The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable systems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell (M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy separately.     

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator \widetilde{J} is presented by constructing a subalgebra \widetilde{G } of the loop algebra \widetilde A₂. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.     

Studies on heteronuclear diatomic molecular dissociation energies using algebraic energy method

The algebraic energy method (AEM) is applied to the study of molecular dissociation energy $D_e$ for 11 heteronuclear diatomic electronic states: $a^3\Sigma^+$ state of NaK, $X^2\Sigma^+$ state of XeBr, $X^2\Sigma^+$ state of HgI, $X^1\Sigma^+$ state of LiH, $A^3\Pi(1)$ state of ICl, $X^1\Sigma^+$ state of CsH, $A(^3\Pi_1)$ and $B0^+(^3\Pi)$ states of ClF, $2^1\Pi$ state of KRb, $X^1\Sigma^+$ state of CO, and $c^3\Sigma^+$ state of NaK molecule. The results show that the values of $D_e$ computed by using the AEM are satisfactorily accurate compared with experimental ones. The AEM can serve as an economic and useful tool to generate a reliable $D_e$ within an allowed experimental error for the electronic states whose molecular dissociation energies are unavailable from the existing literature.     

溶解过程的热效应

溶解热是质-剂微粒间作用(释热)与拆开原先溶质微粒间作用(吸热)、部分溶剂微粒间作用(吸热),在能量方面的代数和,以1 mol溶质溶于大量溶剂(通常是水)计量热效应(kJ/mol)为准。强调“大量溶剂”是因为热效应随溶剂量(在一定范围内)而改变,把1 mol H2SO4、1 mol HCl在水中溶解热和溶剂量的关系列于下表:由H2SO4溶解热前3竖行知:往n(H2SO4)/n(H2O)=1.0/0.5=2.0溶液中加0.5 mol H2O时释热量为:-27.82 kJ/mol-(-15.73 kJ/mol)=-12.09 kJ/mol;若再加0.5 mol H2O,释热量为:-35.90 kJ/mol-(-27.82 kJ/mol)=-8.08 kJ/mol。由HCl(g)溶解热…     

Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.     

Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations

Using the idea of Atanassov, we define the notion of intuitionistic Menger spaces as a netural generalizations of Menger spaces due to Menger. We also obtain a new generalized contraction mapping and utilize this contraction mapping to prove the existance theorems of solutions to differential equations in intuitionistic Menger spaces.     

Nonlinear Dynamical Symmetries of Some Two-Dimensional Quantum Systems

In this paper nonlinear dynamical symmetries of three quantum systems are studied in detail, such as the Kepler-Coulomb system and the isotropic harmonic oscillator in a two-dimensional curved space, and the generalized pseudo-oscillators in the two-dimensional fiat space. Their nonlinear spectrum generating algebras are shown to be relevant to polynomial angular momentum algebras.     

Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras gln and g~ln we construct a family of integrable generalizations of the Toda chains characterized by two integers m and m_. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m =m_,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m ,m_)-Toda chains, which turns out to be very different for different pairs of m and m_. Finally we also obtain the nonabelian generalizations of the (m ,m_)-Toda chains in an explicit form.     

判定线性偏微分方程组解的完备性的一个符号计算方法

从微分代数的角度出发,借助于吴微分特征集理论,对于线性偏微分方程组,给出了判定它的解的完备性的一个符号计算方法.这个算法是一个机械化的算法,借助于符号计算软件Maple,可以在计算机上实现.     

非线型三原子分子势能面的Lie代数方法

利用Lie代数方法得到三原子分子的代数Hamiltonian ,通过对光谱数据的拟合确定其展开系数 ,再用相干态得到代数Hamiltonian的经典极限 ,从而得到三原子分子的势能面 .以H2 O分子为例进行了计算 ,其理论值与实验结果一致. The algebraic Hamiltonian for a triatomic molecule can be obtained by using dynamical Lie algebra method (the expansion coefficients are obtained by fitting spectroscopic data). Triatomic molecular potential energy surface (PES) is obtained by using coherent state to take the classical limits of algebraic Hamiltonian. This PES is applied to H 2O molecule, and the deduced force constant is in good agreement with the experimental data.