A new type of symmetry, ren-symmetry, describing anyon physics and corresponding topological physics, is proposed. Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics, super-symmetric gravity, super-symmetric string theory, super-symmetric integrable systems and so on. Super-symmetry and Grassmann numbers are, in some sense, dual conceptions, and it turns out that these conceptions coincide for the ren situation, that is, a similar conception of ren-number (R-number) is devised for ren-symmetry. In particular, some basic results of the R-number and ren-symmetry are exposed which allow one to derive, in principle, some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems. Training examples of ren-integrable KdV-type systems and ren-symmetric KdV equations are explicitly given.

Imaginarity has proven to be a valuable resource in various quantum information processing tasks. A natural question arises: can the imaginarity of quantum states be broadcast via real operations? In this work, we present explicit structures for nonreal states whose imaginarity can be broadcast and cloned. That is, for a nonreal state, its imaginarity can be cloned if and only if it is a direct sum of several maximally imaginary states under orthogonal transformation, and its imaginarity can be broadcast if and only if it is a direct sum of a real state and some nonreal qubit states which are mixtures of two orthogonal maximally imaginary states under orthogonal transformation. In particular, we show that for a nonreal pure state, its imaginarity cannot be broadcast unless it is a maximally imaginary state. Furthermore, we derive a trade-off relation on the imaginarity broadcasting of pure states in terms of the measure of irreversibility of quantum states concerning real operations and the geometric measure of imaginarity. In addition, we demonstrate that any faithful measure of imaginarity is not superadditive.

In this paper, the (1+1)-dimensional classical Boussinesq–Burgers (CBB) system is extended to a (4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm. The Lax integrability, symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given. Meanwhile, for illustration, an exact solution of a (1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.

Considering the importance of higher-dimensional equations that are widely applied to real nonlinear problems, many (4 + 1)-dimensional integrable systems have been established by uplifting the dimensions of their corresponding lower-dimensional integrable equations. Recently, an integrable (4 + 1)-dimensional extension of the Boiti–Leon–Manna–Pempinelli (4DBLMP) equation has been proposed, which can also be considered as an extension of the famous Korteweg–de Vries equation that is applicable in fluids, plasma physics and so on. It is shown that new higher-dimensional variable separation solutions with several arbitrary lower-dimensional functions can also be obtained using the multilinear variable separation approach for the 4DBLMP equation. In addition, by taking advantage of the explicit expressions of the new solutions, versatile (4 + 1)-dimensional nonlinear wave excitations can be designed. As an illustration, periodic breathing lumps, multi-dromion-ring-type instantons, and hybrid waves on a doubly periodic wave background are discovered to reveal abundant nonlinear structures and dynamics in higher dimensions.

By casting evolution to the Bloch sphere, the dynamics of 2 × 2 matrix non-Hermitian systems are investigated in detail. This investigation reveals that there are four kinds of dynamical modes for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken ${ \mathcal P }{ \mathcal T }$ non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for the degenerate point is two, while the others are one.

Thermalization in many-body systems, especially with strong interactions, is a central question in physics. In this work, we present a novel framework for the thermalization of interacting wave systems, distinguishing between trivial (no momentum exchange) and nontrivial interactions (significant energy redistribution). This distinction leads to a statistically equivalent model with weakened interactions. By applying this to FPUT-like models, we identify a unique double scaling of thermalization times. Crucially, our findings suggest the persistence of prethermalization in strong interactions.

In this work, we have investigated the rotating effect on the thermodynamic properties of a 2D quantum ring. Accordingly, we have considered the radial potential of a 2D quantum ring and solved the Schrödinger equation in the presence of the Aharonov–Bohm effect and a uniform magnetic field for the considered potential. According to the solution of the equation, we calculated the eigenvalues and eigenfunctions of the considered system. Using the calculated energy spectrum, we obtained the partition function and thermodynamic properties of the system, such as the mean energy, specific heat, entropy and free energy. Our results show that the rotating effect has a significant influence on the thermophysical properties of a 2D quantum ring. We also study other effects of the rotating term: (1) the effect of different values of rotating parameters, and (2) the effect of negative rotation on the thermodynamic properties of the system. Our results are discussed in detail.

