Table of contents

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen–Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.

We investigate the symmetry reduction for the two-dimensional incompressible Navier–Stokes equation in conventional stream function form through Lie symmetry method and construct some similarity reduction solutions. Two special cases in [D.K. Ludlow, P.A. Clarkson, and A.P. Bassom, Stud. Appl. Math. 103 (1999) 183] and a theorem in [S.Y. Lou, M. Jia, X.Y. Tang, and F. Huang, Phys. Rev. E 75 (2007) 056318] are retrieved.

In this paper, bilinear form of a negative order AKNS equation hierarchy is given. The soliton solutions are obtained through Hiorta's direct method.

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibility method is investigated. We show the nonclassical symmetries obtained in [J. Math. Anal. Appl. 289 (2004) 55, J. Math. Anal. Appl. 311 (2005) 479] are not all the nonclassical symmetries. Based on a new assume on the form of invariant surface condition, all the nonclassical symmetries for a class of nonlinear partial differential equations can be obtained through the compatibility method. The nonlinear Klein–Gordon equation and the Cahn–Hilliard equations all serve as examples showing the compatibility method leads quickly and easily to the determining equations for their all nonclassical symmetries for two equations.

Classification and reduction of the generalized fourth-order nonlinear differential equations arising from the liquid films are considered. It is shown that these equations have solutions on subspaces of the polynomial, exponential or trigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n = 4, ..., 9. Several examples of exact solutions are presented.

We investigate cooperative behaviors of lattice-embedded scale-free networking agents in the prisoner's dilemma game model by employing two initial strategy distribution mechanisms, which are specific distribution to the most connected sites (hubs) and random distribution. Our study indicates that the game dynamics crucially depends on the underlying spatial network structure with different strategy distribution mechanism. The cooperators' specific distribution contributes to an enhanced level of cooperation in the system compared with random one, and cooperation is robust to cooperators' specific distribution but fragile to defectors' specific distribution. Especially, unlike the specific case, increasing heterogeneity of network does not always favor the emergence of cooperation under random mechanism. Furthermore, we study the geographical effects and find that the graphically constrained network structure tends to improve the evolution of cooperation in random case and in specific one for a large temptation to defect.

We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is time-dependent Hamiltonian system. The quantum unitary operator relevant 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.

We present a two-photon three-dimensional multiparty quantum secret sharing scheme. The secret messages are encoded by performing local operations. This is different from those quantum secret sharing protocols that all sharers must make a state measurement. The merit of our protocol is the high capacity.

A scheme for implementing discrete quantum Fourier transform is proposed via quantum dots embedded in a microcavity, and then some of its applications are investigated, i.e., Deutsch–Jozsa algorithm and Shor's quantum factoring. In particular, the detailed process of implementing one-qubit Deutsch–Jozsa algorithm and the factorization of N = 15 are given. The microcavity mode is only virtually excited in the whole interaction, so the effective decoherent has slight effect on the current scheme. These schemes would be an important step to fabricate a solid quantum computer.

In this paper we develop a variational theory to study the dynamic properties of ultracold Bose gas in a funnel external potential. We obtain one-dimensional nonlinear equation which describes the dynamics of transverse tight confined bosonic gas from three-dimension to one-dimension, and find one-dimensional s-wave scattering length which depends on the shape of transverse confining potential. If the funnel trapping potential is strong enough at zero temperature, all transverse excitations are frozen. We find the dynamic equation which describes the Tonks–Girardeau gas and present a qualitative analysis of the experimental accessibility of the Tonks–Girardeau gas with funnel-trapped alkalic atoms.

In this paper, Killing vectors of spherically spacetimes have been evaluated in the context of teleparallel theory of gravitation. Further, we investigate the Killing vectors of the Friedmann metrics. It is found that for static spherically spacetimes the number of Killing vectors turns out to be seven while for the Friedmann models, we obtain six teleparallel Killing vectors. The results are then compared with those of General Relativity. We conclude that both of these descriptions of gravity do not provide the consistent results in general. However, these results may coincide under certain conditions for a particular spacetime.

