The N-fold Darboux transformation (DT) T_n^[N] of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j (j=1,2…,N) and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j→λ₁ and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically, which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t²).

In this paper, a hybrid method based on the collocation and Newton-Kantorovich methods is used for solving the nonlinear singular Thomas-Fermi equation. At first, by using the Newton-Kantorovich method, the nonlinear problem is converted to a sequence of linear differential equations, and then, the fractional order of rational Legendre functions are introduced and used for solving linear differential equations at each iteration based on the collocation method. Moreover, the boundary conditions of the problem by using Ritz method without domain truncation method are satisfied. In the end, the obtained results compare with other published in the literature to show the performance of the method, and the amounts of residual error are very small, which indicates the convergence of the method.

In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrödinger equation (FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrödinger equation to the fractional one. When applying this form of Schrödinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well, the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrödinger equation.

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations (DTs) for a (2+1)-dimensional nonlinear Schrödinger (NLS) equation is derived, which can be reduced to two (1+1)-dimensional equation: a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave (RW) solutions are constructed by its (1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.

We derive a general phase-matching condition (PMC) for enhancement of sensitivity in SU(1,1) interferometers. Under this condition, the quantum Fisher information (QFI) of two-mode SU(1,1) interferometry becomes maximal with respect to the relative phase of two modes, for the case of an arbitrary state in one input port and an even (odd) state in the other port, and the phase sensitivity is enhanced. We also find that optimal parameters can let the QFI in some areas achieve the Heisenberg limit for both pure and mixed initial states. As examples, we consider several input states: coherent and even coherent states, squeezed vacuum and even coherent states, squeezed thermal and even coherent states. Furthermore, in the realistic scenario of the photon loss channel, we investigate the effect of photon losses on QFI with numerical studies. We find the PMC remains unchanged and is not affected by the transmission coefficients for the above input states. Our results suggest that the PMC can exist in various kinds of interferometers and the phase-matching is robust to even strong photon losses.

Quantum Fisher information (QFI) gap characterizes the stability of QFI to space directions. We study the QFI distributions and QFI gap for quantum states generated from nonlinear Hamiltonians for both spin and bosonic systems. We find that the same spin-squeezing parameter (or principle squeezing parameter) corresponds to two different values QFI gap, and the locations of all extreme points of the QFI are explicitly given.

We employ the variational method to study the properties such as masses, decay constants, oscillation frequency and branching ratios of leptonic decays of heavy flavour mesons with linear cum coulomb Cornell potential. Gaussian function, Coulomb wave function and Airy function are taken as the trial wave-function of variational method in this study. Our analysis suggests that Gaussian trial wave-function provides results which are in close proximity with the experimental results. We also make a comparison with the results from QCD Sum rules and lattice QCD, as well as with recent PDG data.

Recently, we reported an analysis of Proton structure function at small x based on Taylor approximated DGLAP equations assuming a plausible relationship between the singlet and the gluon distributions. In this paper, we report a generalised version of the previous work. A corresponding study of the suggested gluon distribution is also made. The present generalised version of the model for the structure function results in a wider x range of phenomenological validity than the earlier one. A comparison of both the models of the proton structure function and the gluon distribution is made with exact result as well as with the Froissart saturated models of Block, Durand and Mckay.

The first (namely, inner) fission barriers for even-A N=152 nuclei have been studied systematically in the framework of macroscopic-microscopic model by means of potential energy surface (PES) calculations in the threedimensional (β₂, γ, β₄) deformation space. Their collective properties, such as ground-state deformations, are compared with previous calculations and available observations, showing a consistent trend. In addition, it has been found that the microscopic shell correction energy plays an important role on surviving fission in these N=152 deformed shell nuclei. The inclusion of non-axial symmetric degree of freedom γ will pull the fission barrier down more significantly with respect to the calculation involving in hexadecapole deformation β₄. Furthermore, the calculated Woods-Saxon (WS) single particle levels indicate that the large microscopic shell correction energies due to low level densities may be responsible for such a reduction on the inner fission barrier.

