The surface water waves in a water tunnel can be described by systems of the form [Bona and Chen, Physica D116, 191 (1998)] \begin{equation*}\begin{cases} v_t + u_x + (uv)_x + au_{x x x} − bv_{x x t} = 0,\\ u_t + v_x + u u_x + cv_{x x x} − d u_{x x t} = 0,\end{cases} \tag{1}\end{equation*} where 𝑎, 𝑏, 𝑐 and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution systems. In this paper, we obtain exact travelling wave solutions of system (1) using the modified tanh–coth function method with computerized symbolic computation.

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The ($G'/G$)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

In a series of essays, beginning with this article, we are going to develop a new formulation of microphenomena based on the principles of reality and causality. The new theory provides us with a new depiction of microphenomena assuming a unified concept of information, matter and energy. So, we suppose that in a definite microphysical context (including other interacting particles), each particle is enfolded by a probability field whose existence is contingent on the existence of the particle, but it can locally affect the physical status of the particle in a context-dependent manner. The dynamics of the whole particle-field system obeys deterministic equations in a manner such that when the particle is subjected to a conservative force, the field also experiences a conservative complex force, the form of which is determined by the dynamics of the particle. So, the field is endowed with a given amount of energy, but its value is contingent on the physical conditions the particle is subjected to. Based on the energy balance of the particle and its associated field, we argue why the field has a probabilistic objective nature. The basic elements of this new formulation, its application for some stationary states and its nonlinear generalization for conservative systems are discussed here.

We present solutions of the Dirac equation with spin symmetry for vector and scalar modified Pöschl–Teller potentials within the framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov–Uvarov method and the two-component spinor wave functions obtained are in terms of the Jacobi polynomials. It is found that there exist only positive energy states for bound states under spin symmetry, and the energy of a level with fixed value of 𝑛, increases with increase in dimension of space time and the potential range parameter 𝛼.

In this paper, an adaptive fuzzy neural controller (AFNC) for a class of unknown chaotic systems is proposed. The proposed AFNC is comprised of a fuzzy neural controller and a robust controller. The fuzzy neural controller including a fuzzy neural network identifier (FNNI) is the principal controller. The FNNI is used for online estimation of the controlled system dynamics by tuning the parameters of fuzzy neural network (FNN). The Gaussian function, a specific example of radial basis function, is adopted here as a membership function. So, the tuning parameters include the weighting factors in the consequent part and the means and variances of the Gaussian membership functions in the antecedent part of fuzzy implications. To tune the parameters online, the back-propagation (BP) algorithm is developed. The robust controller is used to guarantee the stability and to control the performance of the closed-loop adaptive system, which is achieved always. Finally, simulation results show that the AFNC can achieve favourable tracking performances.

In this paper we have established the equation of state (EOS) for liquids. The EOS was established for hard-sphere (HS) fluid along with Lennard–Jones (LJ) fluid incorporating perturbation techniques. The calculations are based on suitable axiomatic functional forms for surface tension $S_m (r )$, $r \geq d/2$ with intermolecular separation 𝑟, as a variable, and 𝑚 is an arbitrary real number (pole). The results for $\beta P/ \rho$ from the present EOS thus obtained are compared with Percus-Yevick (PY), scaled particle theory (SPT), and Carnahan–Starling (CS). In addition, we have found a simple EOS for the HS fluid in the region which represents the simulation data accurately.

It is observed that, this EOS for HS gives, PY (pressure) for $m = 0$, CS for $m = 4/5$, whereas for $m = 1$ it corresponds to SPT.

We simulated the central reactions of nearly symmetric and asymmetric systems, for energies at which maximum production of intermediate mass fragments (IMF$_s$) occurred ($E^{\text{peak}}_{\text{c.m.}}$). This study was carried out using hard EOS along with Cugnon cross-section employing MSTB method for clusterization. We studied the various properties of fragments. The stability of fragments was checked through persistence coefficient and gain term. The information about the thermalization and stopping in heavy-ion collisions was obtained via relative momentum, anisotropy ratio and rapidity distribution. We found that for a complete stopping of incoming nuclei very heavy systems are required. The mass dependence of various quantities (such as average and maximum central density, collision dynamics as well as the time zone for hot and dense nuclear matter) was also presented. In all cases (i.e., average and maximum central density, collision dynamics as well as the time zone for hot and dense nuclear matter) a power-law dependence was obtained.