By using the solution describing a black hole embedded in the FLRW universe, we obtain the evolving equation of the black hole mass expressed in terms of the cosmological parameters. The evolving equation indicates that in the phantom dark energy universe the black hole mass becomes zero before the Big Rip is reached.

The generalized Thirring model with impurity coupling is defined on two-dimensional noncommutative space-time, a modified propagator and free energy are derived by means of functional integrals method. Moreover, quantum fluctuations and excitation energies are calculated on two-dimensional black hole and soliton background.

In this paper, we investigate a leader-following tracking problem for multi-agent systems with bounded inputs. We propose a distributed bounded protocol for each follower to track a leader whose states may not be completely measured. We theoretically prove that each agent can follow the leader with estimable track errors. Finally, some numerical simulations are presented to illustrate our theoretical results.

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotic circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.

The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated. The effects of pure multiplicative noise on the stationary probability distribution (SPD) and the mean first passage time (MFPT) are analysed based on the approximate Fokker-Planck equation of the system in detail. For the anti-tumor model, with the multiplicative noise intensity D increasing, the tumor population move towards to extinction and the extinction rate can be enhanced. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained.

By truncating the Painlevé expansion at the constant level term, the Hirota bilinear form is obtained for a (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ∊-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota–Satsuma equation for shallow water waves and Boiti–Leon–Manna–Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Furthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.

Based on the general direct method developed by Lou et al. [J. Phys. A: Math. Gen. 38 (2005) L129], the symmetry group theorem is obtained, from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenko–Dubrovsky (KD) equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution.

Using the Schwinger–Dyson equation and perturbation theory, we calculate the two-quark condensates for the light quarks u, d, strange quark s and a heavy quark c with their current masses respectively. The results show that the two-quark condensate will decrease when the quark mass increases, which hints the chiral symmetry may be restored for the heavy quarks.

Based on the Husimi operator in pure state form introduced by Fan et al., which is a squeezed coherent state projector, and the technique of integration within an ordered product (IWOP) of operators, as well as the entangled state representations, we obtain the Husimi functions of the excited squeezed vacuum states (ESVS) and two marginal distributions of the Husimi functions of the ESVS.

In this paper, two-mode displaced excited squeezed vacuum states (TDESVS) are constructed and their normalization and completeness are investigated. Using the entangled state representation and Weyl ordering form of the Wigner operator, the Wigner functions of TDESVS are obtained and the variations of Wigner functions with the parameters m, n and r are investigated. Besides, two marginal distributions of Wigner functions of TDESVS are obtained, which exhibit some entangled properties of the two-particle's system in TDESVS.

We study the scattering process of photons confined in a one-dimensional optical waveguide by a laser controlled atomic ensemble. The investigation leads to an alternative setup of quantum node controlling the coherent transfer of single photon in such one dimensional continuum. To exactly solve the effective scattering equations by using the discrete coordinate approach, we simulate the linear waveguide as a coupled resonator array at the high energy limit. We generally calculate the transmission coefficients and its vanishing at resonance reflects the good controllability of our scheme. We also show that there exist two bound states to describe the localize photons around the cavity.

Thermal transport in the FPU model with FK on-site potential is studied by using fourth-order Runge–Kutta algorithm. The heat flux, local temperature profile, and heat conductivity are simulated and analyzed. It is found that temperature gradient scales behave as N⁻¹ linearly. The divergence of heat conductivity κ with system size N is in term of κ ∝ N^α with α = 0.44. It is shown that thermal transport is mainly dependent on the FPU nonlinear and the FK interactions.

We find that a kind of atomic coherent state, formed as exp[ξJ₊ – ξ*J₋] |00〉, when the SU(2) generators J± are taken as Fan's form, J₊ = (1/2)(a₁ – a₂^†)(a₁^† – a₂), J₋ = (1/2)(a^†₁ + a₂) (a₁ + a₂^†), and J₀ = (1/2)(a₁^†a₂^† – a₁a₂), is simultaneously a two-mode squeezed state. We analyse this squeezed state's physical properites, such as the cross-correlation function, the Wigner function, and its marginal distribution as well as the Husimi function.