In this note, we recalculate the entropy of the Vaidya black hole on the event horizon by considering the generalized uncertainty principle based on the brick-wall model. The result shows that we need not impose a cut-off by hand anymore and the result satisfies the Bekenstein-Hawking law as well.

We investigate cosmological dark energy models where the accelerated expansion of the universe is driven by a field with an anisotropic universe. The constraints on the parameters are obtained by maximum likelihood analysis using observational of 194 Type Ia supernovae (SNIa) and the most recent joint light-curve analysis (JLA) sample. In particular we reconstruct the dark energy equation of state parameter w(z) and the deceleration parameter q(z). We find that the best fit dynamical w(z) obtained from the 194 SNIa dataset does not cross the phantom divide line w(z)=-1 and remains above and close to w(z) =-0.92 line for the whole redshift range 0≤z≤1.75 showing no evidence for phantom behavior. By applying the anisotropy effect on the ΛCDM model, the joint analysis indicates that _σ0=0.0163±0.03, with 194 SNIa, _σ0=-0.0032±0.032 with 238 the SiFTO sample of JLA and _σ0=0.011±0.0117 with 1048 the SALT2 sample of Pantheon at 1σ' confidence interval. The analysis shows that by considering the anisotropy, it leads to more best fit parameters in all models with JLA SNe datasets. Furthermore, we use two statistical tests such as the usual χ_min²/dof and p-test to compare two dark energy models with ΛCDM model. Finally we show that the presence of anisotropy is confirmed in mentioned models via SNIa dataset.

The multipolar velocity field structures are investigated by 2D momentum conservation equation with 3D equilibrium sheared flows in the full toroidal system. Numerical results show that the non-existence of radial velocity field in equilibrium surfaces is suitable only for the zero-order term of our 2D simulation. The non-zero-order radial velocity field is still preserved, even when converted to conventional magnetic surface coordinates. The distribution of velocity field vectors of the order of 1, 2, and 3 are presented respectively in 2, 4, and 6 polar fields with the local vortex structure. The excitation mechanisms of these velocity vortices are the coupling effects of the magneto-fluid structure patterns and the toroidal effects. These results can help us understand the complexity of core physics, the transverse transport across magnetic field by the radial plasma flow and the formation of velocity vortices.

We propose and discuss terahertz (THz) electro-optic modulator induced by periodically patterned graphene microcavity. Due to the joint effect of graphene plasmon resonances and Fabry-Perot (F-P) oscillations, plasmon-induced transparency (PIT) effect is achieved and the operation frequency can be dynamically tuned by graphene Fermi energies and structural parameters. The results reveal that modulation depth (MD) larger than 80% is obtained across a wide frequency range from 4.2 THz to 9.4 THz, and the largest MD and Q factor reaches 95% and 15.8, respectively. In addition, operation frequency range and MD can also be tuned by optimizing the structure parameters. This investigation promises the development of high-performance widely tunable THz modulator in chip integrated photonic circuits.

The dissociation energy and equilibrium bond length as explicit parameters are used to establish an improved five-parameter exponential-type potential energy model for diatomic molecules. We demonstrate that the five-parameter exponential-type potential is identical to the Tietz potential for diatomic molecules. It is observed that the improved five-parameter exponential-type potential can well model the internuclear interaction potential energy curve for the ground electronic state of the carbon monoxide molecule by the utilization of the experimental values of three molecular constants.