Based on the long wave limit method, the general form of the second-order and third-order rogue wave solutions to the focusing nonlinear Schrödinger equation are given by introducing some arbitrary parameters. The interaction solutions between the first-order rogue wave and one-breather wave are constructed by taking a long wave limit on the two-breather solutions. By applying the same method to the three-breather solutions, two types of interaction solutions are obtained, namely the first-order rogue wave and two breather waves, the second-order rogue wave and one-breather wave, respectively. The influence of the parameters related to the phase on the interaction phenomena is graphically demonstrated. Collisions occur among the rogue waves and breather waves. After the collisions, the shape of them remains unchanged. The abundant interaction phenomena in this paper will contribute to a better understanding of the propagation and control of nonlinear waves.

Nonlinear circuits can show multistability when a magnetic flux-dependent memristor (MFDM) or a charge-sensitive memristor (CSM) is incorporated into a one branch circuit, which helps estimate magnetic or electric field effects. In this paper, two different kinds of memristors are incorporated into two branch circuits composed of a capacitor and a nonlinear resistor, thus a memristive circuit with double memristive channels is designed. The circuit equations are presented, and the dynamics in this oscillator with two memristive terms are discussed. Then, the memristive oscillator is converted into a memristive map by applying linear transformation on the sampled time series for the memristive oscillator. The Hamilton energy function for the memristive oscillator is obtained by using the Helmholtz theorem, and it can be mapped from the field energy of the memristive circuit. An energy function for the dual memristive map is suggested by imposing suitable weights on the discrete energy function. The dynamical behaviors of the new memristive map are investigated, and an adaptive law is proposed to regulate the firing mode in the memristive map. This work will provide a theoretical basis and experimental guidance for oscillator-to-map transformation and discrete map energy calculation.

In this paper, we study the N = 2 a = 1 supersymmetric KdV equation. We construct its Darboux transformation and the associated Bäcklund transformation. Furthermore, we derive a nonlinear superposition formula, and as applications we calculate some solutions for this supersymmetric KdV equation and recover the related results for the Kersten–Krasil'shchik coupled KdV-mKdV system.

As a torqued version of the lattice potential Korteweg–de Vries equation, the H1^a is an integrable nonsymmetric lattice equation with only one spacing parameter. In this paper, we present the Cauchy matrix scheme for this equation. Soliton solutions, Jordan-block solutions and soliton-Jordan-block mixed solutions are constructed by solving the determining equation set. All the obtained solutions have jumping property between constant values for fixed n and demonstrate periodic structure.

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, are given.

The nonlinear Schrödinger equation equation is one of the most important physical models used in optical fiber theory to explain the transmission of an optical soliton. The field of chiral soliton propagation in nuclear physics is very interesting because of its numerous applications in communications and ultra-fast signal routing systems. The (1+1)-dimensional chiral dynamical structure that describes the soliton behaviour in data transmission is dealt with in this work using a variety of in-depth analytical techniques. This work has applications in particle physics, ionised science, nuclear physics, optics, and other applied mathematical sciences. We are able to develop a variety of solutions to demonstrate the behaviour of solitary wave structures, periodic soliton solutions, chiral soliton solutions, and bell-shaped soliton solutions with the use of applied techniques. Moreover, in order to verify the scientific calculations, the stability analysis for the observed solutions of the governing model is taken into consideration. In addition, the 3-dimensional, contour, and 2-dimensional visuals are supplied for a better understanding of the behaviour of the solutions. The employed strategies are dependable, uncomplicated, and effective; yet have not been utilised with the governing model in the literature that is now accessible. The resulting outcomes have impressive applications across a large number of study areas and computational physics phenomena representing real-world scenarios. The methods applied in this model are not utilized on the given models in previous literature so we can say that these describe the novelty of the work.

