In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.

For one-dimensional loop space, a nonlinear nonlocal transformation of fields is given to make the action of the self-interacting quantum field to the free one. A specific type of classically broken symmetry is restored in quantum theory. One-dimensional sine-Gordon system and sech interactions are treated as the explicit examples.

Anderson et al have shown that for complex energies, the classical trajectories of real quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that when time is complex $t(t = t_r e^{i\theta_\tau})$, certain real Hermitian systems possess close periodic trajectories only for a discrete set of values of $\theta_\tau$. On the other hand, it is generally true that even for real energies, classical trajectories of non-PT symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper, we show that for given real energy, the classical trajectories of complex quartic Hamiltonians $H = p^2 + ax^4 + bx^k$ (where 𝑎 is real, 𝑏 is complex and $k = 1$ or 2) are closed and periodic only for a discrete set of parameter curves in the complex 𝑏-plane. It was further found that given complex parameter 𝑏, the classical trajectories are periodic for a discrete set of real energies (i.e., classical energy gets discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of complex Hamiltonian $H = p^2 + \mu x^4$, ($\mu= \mu_r$ e$^{i\theta}$) are periodic when $\theta = 4 \tan^{−1}$[($n/(2m + n)$)] for $\forall n$ and $m \epsilon \mathbb{Z}$.

We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrized in terms of Fermi–Dirac integrals. This reproduces the experimental data over the accessible range of fugacity and normalized temperature, but cannot describe the superfluid phase transition found in the MIT experiment [Ku et al, Science 335, 563 (2012)]. The most sensitive data for compressibility and specific heat at phase transition can, however, be fitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis.

In recent years, nonlinear coupled reaction–diffusion (CRD) system has been widely investigated by coupled map lattice method. Previously, nonlinear behaviour was observed dynamically when one or two of the three variables in the discrete system change. In this paper, we consider the chaotic behaviour when three variables change, which is called as four-dimensional chaos. When two parameters in the discrete system are unknown, we first give the existing condition of the chaos in four-dimensional space by the generalized definitions of spatial periodic orbits and spatial chaos. In addition, the chaotic behaviour will vary with the parameters. Then we propose a generalized Lyapunov exponent in four-dimensional space to characterize the different effects of parameters on the chaotic behaviour, which has not been studied in detail. In order to verify the chaotic behaviour of the system and the different effects clearly, we simulate the dynamical behaviour in two- and three-dimensional spaces.

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended ($G'/G$)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.

We analyse the derivative of the reduced cross-section $\dfrac{\partial \sigma^s_r}{\partial ln y}|_x$, using the nonlinear Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (NLDGLAP) evolution equation at small 𝑥. The small 𝑥 behaviour of the structure functions are obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. We show that the strong rise corresponding to the linear QCD evolution equations, can be tamed by screening effects.

We have studied the production of single isolated prompt photons in high-energy proton–proton collisions at the RHIC ($\sqrt{s}$ = 200 GeV) and the LHC ($\sqrt{s}$ = 7 TeV) energies within the framework of perturbative QCD upto next-to-leading order of strong coupling ($\alpha_s$). We have used five different parametrizations of parton distribution function (PDF) starting from the old CTEQ4M to the new CT10 distributions and compared our results with the recent single-prompt photon data from the PHENIX and the CMS Collaborations. The prompt photon cross-section is found to be described equally well by all the PDFs within the experimental errors at the RHIC and the LHC energies. The deviation in the single-prompt photon yield for different PDF sets is within ±20% when compared to CTEQ4M, indicating the upper bound of uncertainty in determining the gluon density. The diphoton measurement could be a potential candidate to constrain the gluon distribution inside the proton.

The operative mechanism for a free-electron laser (FEL) with two consecutive helical wigglers having opposite circular polarization in the presence of an axial magnetic field is proposed and analysed. With the help of fluid theory, a tenth-degree polynomial dispersion equation for electromagnetic and space-charge waves is derived. The results are used to illustrate and discuss the dependence of growth rate on different system parameters. Finally, it is shown that for the same system parameters the growth rate of the proposed structure is more than the growth rate of instability in a conventional FEL.