We study a population model with strong and weak Allee effect driven by internal noise and external noise. Firstly, a single-species population model with Allee effect under environmental colored noise is established, then stable and unstable states are analyzed and interpreted in biology. After that, stationary probability distribution (SPD) of population is derived based on Fokker-Planck equation. Next, mean first-passage time (MFPT) is defined in order to quantify the transition between extinction state and survival state with Allee effect. It is found that population will not extinct when weak Allee effect exists. It is not beneficial to survival of the population with the increase of Allee threshold no matter whether strong Allee effect or weak Allee effect. When strong Allee effect occurs, the correlation time of multiplicative noise plays a positive role in survival of population, while the correlation time of additive noise has a negative effect. Crucially, the phenomenon of resonant activation is firstly discovered in population dynamics with Allee effect. The conclusions we obtain can be applied to the further research of population dynamics in ecology.

A one-dimensional discrete Boltzmann model for detonation simulation is presented. Instead of numerical solving Navier-Stokes equations, this model obtains the information of flow field through numerical solving specially discretized Boltzmann equation. Several classical benchmarks including Sod shock wave tube, Colella explosion problem, and one-dimensional self-sustainable stable detonation are simulated to validate the new model. Based on the new model, the influence of negative temperature coefficient of reaction rate on detonation is further investigated. It is found that an abnormal detonation with two wave heads periodically appears under negative temperature coefficient condition. The causes of the abnormal detonation are analyzed. One typical cycle of the periodic abnormal detonation and its development process are discussed.

Migration is ubiquitous in ecosystem and often plays an important role in biological diversity. In this work, by introducing a time-varying migration rate associated with the difference of subpopulation density into a prey, we study the Hopf bifurcation and the critical phenomenon of predator extinction of the three species prey-predator system, which consists of a predator, a prey and a mobile prey. It is found that the system with migration exhibits richer dynamic behaviors than that without migration, including two Hopf bifurcations and two limit cycles. Interestingly, the parameters of migration have a drastically influence on the critical point of predator extinction, determining the coexistence of species. Moreover, the population evolution dynamics of one-dimensional multiple prey-predator system are also discussed.

The recently developed discrete Boltzmann method (DBM), which is based on a set of uniform linear evolution equations and has high parallel efficiency, is employed to investigate the dynamic nonequilibrium process of Kelvin-Helmholtz instability (KHI). It is found that, the relaxation time always strengthens the global nonequilibrium (GNE), entropy of mixing, and free enthalpy of mixing. Specifically, as a combined effect of physical gradients and nonequilibrium area, the GNE intensity first increases but decreases during the whole life-cycle of KHI. The growth rate of entropy of mixing shows firstly reducing, then increasing, and finally decreasing trends during the KHI process. The trend of the free enthalpy of mixing is opposite to that of the entropy of mixing. Detailed explanations are: (i) Initially, binary diffusion smooths quickly the sharp gradient in the mole fraction, which results in a steeply decreasing mixing rate. (ii) Afterwards, the mixing process is significantly promoted by the increasing length of material interface in the evolution of the KHI. (iii) As physical gradients are smoothed due to the binary diffusion and dissipation, the mixing rate reduces and approaches zero in the final stage. Moreover, with the increasing Atwood number, the global strength of viscous stresses on the heavy (light) medium reduces (increases), because the heavy (light) medium has a relatively small (large) velocity change. Furthermore, for a smaller Atwood number, the peaks of nonequilibrium manifestations emerge earlier, the entropy of mixing and free enthalpy of mixing change faster, because the KHI initiates a higher growth rate.

Numerical investigation of the dusty Williamson fluid with the dependency of time has been done in current disquisition. The flow of multiphase liquid/particle suspension saturating the medium is caused by stretching of porous surface. The influence of magnetic field and heat generation/absorption is observed. It is assumed that particle has a spherical shape and distributed uniformly in fluid matrix. The unsteady two-dimensional problems are modeled for both fluid and particle phase using conservation of mass, momentum and heat transfer. The finalized model generates the non-dimensioned parameters, namely Weissenberg number, unsteadiness parameter, magnetic parameter, heat generation/absorption parameter, Prandtl number, fluid particle interaction parameter, and mass concentration parameters. The numerical solution is obtained. Locality of skin friction and Nusselt number is deliberately focused to help of tables and graphs. While inferencing the current article it is clearly observed that increment of Williamson parameter, unsteadiness parameter, magnetic parameter, volume fraction parameter, and mass concentration parameter reduces the velocity profile of fluid and solid particles as well. And increment of Prandtl number, unsteadiness parameter, volume fraction parameter, and mass concentration parameter reduces the temperature profile of fluid and solid particles as well.

