Table of contents

We study the functional separation of variables to the nonlinear heat equation: u_t = (A(x)D(u)u_xⁿ)_x + B(x)Q(u), A_x ≠ 0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)-dimensional Sawada–Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou's direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada–Kotera and Konopelchenko–Dubrovsky equations, respectively.

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the (2+1)-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.

Using Ablowitz–Ramani–Segur algorithm, the coupled KdV systems are reclassified under the Painlevé integrable sense while the similarity reductions of the model are obtained by using the Clarkson and Kruskal's direct method. Some new types of Painlevé integrable models including a model with different dispersion relations for two layer fluids are found.

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the (2+1)-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in (2+1) dimensions.

Using the truncated Painlevé expansion, symbolic computation, and direct integration technique, we study analytic solutions of (2+1)-dimensional Boussinesq equation. An auto-Bäcklund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ℘ function. Some of them are novel.

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc, are reduced to an integrable ODE expressed by u'' + p(u)(u')² + q(u) = 0 whose general solution can be given. Furthermore, combining complete discrimination system for polynomial, the classifications of all single travelling wave solutions to these equations are obtained. The equation u'' + p(u)(u')² + q(u) = 0 includes the equation (u')² = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.

In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton's principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton's principle. Finally, the Hamilton's principle of high-order Lagrangian function is given.

The double-sine-Gordon equation is studied by means of the so-called mapping method. Some new exact solutions are determined.

In this paper, the Noether–Lie symmetry and conserved quantities of generalized classical mechanical system are studied. The definition and the criterion of the Noether–Lie symmetry for the system under the general infinitesimal transformations of groups are given. The Noether conserved quantity and the Hojman conserved quantity deduced from the Noether–Lie symmetry are obtained. An example is given to illustrate the application of the results.

A manifestly gauge-invariant formulation of non-relativistic quantum mechanics is applied to the case of time-dependent harmonic oscillator in the magnetic dipole approximation. A general equation for obtaining gauge-invariant transition probability amplitudes is derived.

On the condition of electric-LO phonon strong-coupling in a parabolic quantum dot, we obtain the eigenenergy of the ground-state and the first-excited state, the eigenfunctions of the ground-state and the first- excited state by using variational method of Pekar type. This system in quantum dot may be employed as a two-level quantum system-qubit. When the electron is in the superposition state of the ground- and the first-excited state, we obtain the time evolution of the electron density. The relation of the probability density of electron on the Coulomb binding parameter and the relations of the period of oscillation on the Coulomb binding parameter, the electron-LO-phonon coupling constant and the confinement length are derived.

The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr–Sommerfeld quantization rule yields the exact expression for the energy spectrum.

The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix elements for both position and momentum operators are obtained.

Werner states are paradigmatic examples of quantum states and play an innovative role in quantum information theory. In investigating the correlating capability of Werner states, we find the curious phenomenon that quantum correlations, as quantified by the entanglement of formation, may exceed the total correlations, as measured by the quantum mutual information. Consequently, though the entanglement of formation is so widely used in quantifying entanglement, it cannot be interpreted as a consistent measure of quantum correlations per se if we accept the folklore that total correlations are measured (or rather upper bounded) by the quantum mutual information.

Four-qubit entanglement has been investigated using a recent proposed entanglement measure, multiple entropy measures (MEMS). We have performed optimization for the nine different families of states of four-qubit system. Some extremal entangled states have been found.

In this paper we study the entanglement in a two-qubit spin in the XYZ model, and teleport a two-qubit entangled state using this spin chain in the condition of the thermal equilibrium as a quantum channel. We investigate the effects of the interaction of z-component J_Z, the inhomogeneous magnetic field b, the anisotropy γ, and the temperature T on the entanglement and fidelity. In order to characterize the quality of the teleported state, we research the average fidelity F_a. High average fidelity of the teleportation is obtained when the temperat ure is very low. Under some condition, we also find that when inhomogeneity increases to a certain value, the average fidelity can exhibit a larger revival than that for less values of b.

We present a scheme for teleporting multi-qutrit quantum information from a sender to a receiver via the control of many agents in a network. Agents's control parameters are obtained via quantum entanglement swapping. In our scheme, Zhang and Man's QSS protocol [Phys. Rev. A 72 (2005) 022303] based on Bell-state entanglement swapping is generalized to a qutrit case. Our scheme owns the advantage of having higher code capacity and better security than the work [Commun. Theor. Phys. 44 (2005) 847] on controlled teleportation for multi-qubit.

