First-order Solution Research Articles

On thermal conduction in the solar atmosphere: An analytical solution for nonlinear diffusivity without compact support

Context. The scientific community employs complicated multiphysics simulations to understand the physics in solar, stellar, and interstellar media. These must be tested against known solutions to ensure their validity. Several well-known tests exist, such as the Sod shock tube test. However, a test for nonlinear diffusivity is missing. This problem is highly relevant in the solar atmosphere, where various events release energy that subsequently diffuses by Spitzer thermal conductivity. Aims. The aim is to derive an analytical solution for nonlinear diffusivity in 1D, 2D, and 3D, which allows for a nonzero background value. The solution is used to design a test for numerical solvers and study Spitzer conductivity in the solar atmosphere. Methods. There exists an ideal solution assuming zero background value. We performed an analytical first-order perturbation of this solution. The first-order solution was first tested against a dedicated nonlinear diffusion solver, whereupon it was used to benchmark the single- and multifluid radiative magnetohydrodynamics code Ebysus, used to study the Sun. The theory and numerical modeling were used to investigate the role of Spitzer conductivity in the transport of energy released in a nanoflare. Results. The derived analytical solution models nonlinear diffusivity accurately within its region of validity and approximately beyond. Various numerical schemes implemented in the Ebysus code is found to model Spitzer conductivity correctly. The energy from a representative nanoflare is found to diffuse 9 Mm within the first second of its lifetime due to Spitzer conductivity alone, strongly dependent on the electron density. Conclusions. The analytical first-order solution is a step forward in ensuring the physical validity of intricate simulations of the Sun. Additionally, since the derivation and argumentation are general, they can easily be followed to treat other nonlinear diffusion problems.

Exploring N-soliton solutions, multiple rogue wave and the linear superposition principle to the generalized hirota satsuma-ito equation

In this work, numerous rogue wave solution strategy (MRWSS) agreeing to the Hirota bilinear hypothesis is utilized. The said strategy to build different soliton wave solutions for the generalized Hirota-Satsuma-Ito (GHSI) condition is considered. These numerous soliton wave solutions incorporate first-order, second-order, third-order, and fourth-order waves solutions and so on, which are all depending on the starting theory of chosen capacities. For the lump solution with one most extreme or least esteem, the properties of the Hessian lattice are examined. Besides, the arrangements are contained the lump-two kink arrangements, cross-kink wave arrangements, periodic wave arrangements, and periodic-kink wave arrangements. The various classes of arrangements are gotten. We make utilize of the direct superposition procedure to find out N-soliton wave arrangements to the said condition. The appropriateness and viability of the obtained arrangements is detailed through the reenactment comes about within the shape of 3-D, density, and 2-D charts. The gotten comes about in this work are anticipated to open modern viewpoints for the traveling wave theory.

Far-field radiation patterns of distributed acoustic sensing in anisotropic media with an explosive source and vertically straight fiber

Distributed acoustic sensing (DAS) is increasingly used in seismic exploration owing to its wide frequency range, dense sampling and real-time monitoring. DAS radiation patterns help to understand angle response of DAS records and improve the quality of inversion and imaging. In this paper, we solve the 3D vertical transverse isotropic (VTI) Christoffel equation and obtain the analytical, first-order, and zero-order Taylor expansion solutions that represent P-, SV-, and SH-wave phase velocities and polarization vectors. These analytical and approximated solutions are used to build the P/S plane-wave expression identical to the far-field term of seismic wave, from which the strain rate expressions are derived and DAS radiation patterns are thus extracted for anisotropic P/S waves. We observe that the gauge length and phase angle terms control the radiating intensity of DAS records. Additionally, the Bond transformation is adopted to derive the DAS radiation patterns in title transverse isotropic (TTI) media, which exhibits higher complexity than that of VTI media. Several synthetic examples demonstrate the feasibility and effectiveness of our theory.

High-Order Models for Gas Transport in Multiscale Coal Rock.

The coal reservoirs exhibit great heterogeneity and strong anisotropy in multiscale pore/fracture structures. Developing highly accurate multiscale models for real-time prediction of microgaseous flow in complicated porous media with pronounced contrast in transport coefficients is crucial but not yet available, which is time-consuming, expensive, and even computationally impossible. In this study, a multiscale approximate solution of the gas flow and pressure field is derived in this paper for predicting coalbed methane (CBM) transport in macro-microscopic two-scale porous media of typical coal rocks in Guizhou Province, China, and the detailed finite element algorithm is established. The multiscale approximate solutions are constructed from the first- and second-order auxiliary cell functions and low- or high-order derivatives of the homogenized pressure field, which can accurately approximate the solution of the original problem to a certain extent. The numerical accuracy and validity of the multiscale approximate solutions under various working conditions are analyzed in detail by comparing and verifying the results with the direct numerical simulation results of fine-mesh high-precision finite elements. The results show that the homogenized approximate solution can only roughly capture the characteristics of the macroscopic pressure evolution, the first-order approximate solution can partially reproduce the macro-microscopic pressure oscillation phenomenon, and the second-order approximate solution can accurately predict the pressure profile and its evolution law with time in porous media with macro-micro configuration under various complex conditions. The second- and high-order two-scale approximate solutions constructed in this paper exhibit high numerical accuracy and excellent computational efficiency. We also develop multiscale approximate solutions for predicting microgaseous flow at a low reservoir pressure where the slippage effects are very pronounced. Potential applications of our high-order models to the coal reservoirs with a hierarchical matrix and fractures are discussed. The developed multiscale models exhibit excellent applicability to investigating the crossflow effects and coupling mechanisms between the matrix and cleats or fractures with multiple configurations in the CBM and shale gas reservoirs, which is crucial for the effective implementation of hydraulic fracturing technology. This study provides a fundamental understanding of gas transport in multiscale matrix-fracture networks, which helps to improve the optimization design of hydraulic fracturing in tight reservoirs.