Under investigation in this work is the general coupled nonlinear Schrödinger (gCNLS) equation, which can be used to describe a wide variety of physical processes. By using Darboux transformation, the new higher-order rogue wave solutions of the equation are well constructed. These solutions exhibit rogue waves on a multi-soliton background. Moreover, the dynamics of these solutions is graphically discussed. Our results would be of much importance in enriching and predicting rogue wave phenomena arising in nonlinear and complex systems.

In this paper, we first obtain a bilinear form with small perturbation u₀ for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear Bäcklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized (3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parameters α and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.

In a recent article (Commun. Theor. Phys. 67 (2017) 207), three (2+1)-dimensional equations-KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by using different transformation of variables, respectively. In this short note, by adding an adjustment item to original transformation, three more general transformation of variables corresponding to above three equations have been given. Substituting the solutions of the KdV equation into our transformation of variables, more new exact solutions of the three (2+1)-dimensional equations can be obtained.

An extrapolation to the physical limit for the lattice data of ∧_b → ∧_c form factors computed in the nonphysical region is made in this work through a class of fitting functions proposed by us with nonlinear dependence on m_π² derived in the chiral perturbative theory (ChPT) and the heavy quark effective theory (HQET) framework. Then the results are applied to calculate the differential and integrated ∧_b → ∧_c semileptonic decay rates. Meanwhile, a comparison between our results and those obtained through the extrapolation functions with naive linear dependence on m_π² is made. It is shown that the difference between the extrapolated central values of these two cases is about 5%. The total uncertainties (depending on the momentum transfer q²) in the linear case are about 5%~10% (caused by the uncertainties of lattice data) and those in the nonlinear case are about 10%~20% (caused by the uncertainties of both lattice data and input parameters in ChPT and HQET). More accurate lattice data and parameters in ChPT and HQET are needed to reduce the uncertainties of the extrapolated results.

We investigate the spectrum and decay rates of charmonium states within the framework of the non relativistic quark model by employing a Coulomb like potential from the perturbative one gluon exchange and the linear confining potential along with the potential derived from instanton vacuum to account for the hyperfine mass splitting of charmonium states in variational approach. We predict radiative E1, M1, two-photon, lepton and two-gluon decay rates of low lying charmonium states. An overall agreement is obtained with the experimental masses and decay widths. We have estimated the branching ratio of two gluons decaying into light hadrons.

Within the framework of the isospin-dependent transport model, the roles of the reactions N△ → NN and πN → △ are investigated through simulating heavy-ion collisions at 1000 MeV/nucleon. The absorption process N△ → NN plays an important role for heavy impact systems and small impact parameters than for light impact systems and large impact parameters. The resorption process πN → △ is of importance for heavy impact systems and large impact parameters than for light impact systems and small impact parameters. Thus the influences of the reaction N△ → NN (πN → △) on pion production dynamics can be neglected in heavy-ion collisions for smaller (larger) impact parameters and light systems. It is the reaction πN → △ that causes the anti-correlation of pions and nucleons in the rapidity dependence of the directed flow.

We consider the most general static spherically symmetric black hole metric. The accretion of the fluid flow around the Van der Waal's black hole is investigated and we calculate the fluid's four-velocity, the critical point and the speed of sound during the accretion process. We also analyze the nature of the universe's density and the mass of the black hole during accretion of the fluid flow. The density of the fluid flow is also taken into account. We observe that the mass is related to redshift. We compare the accreting power of the Van der Waal's black hole with Schwarzschild black hole for different accreting fluid.

