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In this paper, the thermodynamics of the Friedmann–Lemaître–Robertson–Walker universe have been explored in f(R) theories of gravity with arbitrary matter-geometry coupling. The equivalence between the modified Friedmann equations with any spatial curvature and the first law of thermodynamics is confirmed, where the assumption of the entropy plays a crucial role. Then laws of thermodynamics in our considering case are obtained. They can reduce to the ones given in Einstein’s general theory of relativity under certain conditions. Moreover, a particular model is investigated through the obtained generalized second law of thermodynamics with observational results of cosmographic parameters.

The inherent trade-off among the level of sensing electromagnetic (EM)-energy and the performance parameters, and consequently, the inevitable nonlinearity at high-sensitivity renders the performance-optimization of a micro-fabricated microwave resonant sensor very difficult. This paper presents a shunt capacitor-based EM-energy splitting technique to generate three distinct levels of coupled EM-energies in micro-fabricated microwave LC resonant ethanol sensors and investigate the quantitative correlation among the resulting energy level, sensitivity, linearity, and signal-to-noise ratio (SNR). Gallium arsenide (GaAs)-based micro-fabrication technique is used to implement the proposed microwave LC resonators employing an air-bridged circular spiral inductor and meandered-line coupling capacitors. The experimental results reveal that the performance-optimized sensor rapidly and accurately quantifies the wide range of nanoliter (200 nL) ethanol concentrations (5-50% vol.) with a high-sensitivity (1.817 MHz/% vol.), good linearity, and SNR; thus, highlighting the significance of the proposed method to develop a high-performance ethanol sensor. In addition, ethanol-dependent complex permittivity, which is calculated using the measured central frequencies and loaded-quality factors, reveals a negative correlation between the real-permittivity and ethanol concentration of the aqueous solution.

The dynamics of different kinds of discrete breathers in three types of one-dimensional monatomic chains with on-site and inter-site potentials are investigated. The existence and evolution of symmetric breather, antisymmetric breather, and multibreather in one-dimensional models are proved by using rotating wave approximation, local anharmonic approximation, and the numerical method. The linear stability of these breathers is investigated by using Lyapunov stable analysis. The localization and stability of breathers in three types of models correlate closely to the system nonlinear parameter β.

We study a two-dimensional (2D) diatomic lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DBs) can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the 2D discrete diatomic Klein–Gordon lattice with hard and soft on-site potentials. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers (QDBs) and chaotic discrete breathers (CDBs) by changing the amplitude of the driver. But the DBs and QDBs with symmetric and anti-symmetric profiles that are centered at a heavy atom are more stable than at a light atom, because the frequencies of the DBs and QDBs centered at a heavy atom are lower than those centered at a light atom.

This paper discusses the two-dimensional discrete monatomic Fermi–Pasta–Ulam lattice, by using the method of multiple-scale and the quasi-discreteness approach. By taking into account the interaction between the atoms in the lattice and their nearest neighbours, it obtains some classes of two-dimensional local models as follows: two-dimensional bright and dark discrete soliton trains, two-dimensional bright and dark line discrete breathers, and two-dimensional bright and dark discrete breather.

We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry's linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete two-dimensional monatomic β-FPU lattice.

This paper studies a discrete one-dimensional monatomic Klein–Gordon chain with only quartic nearest-neighbour interactions, in which the compact-like discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. Introducing the trying method, it proves that compact-like discrete breathers exist in this nonlinear system. It also discusses the linear stability of the compact-like discrete breathers, when the coefficient (β) of quartic on-site potential and the coupling constant (K4) of quartic interactive potential satisfy the given conditions, they are linearly stable.

We study the existence of two-dimensional discrete breathers in a two-dimensional face-centred square lattice consisting of alternating light and heavy atoms, with nearest-neighbour coupling containing quartic soft or hard nonlinearity. This study is focused on two-dimensional breathers with frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of existence of two-dimensional gap breathers by using the numerical method, the local anharmonicity approximation and the rotating wave approximation. We obtain six types of two-dimensional gap breathers, i.e., symmetric, mirror-symmetric and asymmetric, no matter whether the centre of the breather is on a light or a heavy atom.

