a solution is a function which is singular enough near the origin (the zero-point of particle energy) but still Lebesgue integrable (so that there is no condensation at the initial time), then the condensation continuously starts to occur from the initial time to every later time. The proof is based on a convex positivity of the cubic collision integral and some properties of a certain Lebesgue derivatives of distributional solutions at the origin. As applications we also study a special type of solutions and present a relation between the conservation of mass and the condensation.     

On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons

In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons with bounded initial data blow up in finite time in the L ^∞ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contain a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.     

A Modified Boltzmann Equation for Bose–Einstein Particles: Isotropic Solutions and Long-Time Behavior

Under some strong cutoff conditions on collision kernels, global existence, local stability, entropy identity, conservation of energy, and moment production estimates are proven for isotropic solutions of a modified (quantum effect) Boltzmann equation for spatially homogeneous gases of Bose–Einstein particles (BBE). Then applying these results with the biting-weak convergence, some results on the long-time behavior of the conservative isotropic solutions of the BBE equation are obtained, including the velocity concentration at very low temperatures and the tendency toward equilibrium states at very high temperatures.     

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H¹ norm.     

A NOTE ON TRAVELING WAVE SOLUTIONS TO THE TWO COMPONENT CAMASSA–HOLM EQUATION

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa–Holm equation are distributional traveling wave solutions to the Camassa–Holm equation provided that the set u^-1(c), where c is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa–Holm equation. However, the two component Camassa–Holm equation has distinct from Camassa–Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of u^-1(c) is not zero. We provide examples of such solutions.     

Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential ${v(x)= \epsilon \chi(x) |x|^{-1}}Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schr?dinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential v(x) = ec(x) |x|^-1{v(x)= \epsilon \chi(x) |x|^{-1}}, where e{\epsilon} is sufficiently small and c ? C₀^￥{\chi \in C_0^{\infty}} even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.     

Long Time Energy Transfer in the Random Schrödinger Equation

We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker–Planck type equation with the wave vector performing a Brownian motion on the (d ? 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrödinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2.     

Six-prong events produced in 5 GeV/cπ−p interactions

Results are presented from a study of 3184 six-prong π^?p interactions at 5 GeV/c in the Dubna 1 m hydrogen bubble chamber. Cross sections for the various reaction channels are given and their energy dependence is discussed. The emphasis of this paper is on exhibiting the semi-inclusive particle spectra is sufficient detail so that it will be possible to use it for testing models and developing new concepts. In particular, invariant distributions are plotted against the scaled longitudinal momentum x, the rapidity y and the longitudinal momentum p_L in the lab system. The net charge per x as well as y interval is also presented. It is found that the isotropic pion component is inconsistent with the Bose-Einstein formula. Comparison with data at other energies is made whenever possible.     

Bose-Einstein solitons in time-dependent linear potential

In this paper, Bose-Einstein soliton solutions of the nonlinear Schrödinger equation with time-dependent linear potential are considered. Based on the F-expansion method, we present a number of Jacobian elliptic function solutions. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric functions, respectively. Specially, for V_ext = ZF(T) = Z[mg + Hcos (ω₁T)], we discussed the Bose-Einstein condensate trapped in the coupling external field with considering the effect of gravity; for F(T) = constant, it describes the wave (Langmuir or electromagnetic) in a linearly inhomogeneous plasma with cubic nonlinearly.     

Can a fourth lepton family exist without its quark counterpart?

Solely within the minimal standard model, we show that it is possible to have a fourth lepton familywithout its quark counterpart provided the Higgs sector is strongly coupled and can accommodate Skyrmion solutions. The triangle and Witten anomalies of the fourth lepton family are cancelled by those of the “weak” Skyrmions if the latter are quantized as fermions. Thus a fourth quark family is not needed to cancel the anomalies of the new leptons. If indeed such a family is discovered in the near future bye ⁺ e ^? machines and if no Higgs boson nor new quarks of masses less than 1TeV are found, this intriguing scenario can be a very viable one. Arguments are presented to show that if such a scheme is realized, it also implies that the Higgs fields are composite.     

Modulation of breathers in cigar-shaped Bose-Einstein condensates

We present new solutions to the nonautonomous nonlinear Schrödinger equation that may be realized through convenient manipulation of Bose-Einstein condensates. The procedure is based on the modulation of breathers through an analytical study of the one-dimensional Gross-Pitaevskii equation, which is known to offer a good theoretical model to describe quasi-one-dimensional cigar-shaped condensates. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Numerical simulations confirm stability of the modulated breathers against random perturbation on the input profile of the solutions.     

Soliton-like solutions of some nonlinear theories and transformations among them

The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ ⁴ theories (the usual and the one with the wrong sign of the mass term), theφ ⁶, theφ ⁸, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ ²ⁿ class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type.     

General Features of the X Part of an ABX Spin System in Isotropic and Liquid Crystalline Phases as Illustrated by theC–{H} Spectra of 2,2′-Difluorobiphenyl

The¹³C–{¹H} NMR spectra of 2,2′-difluorobiphenyl dissolved in isotropic and liquid crystalline solvents have been obtained and analyzed. They are examples of the X part of an ABX spectrum. It is shown that the spectrum of the isotropic solution yieldsJ_AX,J_BX,J_AB, δ_AB, and δ_X, but only if all the transitions are detected, and that intensities as well as frequencies of the transitions are used in the analysis. It is demonstrated that for 2,2′-difluorobiphenyl this requires that for some of the carbons it is necessary to detect very weak transitions. For the spectra of liquid crystalline solutions of ABX systems it is shown that the dipolar couplingsD_AX,D_BX, andD_ABare obtained only if these couplings are in a certain sensitive range of relative values. The sensitive range can be adjusted by using variable angle sample spinning (VASS). It is demonstrated that VASS spectra taken near the magic angle can be used to obtain the absolute signs of the scalar couplings.     

Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates

Under investigation in this paper is a nonlinear Schrödinger equation with an arbitrary linear time-dependent potential, which governs the soliton dynamics in quasi-one-dimensional Bose-Einstein condensates (quasi-1DBECs). With Painlevé analysis method performed to this model, its integrability is firstly examined. Then, the distinct treatments based on the truncated Painlevé expansion, respectively, give the bilinear form and the Painlevé-Bäcklund transformation with a family of new exact solutions. Furthermore, via the computerized symbolic computation, a direct method is employed to easily and directly derive the exact analytical dark- and bright-solitonic solutions. At last, of physical and experimental interests, these solutions are graphically discussed so as to better understand the soliton dynamics in quasi-1DBECs.     

Kinetics of the Bose-Einstein condensation

We study the bosonic Boltzmann-Nordheim kinetic equation, which describes the kinetic regime of weakly interacting bosons with s-wave scattering only. We consider a spatially homogeneous fluid with an isotropic momentum distribution. The issue of the dynamical formation of a Bose-Einstein condensate has been studied extensively. We supply here the completed equations of motion for the coupled system, the energy density distribution of the normal fluid and the density of the condensate. With this information the post-nucleation self-similar solution is investigated in more detail than before.     

The energy tensor of matter in the projectve theory of relativity

In the projective theory of relativity the 5-dimensional field equation \(_{\mu \nu } \) and the resulting equation of motion Tμυ_;μ = 0 are investigated. There Tμυ stands for the 5-dimensional tensor of macroscopic matter. The 4-dimensional field equations and equation of motion obtained by projection are a generalization of Einstein's theory of general relativity and Maxwell's electrodynamics, involving a scalar field φ.They contain a single constant φ₀.The weak field approximation is investigated for the case of an ideal fluid and leads to Newton's mechanics, including Newton's gravitational law, and to Maxwell's electrodynamics. For the constant φ₀ one obtains the approximate value φ₀c⁴/γ_N with Newton's gravitational constant γ_N.For homogeneous and isotropic cosmological models consisting of matter only the general solution for the radius K of curvature is given. This solution is independent of the equation of state of matter For a pure dust universe the general solution for the scalar field φ is given. For a closed universe a power law φ ?K^?1 is valid which leads to Mach's principle. The calculation of the age of a closed universe yields over 7×10⁹y,if one uses mean values of the present cosmological data.     

Extended sources,final state interactions and Bose-Einstein Correlations

I extend the theory of Bose-Einstein correlations to the case where one of a pair of like pions is the daughter of resonance decay and include final state rescattering within the like pion pair. I consider critically the form of the isospin 2s-wave phase shift. The results of numerical calculations yield like pion correlation functions which are at best in marginal agreement with data frome ⁺ e ^? annihilation and proton-proton collisions. I show that the discrepancy which exists is removed by the effects of three (and more) particle rescattering. The quantitative interpretation of Bose-Einstein correlations among hadrons is very complicated unless the effective source size substantially exceeds 1 fm.     

Products of distributions and singular travelling waves as solutions of advection-reaction equations

In this paper, we restrict our attention to the advection-reaction equation u _t + [?(u)] _x = ??(u), where ? and ?? are entire functions. Conditions for the propagation of a distributional wave profile are presented and the wave speed is evaluated. As an example, we prove that, under certain conditions, the propagation of delta-waves in models ruled by the diffusionless Burgers-Fisher equation is possible and compute the speeds of propagation of these waves. In the same setting, the propagation of travelling waves with the shape of a C ¹-function with one jump discontinuity is also studied. These results will be easily explained by our theory of distributional products and are based on a rigorous and consistent concept of a solution that we have already introduced in previous works.     

Bose Condensates in Interaction with Excitations: A Kinetic Model

This paper deals with mathematical questions for Bose gases below the temperature T _BEC where Bose-Einstein condensation sets in. The model considered is of two-component type, consisting of a kinetic equation for the distribution function of a gas of (quasi-)particles interacting with a Bose condensate, which is described by a Gross-Pitaevskii equation. Existence results and moment estimates are proved in the space-homogeneous, isotropic case.     

Scalar,electromagnetic, and gravitational fields interaction: Particlelike solutions

Particlelike static spherically symmetric solutions to massless scalar and electromagnetic field equations combined with gravitational field equations are considered. Two criteria for particlelike solutions are formulated: the strong one (solutions are required to be singularity free) and the weak one (singularities are admitted but the total energy and material field energy should be finite). Exact solutions for the following physical systems are considered with their own gravitational field: (i) linear scalar (minimally coupled or conformal) plus electromagnetic field; (ii) the same fields with a bare mass source in the form of charged incoherent matter distributions; (iii) nonlinear electromagnetic field with an arbitrary dependence on the invariant F_αβF^αβ; and (iv) directly interacting scalar and electromagnetic fields. Case (i) solutions are not particlelike (except those with horizons, in which static regions formally satisfy the weak criterion). For systems (ii), examples of nonsingular models are constructed, in particular, a model for a particle-antiparticle pair of a Wheeler-handle type, without scalar field and explicit electric charges. Besides, a number of limitations upon nonsingular model parameters is indicated. Systems (iii) are proved to violate the strong criterion for any type of nonlinearity but can satisfy the weak criterion (e.g., the Born-Infeld nonlinearity). For systems (iv) some particlelike solutions by the weak criterion are constructed and a regularizing role of gravitation is demonstrated. Finally, an example of a field system satisfying the strong criterion is given.