The SARG04 quantum key distribution protocol can offer greater robustness against photon number splitting attacks than the BB84 protocol that is implemented with weak pulses. In this paper, we propose a tight key analysis for the SARG04 protocol, by considering the one-decoy method and investigating its performance under the influence of a detector afterpulse. Our results demonstrate that an increase in block size leads to a slight increase in both the secure key rate and the maximum transmission distance. Importantly, the detector afterpulse plays a crucial role in practical applications and has a more pronounced effect on the SARG04 protocol compared to the BB84 protocol.

Entanglement-breaking (EB) subspaces determine the additivity of entanglement of formation (EOF), which is a long-standing issue in quantum information. We explicitly construct the two-dimensional EB subspaces of any bipartite system, when system dimensions are equal, and we apply the subspaces to construct EB spaces of arbitrary dimensions. We also present partial construction when system dimensions are different. Then, we present the notion and properties of EB subspaces for some systems, and in particular the absolute EB subspaces. We construct some examples of absolute EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states.

In [Phys. Rev. A 107 012427 (2023)], Baldwin and Jones prove that Uhlmann–Jozsa’s fidelity between two quantum states ρ and σ, i.e., $F(\rho ,\sigma )\,:= \,{\left({\rm{Tr}}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^{2}$, can be written in a simplified form as $F(\rho ,\sigma )={\left({\rm{Tr}}\sqrt{\rho \sigma }\right)}^{2}$. In this article, we give an alternative proof of this result, using a function power series expansion and the properties of the trace function. Our approach not only reinforces the validity of the simplified expression but also facilitates the exploration of novel dissimilarity functions for quantum states and more complex trace functions of density operators.

This article's purpose is to investigate multiple high-order pole solutions for the AB system by the Riemann–Hilbert (RH) approach. We establish the RH problem through using spectral analysis to the Lax pair. Then the RH problem can be resolved and the soliton solution's formula can be given by using the Laurent expansion method. Finally, we get special soliton solutions, including dark solitons, W-type dark solitons and multiple high-pole solutions. In addition, the W-type dark soliton solutions will occur when the spectral parameters are purely imaginary.

Exact analytical solutions are good candidates for studying and explaining the dynamics of solitons in nonlinear systems. We further extend the region of existence of spin solitons in the nonlinearity coefficient space for the spin-1 Bose–Einstein condensate. Six types of spin soliton solutions can be obtained, and they exist in different regions. Stability analysis and numerical simulation results indicate that three types of spin solitons are stable against weak noise. The non-integrable properties of the model can induce shape oscillation and increase in speed after the collision between two spin solitons. These results further enrich the soliton family for non-integrable models and can provide theoretical references for experimental studies.

In the last forty years, the rise of HIV has undoubtedly become a major concern in the field of public health, imposing significant economic burdens on affected regions. Consequently, it becomes imperative to undertake comprehensive investigations into the mechanisms governing the dissemination of HIV within the human body. In this work, we have devised a mathematical model that elucidates the intricate interplay between CD4⁺ T-cells and viruses of HIV, employing the principles of fractional calculus. The production rate of CD4⁺ T-cells, like other immune cells depends on certain factors such as age, health status, and the presence of infections or diseases. Therefore, we incorporate a variable source term in the dynamics of HIV infection with a saturated incidence rate to enhance the precision of our findings. We introduce the fundamental concepts of fractional operators as a means of scrutinizing the proposed HIV model. To facilitate a deeper understanding of our system, we present an iterative scheme that elucidates the trajectories of the solution pathways of the system. We show the time series analysis of our model through numerical findings to conceptualize and understand the key factors of the system. In addition to this, we present the phase portrait and the oscillatory behavior of the system with the variation of different input parameters. This information can be utilized to predict the long-term behavior of the system, including whether it will converge to a steady state or exhibit periodic or chaotic oscillations.

By means of the multilinear variable separation (MLVS) approach, new interaction solutions with low-dimensional arbitrary functions of the (2+1)-dimensional Nizhnik–Novikov–Veselov-type system are constructed. Four-dromion structure, ring-parabolic soliton structure and corresponding fusion phenomena for the physical quantity ${U}=\lambda {(\mathrm{ln}f)}_{{xy}}$ are revealed for the first time. This MLVS approach can also be used to deal with the (2+1)-dimensional Sasa–Satsuma system.

