KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.     

Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltoniansystems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other istime-dependent Hamiltonian system. The quantum unitary operatorrelevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.     

Convolution Theorem of Fractional Fourier Transformation Derived by Representation Transformation in Quantum Mechancis

Based on our previous paper (Commun. Theor. Phys. 39 (2003) 417) we derive the convolution theorem of fractional Fourier transformation in the context of quantum mechanics, which seems a convenient and neat way. Generalization of this method to the complex fractional Fourier transformation case is also possible.     

Three dimensional large deformation analysis of phase transformation in shape memory alloys

Shape memory alloys（SMAs）have been explored as smart materials and used as dampers,actuator elements,and smart sensors.An important character of SMAs is its ability to recover all of its large deformations in mechanical loading-unloading cycles without showing permanent deformation.This paper presents a stress-induced phenomenological constitutive equation for SMAs,which can be used to describe the superelastic hysteresis loops and phase transformation between Martensite and Austenite.The Martensite fraction of SMAs is assumed to be dependent on deviatoric stress tensor.Therefore,phase transformation of SMAs is volume preserving during the phase transformation.The model is implemented in large deformation finite element code and cast in the updated Lagrangian scheme.In order to use the Cauchy stress and the linear strain in constitutive laws,a frame indifferent stress objective rate has to be used.In this paper,the Jaumann stress rate is used.Results of the numerical experiments conducted in this study show that the superelastic hysteresis loops arising with the phase transformation can be effectively captured.     

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, \emph{ J. Phys.} A {\bf 25} (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) \e^{-2\i\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$ and its inverse% $ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) \e^{2\i\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.     

Phase proportioning in CuAlBe shape memory alloys during thermomechanical loadings using electric resistance variation

The microstructure of shape memory alloys changes with the thermomechanical history of the material. During thermomechanical loading, austenite, thermally-induced martensite or stress-induced martensite can be simultaneously present in the material. In applications integrating SMA parts, utilization conditions seriously affect the microstructure and can generate macroscopic strain or stress. Consequently, during thermomechanical loadings, it is important to be able to proportion the different phases and consequently to understand the kinetic transformation. This is very useful in the development of constitutive equations. This study shows, by a series of tests, that the proposed experimental method, based on the measurement of the variation of electric resistance of CuAlBe wires, permits to determine the volume fraction of the different phases present in the material (i.e., austenite, stress-induced martensite and thermally-induced martensite). The proposed method is applied to the most common thermomechanical behavior met in engineering applications of shape memory alloys: pseudoelasticity, pseudoplasticity, recovery-stress and stress-assisted two-way shape memory effect. The proportioning method based on a mixture law integrating the resistivity of pure phases present in the SMA is first performed on different two-phase mixture cases and then applied to a three phase mixture case.     

Crystal plasticity finite element modeling of mechanically induced martensitic transformation (MIMT) in metastable austenite

A new crystal plasticity model incorporating the mechanically induced martensitic transformation in metastable austenitic steel has been formulated and implemented into the finite element analysis. The kinetics of martensite transformation is modeled by taking into consideration of a nucleation-controlled phenomenon, where each potential martensitic variant based on Kurdjumov–Sachs (KS) relationship has different nucleation probability as a function of the interaction energy between externally applied stress and lattice deformation. Therefore, the transformed volume fractions are determined following selective variants given by the crystallographic orientation of austenitic matrix and applied stress in the frame of the crystal plasticity finite element. The developed finite element program is capable of considering the effect of volume change by the Bain deformation and the lattice-invariant shear during the martensitic transformation by effectively modifying the evolution of plastic deformation gradient of the conventional rate-dependent crystal plasticity finite element. The validation of the proposed model has been carried out by comparing with the experimentally measured data under simple loading conditions. Good agreements with the measurements for the stress–strain responses, transformed martensitic volume fractions and the influence of strain rate on the deformation behavior will enable the model to be promising for the future applications to the real forming process of the TRIP aided steel.     

Painlevé Integrability of Nonlinear Schrödinger Equations with bothSpace- and Time-Dependent Coefficients

We investigate the Painlevé integrability of nonautonomous nonlinearSchrödinger (NLS) equations with both space- and time-dependent dispersion, nonlinearity, and external potentials. The Painlevé analysis is carried out without using the Kruskal's simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painlevé analyses and/or become unknown new equations.     

由钛酸盐纳米带水热制备锐钛矿型TiO2纳米带

研究了水热处理具有层状结构的钛酸钠纳米带或钛酸纳米带转化为锐钛矿型TiO₂的制备过程、难易程度和相转化机理. 实验结果表明, 当水热反应温度和时间分别在160 ℃ 和24 h以内, 钛酸钠纳米带很难完全转化为锐钛矿型TiO₂, 若升高反应温度并延长反应时间, 则可制得纯的锐钛矿型TiO₂, 但纳米带形貌被严重破坏; 当水热反应温度和时间分别为160 ℃ 和16 h时, 1次酸洗的钛酸纳米带能够完全转化为锐钛矿型TiO₂, 若钛酸纳米带经过3次强酸浸泡, 则在160 ℃下相转化时间就会缩短到12 h, 所有钛酸纳米带在转化为TiO₂后的形貌仍为纳米带, 但经3次酸浸后生成的TiO₂纳米带表面更光滑. 讨论了钛酸钠纳米带或钛酸纳米带转化为锐钛矿型TiO₂的相转化机理.     

Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel

We newly construct two mutually-conjugate tripartite entangled state representations,based on which we propose the formulation of three-mode entangled fractional Fourier transformation(EFFT) and derive the transformation kernel.The EFFT’s additivity property is proved and the eigenmode of EFFT is derived.As an application,we calculate the EFFT of the three-mode squeezed vacuum state.