In order to investigate the effect of an arbitrary dust size distribution for vortex-like ion distribution dusty plasma, we use a reasonable polynomial-expressed function to represent an arbitrary dust size distribution. The numerical results of linear dispersion relation, nonlinear solitary wave amplitude, width and velocity for polynomial expressed dust size distribution dusty plasma with vortex-like ion distribution have been studied.

Using the reference hypernetted chain (RHNC) integral equation theory and a rigorous stability analysis method, we investigate the phase behavior of a mixture of hard-sphere dipoles and neutral hard spheres based on the correlations of the homogeneous isotropic phase. Lowering the temperature down to the points where the RHNC equations fail to have a solution, several fluctuations strongly increase. At low densities our results indicate the onset of chain formation, which is similar with the pure DHS system. At high densities, the results indicate the appearance of isotropic-to-ferroelectric transitions (small neutral hard spheres concentrations) and demixing transitions (large neutral hard spheres concentrations).

A comprehensive first principles study of III-Antimonide binary compounds is hardly found in literature. We report a broad study of structural and electronic properties of boron antimonide (BSb), aluminium antimonide (AlSb), gallium antimonide (GaSb) and indium antimonide (InSb) in zincblende phase based on density functional theory (DFT). Our calculations are based on Full-Potential Linearized Augmented Plane wave plus local orbitals (FP-L(APW+lo)) method. Different forms of exchange-correlation energy functional and corresponding potential are employed for structural and electronic properties. Our computed results for lattice parameters, bulk moduli, their pressure derivatives, and cohesive energy are consistent with the available experimental data. Boron antimonide is found to be the hardest compound of this group. For band structure calculations, in addition to LDA and GGA, we used GGA-EV, an approximation employed by Engel and Vosko. The band gap results with GGA-EV are of significant improvement over the earlier work.

The mesoscopic nonlinear inductance-capacitance circuit is a typical anharmonic oscillator, due to diodes included in the circuit. In this paper, using the advanced quantum theory of mesoscopic circuits, which based on the fundamental fact that the electric charge takes discrete value, the diode included mesoscopic circuit is firstly studied. Schrödinger equation of the system is a four-order difference equation in representation. Using the extended perturbative method, the detail energy spectrum and wave functions are obtained and verified, as an application of the results, the current quantum fluctuation in the ground state is calculated. Diode is a basis component in a circuit, its quantization would popularize the quantum theory of mesoscopic circuits. The methods to solve the high order difference equation are helpful to the application of mesoscopic quantum theory.

A two-fold Cayley tree with fully q-coordinated sites is constructed and the spin-1 Blume–Capel model in the presence of an external magnetic field is solved exactly. The relevant properties such as magnetization m and square magnetic moment q are investigated as functions of temperature and external magnetic field both for ferromagnetic and antiferromagnetic couplings. In the study, we propose a possible mechanism of the plateau based on the exchange couplings.

By introducing the distribution of the light energy density in GaN-based light-emitting diode (LED), the LED model based on the incoherent regime and the light extraction efficiency are investigated. The energy density as a function of the angle of incidence is calculated to demonstrate the mechanism of the light extraction. The deviation between the tendencies of the transmissivity of the output layer and the extraction efficiency is also demonstrated.

By direct calculation of rotation matrices of SO(3), we show how certain specific sequence of eight consecutive rotations of digital angles can yield a tilting of a facet mirror. We also design a detailed program specifically to tilt an array of mirrors from planar orientation to the required focusing orientation. We describe how to use the 8-step to realize the focusing of the mirror array. We have found, in our designed program, an important feature of row-sharing during the rotations for the columns and similarly the column-sharing during the rotations for the row. This feature can save a lot of operating time during the actual realization of the mechanical movements.

By making use of the decomposition of U(1) gauge potential theory and the φ-mapping method we discuss a mixture of interacting neutral and charged Bose condensates, which is supposed being realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We propose that this system possesses vortex lines and two classes of knotted solitons. The topological charge of the vortex lines are characterized by the Hopf indices and the Brower degrees of φ-mapping, and the knotted solitons are described by nontrivial Hopf invariant and the BF action respectively.