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.

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

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.

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.

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.

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.

In this paper, the Drinfeld–Sokolov–Satsuma–Hirota (DSSH) system is studied by using residual symmetry and the consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system proves to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

We connect magic (non-stabilizer) states, symmetric informationally complete positive operator valued measures (SIC-POVMs), and mutually unbiased bases (MUBs) in the context of group frames, and study their interplay. Magic states are quantum resources in the stabilizer formalism of quantum computation. SIC-POVMs and MUBs are fundamental structures in quantum information theory with many applications in quantum foundations, quantum state tomography, and quantum cryptography, etc. In this work, we study group frames constructed from some prominent magic states, and further investigate their applications. Our method exploits the orbit of discrete Heisenberg–Weyl group acting on an initial fiducial state. We quantify the distance of the group frames from SIC-POVMs and MUBs, respectively. As a simple corollary, we reproduce a complete family of MUBs of any prime dimensional system by introducing the concept of MUB fiducial states, analogous to the well-known SIC-POVM fiducial states. We present an intuitive and direct construction of MUB fiducial states via quantum T-gates, and demonstrate that for the qubit system, there are twelve MUB fiducial states, which coincide with the H-type magic states. We compare MUB fiducial states and SIC-POVM fiducial states from the perspective of magic resource for stabilizer quantum computation. We further pose the challenging issue of identifying all MUB fiducial states in general dimensions.

In the realm of nonlinear integrable systems, the presence of decompositions facilitates the establishment of linear superposition solutions and the derivation of novel coupled systems exhibiting nonlinear integrability. By focusing on single-component decompositions within the potential BKP hierarchy, it has been observed that specific linear superpositions of decomposition solutions remain consistent with the underlying equations. Moreover, through the implementation of multi-component decompositions within the potential BKP hierarchy, successful endeavors have been undertaken to formulate linear superposition solutions and novel coupled KdV-type systems that resist decoupling via alterations in dependent variables.

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.

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.

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 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.

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.

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.

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.

Active matter exhibits collective motions at various scales. Geometric confinement has been identified as an effective way to control and manipulate active fluids, with much attention given to external factors. However, the impact of the inherent properties of active particles on collective motion under confined conditions remains elusive. Here, we use a highly tunable active nematics model to study active systems under confinement, focusing on the effect of the self-driven speed of active particles. We identify three distinct states characterized by unique particle and flow fields within confined active nematic systems, among which circular rotation emerges as a collective motion involving rotational movement in both particle and flow fields. The theoretical phase diagram shows that increasing the self-driven speed of active particles significantly enhances the region of the circular rotation state and improves its stability. Our results provide insights into the formation of high quality vortices in confined active nematic systems.

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.

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.

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.

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.

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.

We present a lattice quantum chromodynamics (QCD) simulation with 2 + 1 + 1 flavor full QCD ensembles using near-physical quark masses and different spatial sizes L, at a ∼ 0.055 fm. The results show that the scalar and pesudoscalar 2-point correlator with a valence pion mass of approximately 230 MeV become degenerated at L ≤ 1.0 fm, and such an observation suggests that the spontaneous chiral symmetry breaking disappears effectively at this point. At the same time, the mass gap between the nucleon and pion masses remains larger than Λ_QCD in the entire L ∈ [0.2, 0.7] fm range.

This article considers a static and spherical black hole (BH) in f(Q) gravity. f(Q) gravity is the extension of symmetric teleparallel general relativity, where both curvature and torsion are vanishing and gravity is described by nonmetricity. In this study, we investigate the possible implications of quasinormal mode (QNM) modified Hawking spectra and deflection angles generated by the model. The Wentzel–Kramers–Brillouin method is used to solve the equations of motion for massless Dirac perturbation fields and explore the impact of the nonmetricity parameter (Q₀). Based on the QNM computation, we can ensure that the BH is stable against massless Dirac perturbations and as Q₀ increases the oscillatory frequency of the mode decreases. We then discuss the weak deflection angle in the weak field limit approximation. We compute the deflection angle up to the fourth order of approximation and show how the nonmetricity parameter affects it. We find that the Q₀ parameter reduces the deflection angle.