The aim of this paper is to examine the structure scalars with account of f(G, T) theory of gravity. We consider the cylindrically symmetric spacetime with dissipative anisotropic background. We have determined the structure scalars by orthogonally decomposing the Riemann curvature tensor and it is shown that these scalars are associated with fundamental properties of fluid. We further investigate the mass function along with the transport equation and discuss their role on the evolutionary stages of relativistic stellar systems. We have also analyzed these structure scalars for static fluid distributions and it is concluded that all possible solutions of field equations can be expressed through these scalars.

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V₀ and a. The wave functions are shrunk towards the origin with the increasing|A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels $\epsilon_{i}$ increase with the increasing potential parameter|A|≥ 1, but the variation of the energy levels becomes complicated for|A|∈ (0, 1), which possesses a double well. It is seen that the energy levels $\epsilon_{i}$ increase with|A|for the parameter interval A ∈ (-1, 0), while they decrease with|A|for the parameter interval A ∈ (0, 1).

The nonlinear features of two-dimensional ion acoustic (IA) solitary and shock structures in a dissipative electron-positron-ion (EPI) quantum plasma are investigated. The dissipation in the system is taken into account by incorporating the kinematic viscosity of ions in plasmas. A quantum hydrodynamic (QHD) model is used to describe the quantum plasma system. The propagation of small but finite amplitude solitons and shocks is governed by the Kadomtsev-Petviashvili-Burger (KPB) equation. It is observed that depending on the values of plasma parameters (viz. quantum diffraction, positron concentration, viscosity), both compressive and rarefactive solitons and shocks are found to exist. Furthermore, the energy of the soliton is computed and possible solutions of the KPB equation are presented numerically in terms of the monotonic and oscillatory shock profiles

The binary perceptron is the simplest artificial neural network formed by N input units and one output unit, with the neural states and the synaptic weights all restricted to ±1 values. The task in the teacher-student scenario is to infer the hidden weight vector by training on a set of labeled patterns. Previous efforts on the passive learning mode have shown that learning from independent random patterns is quite inefficient. Here we consider the active online learning mode in which the student designs every new Ising training pattern. We demonstrate that it is mathematically possible to achieve perfect (error-free) inference using only N designed training patterns, but this is computationally unfeasible for large systems. We then investigate two Bayesian statistical designing protocols, which require 2.3N and 1.9N training patterns, respectively, to achieve error-free inference. If the training patterns are instead designed through deductive reasoning, perfect inference is achieved using N +log₂N samples. The performance gap between Bayesian and deductive designing strategies may be shortened in future work by taking into account the possibility of ergodicity breaking in the version space of the binary perceptron.

An approximation of the orography gravity wave, which is induced by mountainous topography, is considered in this study. By assuming that the horizontal wind is a linear function with respect to the height, the approximating equation for the orography gravity waves is obtained. Four topography functions are considered in this study and the orography gravity wave are obtained. The dynamics of the orography gravity wave is then discussed by considering the effect of the surface topography and background horizontal wind.

A numerical analysis is developed for incompressible hydromagnetic viscous fluid passed through a curved stretching surface. Fluid saturated by porous space is bounded by curved surface. Term of porous medium is characterized by implementation of Darcy-Forchheimer theory. Adequate similarity variables are implemented to develop a system of non-linear ordinary differential system of equations, which govern the flow behavior. The impact of radiation constraint and Eckert number is incorporated in the energy equation. Numerical scheme based on RKF45 technique is implemented to solve the derived flow model. Prescribed heat flux (PHF) and prescribed surface temperature (PST) boundary conditions are utilized on temperature with Prescribed Surface Concentration (PSC) and Prescribed Mass Flux (PMF) on concentration. Flow behavior is discussed for both the slip and no-slip conditions. Dimensionless physical quantities are presented through graphs and tables.