A general one-dimensional discrete monatomic model is investigated by using the multiple-method. It is proven that the discrete bright breathers (DBBs) and discrete dark breathers (DDBs) exist in this model at the anti-continuous limit, and then the concrete models of the DBBs and DDBs are also presented by the multiple-scale approach (MSA) and the quasi-discreteness approach (QDA). When the results are applied to some particular models, the same conclusions as those presented in corresponding references are achieved. In addition, we use the method of the linearization analysis to investigate this system without the high order terms of . It is found that the DBBs and DDBs are linearly stable only when coupling parameter χ is small, of which the limited value is obtained by using an analytical method.

Multi-site compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. We obtain that the two-site compact-like discrete breathers with codes σ = {0,..., 0,1,1, 0,..., 0} and codes σ = {0,...,0,1, - 1,0,..., 0} can exist in discrete one-dimensional monatomic chain with quartic on-site and inter-site potentials. However, the former can only exist in hard quartic on-site potential and cannot exist in soft quartic on-site potential, whereas the latter is just reversed. All of the two-site Compact-like discrete breathers with codes σ = {0,..., 0,1,1, 0,..., 0} and σ = {0,..., 0,1, - 1, 0,..., 0} cannot exist in a pure K4 chain.

A continuous β-Fermi–Pasta–Ulam (FPU) chain is investigated by using the knowledge of elliptic equation and Jacobian elliptic functions. We obtain the new solutions, two-kink soliton solution, breather solution and breather lattice solution, of the continuous β-FPU chain, besides the kink-soliton solution and chaos solution.

Compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. It is proven that compact-like discrete breathers exist not only in soft 4 potential but also in hard 4 potential and K4 chains. The measurements of compact-like discrete breathers' core in soft and hard 4 potential are determined by coupling parameter K4, while those in K4 chains have nothing with coupling parameter K4. The stabilities of compact-like discrete breathers correlate closely to coupling parameter K4 and the boundary condition of lattice.

A theoretical study of Bloch electron transport in a superlattice miniband driven by an electric field parallel to the growth axis is carried out, by Fokker-Planck equation (FPE) in momentum space with the averaged momentum relaxation time () approximation. Steadystate drift-velocity/field characteristics exhibit the expected maximum followed by negative differential conductivity (NDC), and then followed by drift-velocity oscillation when or electric field is large. The oscillation frequency is an increasing function of , and when , the limit of the oscillation frequency is the Bloch frequency as expected.

The formation and propagation of dipole domains in superlattices are studied both by the modified discrete drift model and by the nonlinear Schrodinger equation. The spatiotemporal distribution of the electric field and electron density are presented. The numerical results are compared with the soliton solutions of the nonlinear Schrodinger equation and analysed. It is shown that the numerical solutions agree with the soliton solutions of the nonlinear Schrodinger equation. The dipole electric-field domains in semiconductor superlattices have the properties of solitons.

We investigate the interactions of lattice phonons with Wannier–Mott exciton, the exciton that has a large radius in two-dimensional molecular lattice, by the method of continuum limit approximation, and obtain that the self-trapping can also appear in two-dimensional molecular lattice with a harmonic and nonlinear potential. The exciton effect on molecular lattice does not distort the molecular lattice but only makes it localized and the localization can also react, again through phonon coupling, to trap the energy and prevents its dispersion.

The irradiance of an elliptic Gaussian beam that is high enough to excite high-order nonlinear refraction effect is used to calculate the normalized on-axis transmittance function in the z-scan technique by introducing complex beam parameters which make the calculation simpler. The transmittance formula is applied to the first-, first two-, and first three-order nonlinearities. Numerical evaluation shows that the symmetry no longer holds when using an elliptic Gaussian beam instead of a circular Gaussian beam. A distortion is observed in the central part of the curve, which decreases as ellipticity increases. Moreover, the variation of the normalized peak-valley difference decreases as ellipticity decreases.

Nonlinearity has a crucial impact on the symmetry properties of dynamical systems. This paper studies a one-dimensional mixed Klein–Gordon/Fermi–Pasta–Ulam diatomic chain using the expanded rotating plane-wave approximation and numerical calculations to determine the effect of cubic potentials on the symmetry properties of discrete breathers in this system. The results will be very useful to researchers in the field of numerical calculations on discrete breathers.

We study the features of a single q-breather (SQB) in a Fermi-Pasta-Ulam lattice by the numerical method, and obtain that the stability of SQB correlates to coupling constant K and nonlinear parameter β. No matter whether K or β increases, the periodic SQB can be transformed into a quasiperiodic SQB or a chaotic SQB. We also obtain the conditions of excitation of periodic, quasiperiodic and chaotic SQBs.

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein–Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.