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.

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.

The Stokes–Einstein–Debye (SED) relation is proposed to be broken down in supercooled liquids by many studies. However, conclusions are usually drawn by testing some variants of the SED relation rather than its original formula. In this work, the rationality of the SED relation and its variants is examined by performing molecular dynamics simulations with the Lewis–Wahnstrom model of ortho-terphenyl (OTP). The results indicate the original SED relation is valid for OTP but the three variants are all broken down. The inconsistency between the SED relation and its variants is analyzed from the heterogeneous dynamics, the adopted assumptions and approximations as well as the interactions among molecules. Therefore, care should be taken when employing the variants to judge the validity of the SED relation in supercooled liquids.

This paper aims to develop a direct approach, namely, the Cauchy matrix approach, to non-isospectral integrable systems. In the Cauchy matrix approach, the Sylvester equation plays a central role, which defines a dressed Cauchy matrix to provide τ functions for the investigated equations. In this paper, using the Cauchy matrix approach, we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions. These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem. Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction. These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

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.

Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.

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.

The axion-like particle (ALP) is one kind of the best-motivated new particles. We consider its production from the pseudoscalar mesonic decays $M\to M^{\prime} a$, with M being a pseudoscalar meson B or K. The upper limits on the flavor-conserving ALP–quark coupling parameter g_u are obtained by assuming the ALP to be an invisible particle. We find that the most severe constraint on g_u comes from the decay ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ for 0.05 GeV ≤ M_a ≤ 0.35 GeV, while the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$ and ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ can also generate significant constraints.

The present manuscript uses three Kudryashov-based methods to analytically inspect the class of Gerdjikov–Ivanov equations, which comprises the standard Gerdjikov–Ivanov equation and the perturbed Gerdjikov–Ivanov equation. Various optical solitonic solutions have been constructed. Certainly, as the reported solitonic structures happened to be exponential functions, diverse true solitonic solutions can easily be resorted to upon suitably fixing the involving parameters, including mainly the bright and singular solitons. Lastly, the study graphically examined some of the constructed structures, which were then found to portray some interesting known shapes in the theory of solitary waves and nonlinear Schrödinger equations. Additionally, the Kudryashov-index d has been noted to play a significant role in the propagation of complex waves in the nonlinear media described by Gerdjikov–Ivanov equations.

The phase transition of water molecules in nanochannels under varying external electric fields is studied by molecular dynamics simulations. It is found that the phase transition of water molecules in nanochannels occurs by changing the frequency of the varying electric field. Water molecules maintain the ice phase when the frequency of the varying electric field is less than 16 THz or greater than 30 THz, and they completely melt when the frequency of the varying electric field is 24 THz. This phenomenon is attributed to the breaking of hydrogen bonds when the frequency of the varying electric field is close to their inherent resonant frequency. Moreover, the study demonstrates that the critical frequency varies with the confinement situation. The new mechanism of regulating the phase transition of water molecules in nanochannels revealed in this study provides a perspective for further understanding of the phase transition of water molecules in nanochannels, and has great application potential in preventing icing and deicing.

Introducing ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials into the three-coupled nonlinear Gross–Pitaevskii equations offers a new way to seek stable soliton states in quasi-one-dimensional spin-1 Bose–Einstein condensates. In scenarios where the spin-independent parameter c₀ and the spin-dependent parameter c₂ vary, we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross–Pitaevskii equations with ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials. We obtain analytical soliton states and find that simply modulating c₂ may change the analytical soliton states from unstable to stable. Additionally, we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations, exhibiting distinct behavior in energy exchange. For further investigation, we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components. These findings may contribute to a deeper understanding of solitons in Bose–Einstein condensates with ${ \mathcal P }{ \mathcal T }$-symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.