Symmetries of the interacting boson model

This contribution reviews the symmetry properties of the interacting boson model of Arima and Iachello. While the concept of a dynamical symmetry is by now a familiar one, this is not necessarily so for the extended notions of partial dynamical symmetry and quasi dynamical symmetry, which can be beautifully illustrated in the context of the interacting boson model. The main conclusion of the analysis is that dynamical symmetries are scarce while their partial and quasi extensions are ubiquitous.     

Partial Wave Crossing Relations for Meson-Baryon Scattering

Three different types of relations for meson-baryon partial wave amplitudes following from the s ↔ u crossing symmetry of the invariant amplitudes are investigated: i) the partial wave crossing relations directly derived from the s ↔ u crossing symmetry and the partial wave projection, ii) the partial wave crossing relations following from a combined use of s ↔ u crossing symmetry and analyticity, and iii) the crossing sum rules, which are a direct consequence of the s ↔ u crossing symmetry. The kernels entering the crossing relations i) and ii) are given in an analytic form, valid for arbitrary angular momentum. Their properties are studied in detail. By means of the partial wave crossing relations ii) it is demonstrated that u-channel exchange of a resonance leads to a resonance-like behaviour of the partial wave amplitudes on the crossed physical cut. This phenomenon is described in terms of “pseudoresonances”. Finally the crossing sum rules iii) and their practical use are thoroughly discussed, and they are compared with the partial wave crossing relations i) and ii). Special attention is paid to the crossing symmetry constraints on the S-wave amplitudes. As a result of these investigations we present a crossing symmetric ansatz for these amplitudes which gives a more reliable parametrization than the usual effective range approximation.     

对称分类在非线性偏微分方程组边值问题中的应用

研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用.首先,利用偏微分方程(组)完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类;其次,利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题;最后,利用龙格-库塔法求解了常微分方程组初值问题的数值解.     

广义对称性与辐射直接交换面积

借助广义对称性,将辐射直接交换面积的积分函数归结为四类基本形式,并应用于10种运算.研究分析了辐射直接交换面积矩阵的各种分块,将整个矩阵按特性划分为4种分区.定量地给出了应用广义对称性之后,三维和二维体系下直接交换面积计算量减少的评估算式.     

Generalized partial dynamical symmetry in nuclei

We introduce the notion of a generalized partial dynamical-symmetry for which part of the eigenstates have part of the dynamical symmetry. This general concept is illustrated with the example of Hamiltonians with a partial dynamical O(6) symmetry in the framework of the interacting boson model. The resulting spectrum and electromagnetic transitions are compared with empirical data in 162Dy.     

Electromagnetic retarded interaction and symmetry violation of time reversal in high order stimulated radiation and absorption processes of light

It has been proved that when the retarded effect (or multiple moment effect) of radiation fields is taken into account, the high order stimulated radiation and stimulated absorption probabilities of light are not the same so that time reversal symmetry would be violated, though the Hamiltonian of electromagnetic interaction is still unchanged under time reversal. The reason to cause time reversal symmetry violation is that certain filial or partial transition processes of bound atoms are forbidden or cannot be achieved due to the law of energy conservation and the special states of atoms themselves. These restrictions would cause the symmetry violation of time reversal of other filial or partial transition processes which can be actualized really. The symmetry violation is also relative to the asymmetry of initial states of bound atoms before and after time reversal. For the electromagnetic interaction between non-bound atoms and radiation field, there is no such kind of symmetry violation of time reversal. In this way, the current formula on the parameters of stimulated radiation and absorption of light with time reversal symmetry should be revised. A more reliable foundation can be established for the theories of laser and nonlinear optics in which non-equilibrium processes are involved.     

Application of Differential Forms to Construction of Nonlocal Symmetries

Abstract

Differential forms are used for construction of nonlocal symmetries of partial differential equations with conservation laws. Every conservation law allows to introduce a nonlocal variable corresponding to a conserved quantity. A prolongation technique is suggested for action of symmetry operators on these nonlocal variables. It is shown how to introduce these variables for the symmetry group to remain the same. A new hidden symmetry and corresponding group-invariant solution are found for gas dynamic equations.     

