Asymptotic Solitons of the Johnson Equation

Abstract

We prove the existence of non-decaying real solutions of the Johnson equation, vanishing as x → +∞. We obtain asymptotic formulas as t → ∞ for the solutions in the form of an infinite series of asymptotic solitons with curved lines of constant phase and varying amplitude and width.     

Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Abstract

We analyze asymptotic scaling properties of a model class of anomalous reaction-diffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to analyze asymptotic symmetry properties; this provides an analytical explanation of the observed asymptotic scaling properties for the considered ARD equations.     

幂函数叠加势的径向薛定谔方程的解析解

研究多种正幂势函数与逆幂势函数紧密耦合条件下薛定谔径向方程解析解的求解方法.对势函数为V（r）=α₁r⁸＋α₂r³+α₃r²+β₃r^-1＋β₂r^-3＋β₁r^-4的径向薛定谔方程存在解析解的条件以及精确的解析解进行了研究. 根据量子系统波函数必须满足单值、有界和连续的标准条件,首先求出径向坐标r→∞以及r→0时的渐近解,然后采用非正则奇点邻域附近的波函数级数解法与求得的渐近解相结合,通过幂级数系数比较法得到径向薛定谔方程在势函数系数紧密耦合条件下的一系列定态波函数解析解以及相应的能级结构,并作适当讨论与结论. 关键词： 级数解法 幂势函数 径向波函数 渐近解     

Unconventional spin solutions of the Schwinger and Thirring models

A satisfactory operator solution of the Schwinger-Thirring model is found which ase→0 org→0 tends respectively to the operator solution of the Thirring or the Schwinger model. We also propose the general solutions of the Schwinger-Thirring model which can be associated with the solution of the Thirring model with unconventional statistics. In particular our method allows for a generalization of the Schwinger model for arbitrary spins giving rise to a new solution with the “photon” mass dependent ons. The following consequences for a bosonization procedure is also described.

     

Pulse propagation of acoustic waves scattered in a three-dimensional channel

Abstract

In a previous paper (Whitman et al 1999 Waves Random Media 9 1–11) we discussed the scattering of acoustic waves by random sound-speed fluctuations in a two-dimensional channel and presented an asymptotic form for an acoustic pulse propagating in the channel. Here we include the three-dimensional effect of transverse scattering. We find an asymptotic solution in which initially the two-dimensional mode-transfer effect is more important than the transverse scattering effect. However, for large enough propagation distances the transverse scattering effect dominates the pulse spread. In this paper we shall show the form of the pulse shape in both propagation ranges as well as in the transition regime. We shall begin with a discussion of the physics of the problem and then present a mathematical discussion.     

Gentile statistics and restricted partitions

In a recent paper (Tranet al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitionsp _k ^s (n), which is the number of partitions of an integern into the summand of sth powers of integers such that each power of a given integer may occur utmostk times. While the method is not rigorous, it reproduces the well-known asymptotic results fors = 1 apart from yielding more general results for arbitrary values ofs.     

Distinguishing Three-Dimensional Lens Spaces L(7,1) and L(7,2) by Means of Classical Pentagon Equation

Abstract

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a “2 tetrahedra → 3 tetrahedra” local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.     

Calculation of the energy density function by the method of steepest descent

We present a unified theory of diatomic molecules which reconciles bound state spectroscopy and atomic scattering theory. The total wave-function is expanded in a complete set of atomic channel states which is entirely equivalent to an expansion in Hund's case (e) electronic-rotational states. An analysis of the coupled radial, that is vibrational, functions places strong constraints on the asymptotic properties of the molecular wave-functions. These are presented in terms of the reactance K and scattering S matrices of atomic scattering theory which offers a uniform treatment for open channels (inelastic scattering and continuum spectroscopy), closed channels (bound state spectroscopy) and mixtures of both (predissociation). The normalization of the total wavefunction is derived and related to the asymptotic boundary conditions both for continuum and bound states.     