Collective quantum states, such as subradiant and superradiant states, are useful for controlling optical responses in many-body quantum systems. In this work, we study novel collective quantum phenomena in waveguide-coupled Bragg atom arrays with inhomogeneous frequencies. For atoms without free-space dissipation, collectively induced transparency is produced by destructive quantum interference between subradiant and superradiant states. In a large Bragg atom array, multi-frequency photon transparency can be obtained by considering atoms with different frequencies. Interestingly, we find collectively induced absorption (CIA) by studying the influence of free-space dissipation on photon transport. Tunable atomic frequencies nontrivially modify decay rates of subradiant states. When the decay rate of a subradiant state equals to the free-space dissipation, photon absorption can reach a limit at a certain frequency. In other words, photon absorption is enhanced with low free-space dissipation, distinct from previous photon detection schemes. We also show multi-frequency CIA by properly adjusting atomic frequencies. Our work presents a way to manipulate collective quantum states and exotic optical properties in waveguide quantum electrodynamics (QED) systems.

In this article a new achievement of fractional-order 3 × n Fan networks is presented. In the first step, the RT-I method is used to derive the general formulae of the equivalent impedance of fractional-order 3 × n Fan networks. In the second part, the effects of five system parameters (L, C, n, α and β) on amplitude-frequency and phase-frequency characteristics are analyzed. At the same time, the amplitude-frequency and phase-frequency characteristics of the fractional order 3 × n Fan network are revealed by Matlab drawing. This work has important theoretical and practical significance for resistor network models in the field of natural science and engineering technology.

We establish tighter uncertainty relations for arbitrary finite observables via (α, β, γ) weighted Wigner–Yanase–Dyson ((α, β, γ) WWYD) skew information. The results are also applicable to the (α, γ) weighted Wigner–Yanase–Dyson ((α, γ) WWYD) skew information and the weighted Wigner–Yanase–Dyson (WWYD) skew information. We also present tighter lower bounds for quantum channels and unitary channels via (α, β, γ) modified weighted Wigner–Yanase–Dyson ((α, β, γ) MWWYD) skew information. Detailed examples are provided to illustrate the tightness of our uncertainty relations.

The third-harmonic generation (THG) coefficient for a spherical quantum dot system with inversely quadratic Hellmann plus inversely quadratic potential is investigated theoretically, considering the regulation of quantum size, confinement potential depth and the external environment. The numerical simulation results indicate that the THG coefficient can reach the order of 10⁻¹² m² V^–2, which strongly relies on the tunable factor, with its resonant peak experiencing a redshift or blueshift. Interestingly, the effect of temperature on the THG coefficient in terms of peak location and size is consistent with the quantum dot radius but contrasts with the hydrostatic pressure. Thus, it is crucial to focus on the influence of internal and external parameters on nonlinear optical effects, and to implement the theory in practical experiments and the manufacture of optoelectronic devices.

We construct the Riemann–Hilbert problem of the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions, and use the Laurent expansion and Taylor series expansion to obtain the exact formulas of the soliton solutions in the case of a higher-order pole and multiple higher-order poles. The dynamic behaviors of a simple pole, a second-order pole and a simple pole plus a second-order pole are demonstrated.

The aim of this work is to investigate anisotropic compact objects within the framework of f(G) modified theory of gravity. For our present work, we utilize Krori–Barua metrics, i.e., λ(r) = Xr² + Y and β(r) = Zr². We use some matching conditions of spherically symmetric spacetime with Bardeen's model as an exterior geometry. Further, we establish some expressions of energy density and pressure components to analyze the stellar configuration of Bardeen compact stars by assuming viable f(G) models. We examine the energy conditions for different stellar structures to verify the viability of our considered models. Moreover, we also investigate some other physical features, such as equilibrium condition, equation of state parameters, adiabatic index, stability analysis, mass function, surface redshift, and compactness factor, respectively. It is worthwhile to mention here for the current study that our stellar structure in the background of Bardeen's model is more viable and stable.