By using the partial transpose and realignment method, we study the time evolution of the bound entanglement under the bilinear-biquadratic Hamiltonian. For the initial Horodecki's bound entangled state, it keeps bound entangled for some time, while for the initial bound entangled states constructed from the unextendable product basis, they become free once the time evolution begins. The time evolution provides a new way to construct bound entangled states, and also gives a method to free bound entanglement.

Transmitting quantum states by channels of analogous Bell states is studied in this paper. We analyze the transmitting process, constructed the probabilitic unitary operator, and gain the largest successful transfer quantum state probability.

A scheme is presented for the generation of entangled states for two atoms trapped in two distant cavities. In the scheme each atom is resonantly coupled with the respective cavity mode and driven by a strong classical field. The detection of a photon decaying from the cavities and passing through a beam-splitter collapses the atoms to an entangled state. The required atom-field interaction time is very short and thus the decoherence effect is suppressed. Our scheme is within the reach of presently available cavity QED techniques.

A scheme for approximately and conditionally teleporting an unknown atomic state in dissipative cavity QED is proposed. It is the extension of the scheme of [Phys. Rev. A 69 (2004) 064302], where the cavity mode decay has not been considered and only a time point of system evolution and the corresponding fidelity implementing the teleportation are given. In fact, the cavity mode decay exists really and must be delt with. In this paper, we investigate the influence from the cavity mode decay on the implementation of the approximate and conditional teleportation by means of the dissipative Jaynes–Cummings model and then show the analytical expression of the fidelity of realization of the teleportation. Alternatively, our scheme does not involve an additional atom, only requiring two atoms and one single-mode cavity.

We propose a remote state preparation (RSP) scheme of three-particle Greenberger–Horne–Zeilinger (GHZ) class states, where quantum channels are composed of two maximally entangled states. With the aid of forward classical bits, the preparation of the original state can be successfully realized with the probability 1/2, the necessary classical communication cost is 0.5 bit on average. If the state to be prepared belongs to some special states, the success probability of preparation can achieve 1 after consuming one extra bit on average. We then generalize this scheme to the case that the quantum channels consist of two non-maximally entangled states.

Quantum dense coding (QDC) is a process originally proposed to send two classical bits information from a sender to a receiver by sending only one qubit. Our scheme of QDC is proposed following some ideas on secret sharing with entanglement in cavity QED. Based on the theory of secret sharing the QDC process can be more secure.

An experimentally feasible scheme for teleportation of an unknown two-atom entangled state is proposed. Our scheme uses a cluster state as the quantum channel, where we do not need any joint Bell-state measurement. Moreover the successful probability and fidelity of teleportation can both reach 1.0. The current scheme can be realized within the current experimental technology.

We propose a physical scheme for implementing the Deutsch-Jozsa algorithm with superconducting quantum interference devices (SQUIDs) in cavity-QED. The scheme is based on SQUID coupled to a single-mode microwave cavity field or classical microwave pluses. The scheme is very simple and may be realizable experimentally.

We derive an approximate analytical expression for the ground state of double-well BEC's, which reproduces highly accurately the numerical solution for the whole parameter regimes of the two-body repulsive interaction strength, the total number of atoms, and the hopping parameter.

Using Damour–Ruffini's method, Hawking radiation from a general stationary black hole is investigated again deeply. Considering the back reaction of the particle to the space-time and energy conservation, we find that the radiation is not exactly thermal and can take out information from the black hole. This can be used to explain the information loss paradox, and the result is consistent with the works finished before.

By appropriately choosing additional dimensions of space-time, the mass spectrum of mesons is obtained, and the calculated results agree with the experimental data.

The phenomenon of the resonant activation (RA) of a particle over a fluctuating potential barrier with a four-value noise is investigated. It is shown that the mean first passage time (MFPT) displays six minima as the function of the transition rates γ₁, γ₂, γ₃, γ₄, γ₅ and γ₆ of the four-value noise, respectively. In addition, the effect of other parameters of the system, such as the noise strength D of the additive Gaussian white noise and the parameter value a, b, c, and d of the four-value noise, on the RAs is also investigated.

The phenomena of GS in a drive-response system have been studied. Several kinds of GS are found: the driver-induced GS, the responser-induced GS, and the intermediate GS. The mechanism generating these types of GS is given and the roles played by response and drive dynamics in realizing different GS states are discussed.