Dynamics of geometric discord and measurement-induced nonlocality at finite temperature

By using geometric measure of discord (GMOD) [B. Dakić, V. Vedral, Č. Brukner, Phys. Rev. Lett. 105, 109502 (2010)] and measurement-induced nonlocality (MIN) [S. Luo, S. Fu, Phys. Rev. Lett. 106, 120401 (2011)], we investigate quantum correlation of a pair of two-level systems, each of which is interacting with a reservoir at finite temperature T. We show that, for a broad class of states of the system, GMOD and MIN can endure sudden death, and there is no asymptotic decay for MIN while asymptotic decay exists for GMOD. We also give the dynamics of GMOD and MIN with respect to the temperature and illustrate their different characteristics.     

Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy

The total space of the spinor bundle on the four-dimensional sphere S⁴ is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang–Mills instanton on this eight-dimensional gravitational instanton. This is a higher dimensional generalization of (anti-) self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang–Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

微扰的耦合非线性薛定谔方程的近似求解

将直接微扰方法应用于可积的含修正项的非线性薛定谔方程，通过近似解与精确解的比较确定了直接微扰方法的可靠性.继而，将该方法应用于微扰的耦合非线性薛定谔方程，并获得了该微扰方程的可靠的近似解. 关键词： 直接微扰方法 微扰 耦合非线性薛定谔方程 近似解     

Thermal conductivity of nonlinear waves in disordered chains

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity κ(T). We find indications for an asymptotic low-temperature κ ∼ T ⁴ and intermediate temperature κ ∼ T ² laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (Laptyeva et al, Europhys. Lett. 91, 30001 (2010)).     

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.     

Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008). In memoriam V.A. Borovikov     

Monitoring the pH-Dependent Oximation of Methyl Ethyl Ketone and Benzaldehyde by in situ Fourier Transform IR Spectroscopy in a Heterogeneous Liquid–Liquid Two-Phase System

In situ Fourier transform IR spectroscopy was a useful tool in monitoring the pH-dependent and two-step oximation of methyl ethyl ketone and benzaldehyde in a liquid–liquid two-phase system, one phase of which was water. Carrying out the oximations at pH 8, the oximation was complete within 45 min and the corresponding carbonyl compound–hydroxylamine adduct (2) could be detected as an intermediate, but at pH 10, the corresponding oxime (3) was formed faster, practically without the appearance of the intermediate. At pH ～ 2.5 using methyl ethyl ketone, the protonated form of the corresponding oxime (3'a) developed gradually without the transient appearance of the intermediate (2). pH dependence of the 3a → 3'a and 3'a → 3a interconversions were studied in separate experiments.     

High-precision measurement of separatrix splitting in a nonlinear resonance

We present the results of numerical modeling and a theoretical analysis of the splitting of a nonlinear-resonance separatrix in the intermediate asymptotic region for the standard-map model. Direct measurements of the splitting angle α(K), where K is the small parameter of the system, have been carried out over a huge range, 0.1≳α≳10⁻²⁰⁸ (1⩾K⩾0.0004), with a relative accuracy greater than one part in 10⁻²⁵ and an average accuracy of roughly one part in 10⁻³⁰. This made it possible to compare in detail our results with those of the existing asymptotic theory and to detect a number of new effects. We find a relatively simple empirical expression for the α vs. K dependence in the intermediate asymptotic region, and this region proves to be surprisingly broad: K≲10⁻². We also study the effect of noise, in particular, errors in measuring the angle, which proved to be much more significant and complicated than expected. Finally, we point out unresolved questions and possible directions of research involving this problem. Zh. éksp. Teor. Fiz. 114, 1516–1531 (October 1998)     

Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation

This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(\lambda)$ which act in the highest weight Virasoro module and commute for different values of the parameter λ. These operators appear to be the CFT analogs of the Q - matrix of Baxter [2], in particular they satisfy Baxter's famous T- Q equation. We also show that under natural assumptions about analytic properties of the operators as the functions of λ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q(λ) contains the “dual” nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the Q - operators and the stationary transport properties in the boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum Hall system. Received: 2 December 1996 / Accepted: 11 March 1997