Symmetry Groups and New Exact Solutions to (2+1)-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2 1)-dimensional variable coefficient canonical generalized KP (VCCGKP) equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

Partial dynamical symmetries

This overview focuses on the notion of partial dynamical symmetry (PDS), for which a prescribed symmetry is obeyed by a subset of solvable eigenstates, but is not shared by the Hamiltonian. General algorithms are presented to identify interactions, of a given order, with such intermediate symmetry structure. Explicit bosonic and fermionic Hamiltonians with PDS are constructed in the framework of models based on spectrum generating algebras. PDSs of various types are shown to be relevant to nuclear spectroscopy, quantum phase transitions and systems with mixed chaotic and regular dynamics.     

Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient

Transient heat transfer through a longitudinal fin of various profiles is studied. The thermal conductivity and heat transfer coefficients are assumed to be temperature dependent. The resulting partial differential equation is highly nonlinear. Classical Lie point symmetry methods are employed and some reductions are performed. Since the governing boundary value problem is not invariant under any Lie point symmetry, we solve the original partial differential equation numerically. The effects of realistic fin parameters such as the thermogeometric fin parameter and the exponent of the heat transfer coefficient on the temperature distribution are studied.     

Mechanisms of binary and ternary low-energy fission of nuclei with allowance for nonsphericity effects

Within the quantum-mechanical theory of fission, wave functions for fragments of binary nuclear fission and amplitudes of partial fission widths are constructed with allowance for a strong nonsphericity of fragment-interaction potentials. It is shown that, in the strong-coupling approximation, the symmetry axes of fission fragments are oriented along the symmetry axis of a fissile nucleus. The structure of the fragment-interaction potentials is analyzed, and the mechanism that is responsible for the alignment of the spins and relative orbital angular momenta of fission fragments and which explains the emergence of high fragment-spin values in experiments is substantiated. The mechanisms in question are generalized to the case of ternary nuclear fission. The fragment-interaction potentials and fragment wave functions are investigated, along with the partial fission widths with respect to the ternary fission of nuclei.     

二维运动电荷的Mei对称性

把Mei对称性方法用于电磁场中带电粒子的运动，从二维运动电荷的Mei对称性出发,运用比较系数法，得到与Mei对称性相应的生成元的普遍表达式及电磁场所满足的偏微分方程组.对一特例进行了讨论. 关键词： 二维运动电荷 电磁场 Mei对称性     

Generalized conditional symmetries of nonlinear differential-difference equations

Generalized conditional symmetry method for tackling nonlinear partial differential equations is extended to differential-difference equations. As the applications, some exact solutions to several nonlinear differential-difference equations are obtained.     

A new method to obtain approximate symmetry of nonlinear evolution equation from perturbations

A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter series, we obtain more affluent approximate symmetries. The method is applied to two perturbed nonlinear partial differential equations and new approximate solutions are derived.     

二维各向异性谐振子的第三个独立守恒量及其对称性

二维各向异性谐振子和两分振子的能量是守恒的, 但三个守恒量中只有其中两个是独立的. 当频率比ω₁/ω₂ 为有理数时, 系统存在第三个独立的守恒量.本文用扩展Prelle-Singer 法得到五个典型谐振子的第三个独立守恒量, 并讨论了与守恒量相应的Noether对称性与Lie对称性.     

Similarity Reductions for a Nonlinear Diffusion Equation

Abstract

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.     

An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.     

Relation of permutation symmetry of the tensor of nonlinear susceptibility to the conservation laws of photon number in nonlinear quadratic nondissipative and partially dissipative media

Relation between permutation symmetry for components of the tensor of the nonlinear quadratic susceptibility and the conservation laws of the number of photons in a nondissipative nonlinear quadratic medium is pointed out in this work. On the basis of validity of partial conservation laws of the number of photons some relations of the permutation symmetry for components of the tensor of the nonlinear quadratic susceptibility are derived even in the case of a partially dissipative medium.     

哈密顿Ermakov系统的形式不变性

把形式不变性的方法用于研究哈密顿Ermakov系统，从哈密顿Ermakov系统的形式不变性出发，运用比较系数法得到与形式不变性相应的点对称变换生成元的表达式及势能所满足的偏微分方程.结果表明，在点对称变换下，只有自治的哈密顿Ermakov系统才具有形式不变性. 关键词： 哈密顿Ermakov系统 拉格朗日函数 点对称变换 形式不变性