An integrable (2+1)-dimensional coupled mKdV equation is decomposed into two (1+1)-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two (1+1)-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.

Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.

Complex behavior in a selective aging simple neuron model based on small world networks is investigated. The basic elements of the model are endowed with the main features of a neuron function. The structure of the selective aging neuron model is discussed. We also give some properties of the new network and find that the neuron model displays a power-law behavior. If the brain network is small world-like network, the mean avalanche size is almost the same unless the aging parameter is big enough.

The new members of the charm-strange family D^*_sJ(2317), D_sJ(2460), and D_s(2632), which have the surprising properties, are challenging the present models. Many theoretical interpretations have been devoted to this issue. Most authors suggest that they are not the conventional c quark model states, but possibly are four-quark states, molecule states, or mixtures of a P-wave c and a four-quark state. In this work, we follow the four-quark-state picture, and study the masses of cn/cs states (n is u or d quark) in the chiral SU(3) quark model. The numerical results show that the mass of the mixed four-quark state (cn/cs) with spin parity J^P = 0⁺ might not be D_s(2632). At the same time, we also conclude that D^*_sJ (2317) and D_sJ (2460) cannot be explained as the pure four-quark state.

The neutral gauge boson B_H with the mass of hundreds GeV is the lightest particle predicted by the littlest Higgs (LH) model, and such particle should be the first signal of the LH model at the planed ILC if it exists indeed. In this paper, we study some processes of the B_H production associated with the fermion pair at the ILC, i.e., γγ → fB_H. The studies show that the most promising processes to detect B_H among γγ → fB_H are γγ → l'⁺l'⁻B_Hl' = e, μ), and they can produce the sufficient signals in most parameter space preferred by the electroweak precision data at the ILC. On the other hand, the signal produced via the certain B_H decay modes is typical and such signal can be easily identified from the SM background. Therefore, B_H, the lightest gauge boson in the LH model, would be detectable at the photon collider realized at the ILC.

Utilizing the results that the Faddeev model is equivalent to the mesonic sector of the SU(2) Skyrme model, where the baryon number current vanishes everywhere, some exact solutions including the vortex solutions of the Faddeev model are discussed. The solutions are classified by the first Chern number. When the Chern number equals 2, the new multisoliton solutions are obtained and the pipe shape distribution of the energy density of the solutions are found.

We focus on a new gauge symmetry keeping regularization scheme for momentum integration and point out that dropping out momentum space asymptotic non-logarithmic total derivative divergent integrations in quantum field theory is a simple and natural way to keep the computation program gauge-covariant.

In equilibrium statistical field theory, the partition function has fundamental importance. In this paper we propose a direct and general method for calculating the partition function and equation of state of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phases is obtained. This general method is applied to two specific cases: the free quark theory and QCD with a model dressed quark propagator having confinement features. In the first case, the standard Fermi distribution at T = 0 is reproduced. In the second case, we apply the conclusion in previous works to obtain the dressed quark propagator at finite chemical potential and find the unphysical result that the baryon number density vanishes for all values of chemical potential. The reason for this result is discussed.

A Higgs–Yang–Mills monopole scattering spherical symmetrically along light cones is given. The left incoming anti-self-dual a plane fields are holomorphic, but the right outgoing SD β plane fields are antiholomorphic, meanwhile the diffeomorphism symmetry is preserved with mutual inverse affine rapidity parameters μ and μ⁻¹. The Dirac wave function scattering in this background also factorized respectively into the (anti)holomorphic amplitudes. The holomorphic anomaly is realized by the center term of a quasi Hopf algebra corresponding to an integrable conformal affine massive field. We find explicit Nahm transformation matrix (Fourier-Mukai transformation) between the Higgs YM BPS (flat) bundles (D modules) and the affinized blow up ADHMN twistors (perverse sheafs). Thus we establish the algebra for the 't Hooft–Hecke operators in the Hecke correspondence of the geometric Langlands program.