An attempt has been made to modify the original proximity potential using up-to-date knowledge of the universal function and surface energy coefficient available in the literature. A new radius formula has also been obtained using the recent data on charge distribution. The detailed investigation of over 395 reactions reveal that the new proximity potential reproduces the experimental data better than earlier versions.

Small amplitude quantum ion-acoustic solitary waves are studied in an unmagnetized twospecies relativistic quantum plasma system, comprised of electrons and ions. The one-dimensional quantum hydrodynamic model (QHD) is used to obtain a deformed Korteweg–de Vries (dKdV) equation by reductive perturbation method. A linear dispersion relation is also obtained taking into account the relativistic effect. The properties of quantum ion-acoustic solitary waves, obtained from the deformed KdV equation, are studied taking into account the quantum mechanical effects in the weak relativistic limit. It is found that relativistic effects significantly modify the properties of quantum ion-acoustic waves. Also the effect of the quantum parameter 𝐻 on the nature of solitary wave solutions is studied in some detail.

The pulsational mode of gravitational collapse (PMGC) in a hydrostatically bounded dust molecular cloud is responsible for the evolution of tremendous amount of energy during star formation. The source of free energy for this gravito-electrostatic instability lies in the associated self-gravity of the dispersed phase of relatively huge dust grains of solid matter over the gaseous phase of background plasma. The nonlinear stability of the same PMGC in an infinite dusty plasma model (plane geometry approximation for large wavelength fluctuation in the absence of curvature effects) is studied in a hydrostatic kind of homogeneous equilibrium configuration. By the standard reductive perturbation technique, a Korteweg–de Vries (KdV) equation for investigating the nonlinear evolution of the lowest order perturbed self-gravitational potential is developed in a time-stationary (steady-state) form, which is studied analytically as well as numerically. Different nonlinear structures (soliton-like and soliton chain-like) are found to exist in different situations. Astrophysical situations, relevant to it, are briefly discussed.

Good-quality hexagonal NbSe₂ single crystals were prepared. In 2H-NbSe₂, superconducting and charge density wave (CDW) transitions were found at $T_s = 7.4$ K and $T_c = 35$ K respectively as reported previously. We have noticed that these two transitions are changed to $T_c = 42$ K and $T_s = 6.5$ K, in 4H-NbSe₂. Thermopower has shown clear anomaly at CDW transitions. The anisotropic upper critical field was calculated as $\sim 3$ and 6.3 for 2H- and 4H-single crystals around $t = 0.81$, where $t = T/T_s$, from resistivity and explained in terms of coherence length. From the relation, $H_{c2}(T) = H_{c2}(0)[1 − t^2]$, $H^l_{c2}(0)$ was calculated as $\sim 8.15$ T and 16.98 T at $t = 0.84$ in 2H-NbSe₂ and 4H-NbSe₂ respectively. However, $H^t_{c2}(0) = 2.68$ T for both single crystals.

Electronic properties of single-walled boron nitride nanotube in zig-zag form are numerically investigated by replacing B atoms with C atoms. Using a tight-binding Hamiltonian, the methods based on Green’s function theory, Landauer formalism and Dyson equation, the electronic density of states and electronic conductance in boron nitride nanotube and boron carbonitride nanotube are calculated. Our calculations indicate that in a boron nitride nanotube, the localized states associated with C impurities appear as the concentration of C atoms increases. The boron carbonitride nanotube thus behaves like a semiconductor. Also, by increasing the C atom concentration, the voltage in the first step on the $I–V$ characteristics decreases, whereas the corresponding current increases.

Zinc oxide nanothin films were prepared on glass substrate by sol–gel dip-coating method using zinc acetate dihydrate, methanol, and monoethanolamine as precursor, solvent, and stabilizer, respectively. The relationship between drying conditions and the characteristics of ZnO nanocrystalline films (𝑐-axis orientation, grain size, roughness and optical properties) was studied. The films were dried in an oven at different temperatures and by IR radiation. Then, the films were annealed at $500^{\circ}$C in a furnace. The chemical composition, transmission spectra, structure, and morphology of the samples were studied using infrared (IR) and UV–visible spectroscopy, X-ray diffraction (XRD), and atomic force microscopy (AFM), respectively. The XRD results show that the drying conditions affect the orientation of crystallization along the (0 0 2) plane. AFM images show that the thicknesses of the films decrease from 128 to 93 nm by changing the drying conditions. The photoluminescence (PL) of ZnO nanothin films shows the UV emission at near band edge and broad green radiation at about 465 nm wavelength.