Recently, the bound state solutions of a confined Klein-Gordon particle under the mixed scalar-vector generalized symmetric Woods-Saxon potential in one spatial dimension have been investigated. The obtained results reveal that in the spin symmetric limit discrete spectrum exists, while in the pseudo-spin symmetric limit it does not. In this manuscript, new insights and information are given by employing an analogy of the variational principle. The role of the difference of the magnitudes of the vector and scalar potential energies, namely the differentiation parameter, on the energy spectrum is examined. It is observed that the differentiation parameter determines the measure of the energy spectrum density by modifying the confined particle's mass-energy in addition to narrowing the spectrum interval length.

The BC_r-KP hierarchy is an important sub-hierarchy of the KP hierarchy. In this paper, the BC_r-KP hierarchy is investigated from three aspects. Firstly, we study the gauge transformation for the BC_r-KP hierarchy. Different from the KP hierarchy, the gauge transformation must keep the constraint of the BC_r-KP hierarchy. Secondly, we study the gauge transformation for the constrained BC_r-KP hierarchy. In this case, the constraints of the Lax operator must be invariant under the gauge transformation. At last, the compatibility between the additional symmetry and the gauge transformation for the BC_r-KP hierarchy is explored.

A cold atomic medium (Rydberg medium) with cascade configuration under the blockade mechanism is considered. A partial coherent light (PCL) beam is incident on the medium, which makes an angle θ with z-axis. We study the influence of PCL field on the transmission spectrum and find high transmission of probe field for PCL field. Conversely, it is investigated that the transparency of probe field decrease for coherent light field. The transmission of probe field is also studied via beam width of PCL field and investigated high transmission of probe field for small beam width and vice versa. Interestingly, the Goos-Hänchen shift (GHS) in the transmitted light (TL) is studied for PCL field. Large negative and positive GHS in the TL are investigated for PCL field and small beam width of PCL field.

CP violation in the lepton sector, and other aspects of neutrino physics, are studied within a high scale supersymmetry model. In addition to the sneutrino vacuum expectation values (VEVs), the heavy vector-like triplet also contributes to neutrino masses. Phases of the VEVs of relevant fields, complex couplings, and Zino mass are considered. The approximate degeneracy of neutrino masses mν1 and mν2 can be naturally understood. The neutrino masses are then normal ordered, ～ 0.020 eV, 0.022 eV, and 0.054 eV. Large CP violation in neutrino oscillations is favored. The effective Majorana mass of the electron neutrino is about 0.02 eV.

We have systematically analyzed the experimental β^--decay half-lives of waiting point heavy nuclei around neutron number N=126. A new set of parameters for the exponential formula of β^--decay half-lives is proposed. The forbidden transition effects are included in the new set of parameters self-consistently. Theoretical β^--decay half-lives of nuclei around N=126 are compared with recent theoretical results and experimental data. It is found that the new theoretical results are in better agreement with experimental data. The unknown β^--decay half-lives of some nuclei in this region are predicted for studies on nuclear structure far from stability and the nucleosynthesis in stars.

In this paper, we have completely classified the locally rotationally symmetric (LRS) Bianchi type I spacetimes via Noether symmetries (NS). The usual Lagrangian corresponding to LRS Bianchi type I metric is used to find the set of determining equations. To achieve a complete classification, these determining equations are generally integrated to find the components of NS vector field and the metric coefficients. During this procedure, several cases arise which give different Noether algebras of dimension 5,..., 9, 11, and 17. A comparison is established between the obtained NS and the Killing and homothetic vectors. Corresponding to all NS generators, the conservation laws are stated by using Noether's theorem. The metrics which we have obtained as a result of our classification are shown to be anisotropic or perfect fluids which satisfy certain energy conditions.

The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the "aether". In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory (NAT)-a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A) dS type solutions, Reissner-Nordström-(A) dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT:We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions-AdS-plane waves and pp-waves-in this theory in any dimension D ≥ 3. Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a "mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.