Based on the diquark model, we assume that the light scalar mesons are q²² states rather than q. The chiral effective Lagrangian for the light scalar meson is constructed, and the mass relations are obtained: the isotriplet (a₀) and the isosinglet (f₀) are the heaviest and are degenerate, the isodoublets (κ) are heavier and the other isosinglet (σ) is the lightest; and 2M_κ²= M_a0² + M_σ². Using experimental value for a₀ and σ mass, we obtain M_κ = 794 MeV, which is consistent with the experimental value. Then taking Γ(a₀⁰ → ηπ⁰) = 90 MeV and Γ(f₀⁰ → π⁰π⁰) = 20 MeV, we get the width of σ is: Γ(σ → π⁺π⁻) = 150 MeV.

Based on the quark-gluon structure of nucleon and the possible existence of Odderon in strong interaction process due to gluon self-interaction, the elastic scatterings of pp and p at high energies are studied. The contributions from individual terms of quark-quark, gluon-gluon interactions, quark-gluon interference, and the Odderon terms to the nuclear slope parameter B(s) are analyzed. Our results show that the QCD inspired model gives a good fit to the LHC experimental data of the nuclear slope parameter.

Deconfinement phase transition and neutrino trapping in (proto)neutron star matter are investigated in a chiral hadronic model (also referred to as the FST model) for the hadronic phase (HP) and in the color-flavor-locked (CFL) quark model for the deconfined quark phase. We include a perturbative QCD correction parameter α_s in the CFL quark matter equation of states. It is shown that the CFL quark core with K⁰ condensation forms in neutron star matter with the large value of α_s. If the small value of α_s is taken, hyperons suppress the CFL quark phase and the HP is dominant in the high-density region of (proto)neutron star matter. Neutrino trapping makes the fraction of the CFL quark matter decrease compared with those without neutrino trapping. Moreover, increasing the QCD correction parameter α_s or decreasing the bag constant B and the strange quark mass m_s can make the fraction of the CFL quark matter increase, simultaneously, the fraction of neutrino in protoneutron star matter increases, too. The maximum masses and the corresponding radii of (proto)neutron stars are not sensitive to the QCD correction parameter α_s.

Multielectron atoms near a metal surface are essentially more complicated than hydrogen atom with regard to theoretical treatments. By using the semicalssical closed orbit theory generalized to the multielecton atoms, we study the dynamical properties of the Rydberg lithium atom near a metal surface. The photoabsorption spectra and recurrence spectra of this system have also been calculated. Considering the effect of the ionic core potential of the Rydberg lithium atom, the number of the closed orbits increases, which leads to more peaks in the recurrence spectra than the case of hydrogen atom near a metal surface. This result shows that the core-scattered effects play an important role in nonhydrogenic atoms. This study is a new application of the closed-orbit theory and is of potential experimental interest.

We propose a scheme of generating multi-component entangled coherent states of cavity fields. In this scheme, the atoms pass through cavities one by one, simultaneously driven by a strong classical field in each cavity. Then the detection of the atomic states collapses the cavity fields onto multi-component entangled coherent states. It is shown that, with a judicious choice of the parameters of the classical field, we can conditionally produce macroscopic multi-dimensional maximal entanglement for the cavity modes.

We show that Wu-Yang theory of fully quantized four-wave mixing can be generalized to the six-wave mixing and derive the analytical solution of the coupled equations describing the quantum dynamics of six-wave mixing.

In this paper, if the condition of variation δt = 0 is satisfied, the higher-order Lagrangian equations and higher-order Hamilton's equations, which show the consistency with the results of traditional analytical mechanics, are obtained from the higher-order Lagrangian equations and higher-order Hamilton's equations. The results can enrich the theory of analytical mechanics.

By deriving a Korteweg-de Vries (KdV) equation for particular dusty plasma, we find the existing regions of solitary waves. The regions of both rarefactive and compressive solitary waves have also been given in this paper for a two-temperature ions and a hot adiabatic dusty fluid, where the dust charges can vary.

The transmission properties of terahertz (THz) wave passing through semiconductor aperture have been investigated. The dispersion relationship for surface plasmon polariton (SPP) at different temperatures has been numerically calculated. The results show that the dispersion relationship increases with the increasing of frequency and the decreasing of temperature, the thickness of slab has to be taken into consideration because of the large skin depth for semiconductor slab. In addition, the propagation constant increases with the increasing of frequency and the decreasing of temperature.

The elastic properties of the wurtzite-type aluminum nitride (w-AlN) are investigated by ab initio plane-wave pseudopotential density functional theory method. The pressure dependences of the normalized primitive cell volume V/V₀, the elastic constants c_ij, the aggregate elastic modulus (B, G, E), the Poisson's ratio (ν), and the Debye temperature θ_D are successfully obtained. From the elastic constants of the w-AlN under pressure, we find that the w-AlN should be unstable at higher pressure than 61.33 GPa.