A quantum network concerns several independent entangled resources and can create strong quantum correlations by performing joint measurements on some observers. In this paper, we discuss an n-partite chain network with each of two neighboring observers sharing an arbitrary Bell state and all intermediate observers performing some positive-operator-valued measurements with parameter λ. The expressions of all post-measurement states between any two observers are obtained, and their quantifications of Bell nonlocality, Einstein–Podolsky–Rosen steering and entanglement with different ranges of λ are respectively detected and analyzed.

This study introduces an innovative dual-tunable absorption film with the capability to switch between ultra-wideband and narrowband absorption. By manipulating the temperature, the film can achieve multi-band absorption within the 30–45 THz range or ultra-wideband absorption spanning 30–130 THz, with an absorption rate exceeding 0.9. Furthermore, the structural parameters of the absorption film are optimized using the particle swarm optimization (PSO) algorithm to ensure the optimal absorption response. The absorption response of the film is primarily attributed to the coupling of guided-mode resonance and local surface plasmon resonance effects. The film's symmetric structure enables polarization incoherence and allows for tuning through various means such as doping/voltage, temperature and structural parameters. In the case of a multi-band absorption response, the film exhibits good sensitivity to refractive index changes in multiple absorption modes. Additionally, the absorption spectrum of the film remains effective even at large incidence angles, making it highly promising for applications in fields such as biosensing and infrared stealth.

In this paper, we investigate the Cauchy problem of the Sasa–Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the $\overline{\partial }$-nonlinear steepest descent method of the mixed $\bar{\partial }$-Riemann–Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the 'sender-receiver' model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver's density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.

The effective liquid drop model (ELDM) is improved by introducing an accurate nuclear charge radius formula and an analytic expression for assaulting frequency. Within the improved effective liquid drop model (IMELDM), the experimental cluster radioactivity half-lives of the trans-lead region are calculated. It is shown that the accuracy of the IMELDM is improved compared with that of the ELDM. At last, the cluster radioactivity half-lives that are experimentally unavailable for the trans-lead nuclei are predicted by the IMELDM. These predictions may be useful for searching for new candidates for cluster radioactivity in future experiments.

This paper theoretically studies the impurity states and the effects of impurity concentration and configuration on the optical, electrical, and statistical properties of CdSe nanoplatelets (NPLs). An image charge-based model of electron-impurity interaction is proposed. The charge-carrier energy spectra and corresponding wave functions depending on the impurity number and configuration are calculated. The electron binding energies are calculated for different NPL thicknesses. It is shown that the image charge-based interaction potential that arises due to the dielectric constants mismatch is much stronger than the interaction potential that does not take such a mismatch into account. Also, it is demonstrated that the binding energies are increasing with the number of impurities. We calculate the canonical partition function using the energy levels of the electron, which in turn is used to obtain the mean energy, heat capacity, and entropy of the non-interacting electron gas. The thermodynamic properties of the non-interacting electron gas that depend on the geometric parameters of the NPL, impurity number, configuration, and temperature are studied.

Quantifying entanglement measures for quantum states with unknown density matrices is a challenging task. Machine learning offers a new perspective to address this problem. By training machine learning models using experimentally measurable data, we can predict the target entanglement measures. In this study, we compare various machine learning models and find that the linear regression and stack models perform better than others. We investigate the model's impact on quantum states across different dimensions and find that higher-dimensional quantum states yield better results. Additionally, we investigate which measurable data has better predictive power for target entanglement measures. Using correlation analysis and principal component analysis, we demonstrate that quantum moments exhibit a stronger correlation with coherent information among these data features.