Nonlinear Mach–Zehnder interferometer (NMZI) created with photonic crystal waveguides (PCW) and with Kerr-type nonlinearity has been investigated in this paper. The NMZI has been simulated using two-dimensional finite difference time domain (2D-FDTD) method. Input verses output (I/O) characteristics have been obtained for different lengths of the nonlinear arm, nonlinear coefficients of the nonlinear arm, wavelengths of the input beam, sizes of defect rods and NMZI offset. The results obtained are compared with earlier published results of NMZI created with conventional step index waveguides (SIW). It is shown that all useful features of light switching offered by SIW-based NMZIs are also possible with PCW-based NMZIs of extremely small dimensions. Moreover, PCW-based NMZIs offer additional useful feature not available with SIW-based NMZIs.

This paper presents the experimental studies on self-breakdown-based single-gap plasma cathode electron (PCE) gun (5–20 kV/50–160 A) in argon, gas atmosphere and its performance evaluation based on particle-in-cell (PIC) simulation code `OOPIC-Pro’.The PCE-Gun works in conducting phase (low energy, high current) of pseudospark discharge. It produces an intense electron beam, which can propagate more than 200 mm in the drift space region without external magnetic field. The profile of this beam in the drift space region at different breakdown conditions (i.e., gas pressures and applied voltages) has been studied and the experimental results are compared with simulated values. It is demonstrated that ∼30% beam current is lost during the propagation possibly due to space charge neutralization and collisions with neutral particles and walls.

The pump–probe experiment is typically used to study relaxation phenomena in nonlinear optical systems. Here we use it as a tool to study the phenomenon of anomalous Rabi oscillations in graphene that was predicted recently in single-layer graphene. Unlike conventional Rabi oscillations, anomalous Rabi oscillations are unique to graphene (and possibly to surface states of topological insulators (TIs)), attributable to the pseudospin (conventional spin for TI) degree of freedom and Dirac-fermion character of the graphene system. A pump pulse of a finite duration long enough to contain a large number of cycles induces a current density that oscillates with the frequency of the pump pulse. The amplitude associated with these fast oscillations is seen to exhibit much slower oscillations with a frequency given by $2\omega^2_R/\omega$–the anomalous Rabi frequency, where $\omega_R$ is the conventional Rabi frequency and 𝜔 is the frequency of the external pump field. This effect is easily probed by a probe pulse subsequent to the pump, where it manifests itself as periodic oscillations of the probe susceptibility as a function of pump duration at each probe frequency. Alternatively, it is also seen as an oscillatory function of the pump–probe delay with other variables remaining fixed. This period corresponds to the anomalous Rabi frequency. An analysis of the previously reported experimental data confirms the presence of anomalous Rabi oscillations in graphene.

We present first-principle calculations of 2D nanostructures of graphene functionalized with hydrogen and fluorine, respectively, in chair conformation. The partial density of states, band structure, binding energy and transverse displacement of C atoms due to functionalization (buckling) have been calculated within the framework of density functional theory as implemented in the SIESTA code. The variation in band gap and binding energy per add atom have been plotted against the number of add atoms, as the number of add atoms are incremented one by one. In all, 37 nanostructures with 18C atoms, $3 \times 3 \times 1$ (i.e., the unit cell is repeated three times along 𝑥-axis and three times along 𝑦-axis) supercell, have been studied. The variation in C–C, C–H and C–F bond lengths and transverse displacement of C atoms (due to increase in add atoms) have been tabulated. A large amount of buckling is observed in the carbon lattice, 0.0053–0.7487 Å, due to hydrogenation and 0.0002–0.5379 Å, due to fluorination. As the number of add atoms (hydrogen or fluorine) is increased, a variation in the band gap is observed around the Fermi energy, resulting in change in behaviour of nanostructure from conductor to semiconductor/insulator. The binding energy per add atom increases with the increase in the number of add atoms. The nanostructures with 18C+18H and 18C+18F have maximum band gap of 4.98 eV and 3.64 eV, respectively, and binding energy per add atom –3.7562 eV and –3.3507 eV, respectively. Thus, these nanostructures are stable and are wide band-gap semiconductors, whereas the nanostructures with 18C+2H, 18C+4H, 18C+4F, 18C+8F, 18C+10F and 18C+10H atoms are small band-gap semiconductors with the band gap lying between 0.14 eV and 1.72 eV. Fluorine being more electronegative than hydrogen, the impact of electronegativity on band gap, binding energy and bond length is visible. It is also clear that it is possible to tune the electronic properties of functionalized graphene, which makes it a suitable material in microelectronics.