The confined longitudinal-optical (LO) phonon and surface-optical (SO) phonon modes of a free-standing annular cylindrical quantum dot are derived within the framework of dielectric continuum approximation. It is found that there exist two types of SO phonon modes: top SO (TSO) mode and side SO (SSO) mode in a cylindrical quantum annulus. Numerical calculation on CdS annulus system has been performed. Results reveal that the two different solutions of SSO mode distribute mainly at the inner or outer surfaces of the annulus. The dispersion relations and the coupling intensions of phonons in a quantum annulus are compared with those in a cylindrical quantum dot. It is found that the dispersion relations of the two different structures are similar, but the coupling intension of the phonon-electron interaction in quantum annulus is larger than that in quantum dot. The Hamiltonians describing the free phonon modes and their interactions with electrons in the system are also derived.

With the help of supercell method, the first-principle calculations were performed for the study of doping crystal Mg_1-xAl_xB₂ and Mg(B_1-yC_y)₂. Analyzing the variations of the charge distribution and the partial densities of states, we found that the compounds with doping Al to MgB₂ compound and/or replacing boron by carbon exhibit new covalent bond effects and unexpected electronic properties, related to superconductivity. The study of the density of states indicates that superconductivity decreases with the increase of Al fraction and carbon concentration. There exists a transition of superconductor to non-superconductor with the change of Al doping fraction. The substitution of boron by carbon results in the decrease of the transition temperature since the decrease of the electron concentration and the lattice constant. The theoretical predictions agree with experimental observations.

The properties of the low-lying states of a negative donor center trapped by a spherical quantum dot, which is subjected to a parabolic potential confinement, are investigated in the absence of magnetic field. The calculations have been performed by means of the exact diagonalization of the Hamiltonian matrix within the effective-mass approximation. We find that there is only one bound state the D⁻ center in a spherical parabolic quantum dot in the absence of magnetic field. The binding energy of the ground state is obtained as a function of the dot size. Moreover, the critical confined potential radius value at which the negative donor center changes from unbound to bound is obtained.

Within the effective field theory (EFT), staggered quadrupolar phase and bicritical point of spin-1 bond and anisotropy dilution Blume–Emery–Griffiths model is studied on simple cubic lattice in the restricted range of biquadratic interaction and bilinear interaction ratio α ⩽ −1. The phase diagrams present a line of staggered quadrupolar-paramagnetic (SQ-P) phase and a line of ferromagnetic-paramagnetic (F-P) phase, separated by a bicritical point (BCP). A large negative ratio and two different dilution factors magnify the range of the SQ phase and reduce range of ferromagnetic phase in T-α or T-D plane. These parameters can assist the reentrant behavior of the SQ-P line and suppress that of the F-P line. The influence of bond dilution on the BCP is dissimilar to that of anisotropy dilution. There is a degenerate pattern at ground state in T-D plane. Some results obtained by the pure BEG model is in qualitative agreement with the results of Monte Carlo simulations.

We analyze the time evolution of entanglement of two-qutrit system within the framework of Milburn's model of intrinsic decoherence. The entanglement evolution relies not only on the parameters of system, but also on the concrete states either pure or mixed. The linear entropy used to measure the extent to which the intrinsic decoherence affects quantum states is evaluated.

There is a phase transition between quasi-periodic state and intermittent chaos in GOY model with a critical value δ₀. When we add a modulated periodic external force to the system, the phase transition can also be found with a critical value δ_e. Due to coupling between the force and the intrinsic fluctuation of the velocity on shells in GOY model, the stability of the system has been changed, which results in the variation of the critical value. For proper intensity and period of the force, δ_e is unequal to δ₀. The critical value is a nonlinear function of amplitude of the force, and the fluctuation of the velocity can resonate with the external force for certain period T_e.

We propose an Ashkin–Teller-like model for elastic response of DNA molecule to external force and torque. The base-stacking interaction is described in a simple and uniform way. We obtain the phase diagram of dsDNA, and in particular, the transition from B form to the S state induced by stretching and twisting. The elastic response of the ssDNA is presented also in a unified formalism. The close relation of dsDNA molecule structure with elastic response is shown clearly. The calculated folding angle of the dsDNA molecule is 59.2°.