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

We present a flexible manipulation and control of solitons via Bose–Einstein condensates. In the presence of Rashba spin–orbit coupling and repulsive interactions within a harmonic potential, our investigation reveals the numerical local solutions within the system. By manipulating the strength of repulsive interactions and adjusting spin–orbit coupling while maintaining a zero-frequency rotation, diverse soliton structures emerge within the system. These include plane-wave solitons, two distinct types of stripe solitons, and odd petal solitons with both single and double layers. The stability of these solitons is intricately dependent on the varying strength of spin–orbit coupling. Specifically, stripe solitons can maintain a stable existence within regions characterized by enhanced spin–orbit coupling while petal solitons are unable to sustain a stable existence under similar conditions. When rotational frequency is introduced to the system, solitons undergo a transition from stripe solitons to a vortex array characterized by a sustained rotation. The rotational directions of clockwise and counterclockwise are non-equivalent owing to spin–orbit coupling. As a result, the properties of vortex solitons exhibit significant variation and are capable of maintaining a stable existence in the presence of repulsive interactions.

Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage, widely employed in the era of noisy intermediate-scale quantum computing. This study presents an advanced variational hybrid algorithm (EVQLSE) that leverages both quantum and classical computing paradigms to address the solution of linear equation systems. Initially, an innovative loss function is proposed, drawing inspiration from the similarity measure between two quantum states. This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver. Subsequently, a specialized parameterized quantum circuit structure is presented for small-scale linear systems, which exhibits powerful expressive capabilities. Through rigorous numerical analysis, the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm, and it obtained the best score compared to the others. Moreover, the expansion in system size is accompanied by an increase in the number of parameters, placing considerable strain on the training process for the algorithm. To address this challenge, an optimization strategy known as quantum parameter sharing is introduced, which proficiently minimizes parameter volume while adhering to exacting precision standards. Finally, EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 2²⁰.

We present the first numerical solution that corresponds to a pair of Cho-Maison monopoles and antimonopoles (MAPs) in the SU(2) × U(1) Weinberg-Salam (WS) theory. The monopoles are finitely separated, while each pole carries a magnetic charge ±4π/e. The positive pole is situated in the upper hemisphere, whereas the negative pole is in the lower hemisphere. The Cho-Maison MAP is investigated for a range of Weinberg angles, $0.4675\leqslant \tan {\theta }_{{\rm{W}}}\leqslant 10$, and Higgs self-coupling, 0 ≤ β ≤ 1.7704. The magnetic dipole moment (μ_m) and pole separation (d_z) of the numerical solutions are calculated and analyzed. The total energy of the system, however, is infinite due to point singularities at the locations of monopoles.

In this paper, the proton structure function ${F}_{2}^{p}(x,{Q}^{2})$ at small-x is investigated using an analytical solution to the Balitsky-Kovchegov (BK) equation. In the context of the color dipole description of deep inelastic scattering (DIS), the structure function ${F}_{2}^{p}(x,{Q}^{2})$ is computed by applying the analytical expression for the scattering amplitude N(k, Y) derived from the BK solution. At transverse momentum k and total rapidity Y, the scattering amplitude N(k, Y) represents the propagation of the quark-antiquark dipole in the color dipole description of DIS. Using the BK solution we extracted the integrated gluon density xg(x, Q²) and then compared our theoretical estimation with the LHAPDF global data fits, NNPDF3.1sx and CT18. Finally, we have investigated the behavior of ${F}_{2}^{p}(x,{Q}^{2})$ in the kinematic region of 10⁻⁵ ≤ x ≤ 10⁻² and 2.5 GeV² ≤ Q² ≤ 60 GeV². Our predicted results for ${F}_{2}^{p}(x,{Q}^{2})$ within the specified kinematic region are in good agreement with the recent high-precision data for ${F}_{2}^{p}(x,{Q}^{2})$ from HERA (H1 Collaboration) and the LHAPDF global parametrization group NNPDF3.1sx.

The current investigation examines the fractional forced Korteweg–de Vries (FF-KdV) equation, a critically significant evolution equation in various nonlinear branches of science. The equation in question and other associated equations are widely acknowledged for their broad applicability and potential for simulating a wide range of nonlinear phenomena in fluid physics, plasma physics, and various scientific domains. Consequently, the main goal of this study is to use the Yang homotopy perturbation method and the Yang transform decomposition method, along with the Caputo operator for analyzing the FF-KdV equation. The derived approximations are numerically examined and discussed. Our study will show that the two suggested methods are helpful, easy to use, and essential for looking at different nonlinear models that affect complex processes.