Modulation instability and rogue waves for two and three dimensional nonlinear Klein-Gordon equation.

We perform the modulation instability analysis of the 2D and 3D nonlinear Klein-Gordon equation. The instability region depends on dispersion and wavenumbers of the plane wave. The N-breathers of the nonlinear Klein-Gordon equation are constructed directly from its 2N-solitons obtained in history. The regularity conditions of breathers are established. The dynamic behaviors of breathers of the 2D nonlinear Klein-Gordon equation are consistent with modulation instability analysis. Furthermore, by means of the bilinear method together with improved long-wave limit technique, we obtain general high order rogue waves of the 2D and 3D nonlinear Klein-Gordon equation. In particular, the first- and second-order rogue waves and lumps of the 2D nonlinear Klein-Gordon equation are investigated by using their explicit expressions. We find that their dynamic behaviors are similar to the nonlinear Schrödinger equation. Finally, the first-order rational solutions are illustrated for the 3D nonlinear Klein-Gordon equation. It is demonstrated that the rogue waves of the 2D and 3D nonlinear Klein-Gordon equation always exist by choosing dispersion and wavenumber of plane waves.

Mechanistic definition and prediction of the mass exchange coefficient between rivers and hyporheic zones: The [formula omitted] of two [formula omitted]s

Solute transport in interconnected rivers and hyporheic zones is typically modeled through dual-domain models where first-order solute mass transfer between the two domains, ΩR and ΩHZ, is represented by a coefficient α. The transient storage model (TSM) is an example of such an approach. In practice, α is determined by fitting the tails of solute tracer breakthrough curves using a TSM. This approach has led to ambiguity regarding α’s physical meaning and transferability. Here, we investigated the physical basis for α and tested it with virtual experiments through the fully coupled multiphysics model hyporheicFoam for the ΩR−ΩHZ system. hyporheicFoam explicitly simulated coupled flow and solute transport over a kilometer with centimeter-scale resolution. Model results were analyzed to calculate α following its theoretical definition directly. Using the determined α within a TSM enables accurate reproduction of solute transport, underscoring α’s physical relevance and precision.

The Effect of Proportional, Proportional-Integral, and Proportional-Integral-Derivative Controllers on Improving the Performance of Torsional Vibrations on a Dynamical System

The primary goal of this research is to lessen the high vibration that the model causes by using an appropriate vibration control. Thus, we begin by implementing various controller types to investigate their impact on the system’s reaction and evaluate each control’s outcomes. The controller types are presented as proportional (P), proportional-integral (PI), and proportional-integral-derivative (PID) controllers. We employed PID control to regulate the torsional vibration behavior on a dynamical system. The PID controller aims to increase system stability after seeing the impact of P and PI control. This kind of control ensures that there are no unstable components in the system. By using the multiple time scale perturbation (MTSP) technique, a first-order approximate solution has been obtained. Using the frequency response function approach, the stability and steady-state response of the system at the primary resonance scenario (Ω1≅ω1,Ω2≅ω2) are considered as the worst resonance and addressed. Additionally examined are the nonlinear dynamical system’s chaotic response and the numerical solution for various parameter values. The MATLAB programs are utilized to attain simulation outcomes.

Approximate analytical solution using power series method for the propagation of blast waves in a rotational axisymmetric non-ideal gas

In this paper, the propagation of the blast (shock) waves in non-ideal gas atmosphere in rotational medium is studied using a power series method in cylindrical geometry. The flow variables are assumed to be varying according to the power law in the undisturbed medium with distance from the axis of symmetry. To obtain the similarity solution, the initial density is considered as constant in the undisturbed medium. Approximate analytical solutions are obtained using Sakurai’s method by extending the power series of the flow variables in power of a0U2, where U and a0 are the shock velocity and speed of sound, respectively, in undisturbed fluid. The strong shock wave is considered for the ratio a0U2 which is considered to be a small quantity. With the aid of that method, the closed-form solutions for the zeroth-order approximation is given as well as first-order approximate solutions are discussed. Also, with the help of graphs behind the blast wave for the zeroth-order, a comparison between the numerical solution and the approximate analytical solution are shown. The distributions of flow variables such as density, radial velocity, pressure and azimuthal velocity are analysed. The results for the rotationally axisymmetric non-ideal gas environment are compared to those for the ideal gas atmosphere. The velocity-distance and time-distance curves are also shown to analyse the decaying characteristic of a blast wave.

First-order compartment model solutions – Exponential sums and beyond

First-order compartment models are common tools for modelling many biological processes, including pharmacokinetics. Given the compartments and the transfer rates, solutions for the time-dependent quantity (or concentration) curves can normally be described by a sum of exponentials. This paper investigates cases that go beyond simple sums of exponentials. With specific relations between the transfer rate constants, two exponential rate constants can be equal, in which case the normal solution cannot be used. The conditions for this to occur are discussed, and advice is provided on how to circumvent these cases. An example of an analytic solution is given for the rare case where an exact equality is the expected result. Furthermore, for models with at least three compartments, cases exist where the solution to a real-valued model involves complex-valued exponential rate constants. This leads to solutions with an oscillatory element in the solution for the tracer concentration, i.e., there are cases where the solution is not a simple sum of (real-valued) exponentials but also includes sine and cosine functions. Detailed solutions for three-compartment cases are given. As a tentative conclusion of the analysis, oscillatory solutions appear to be tied to cases with a cyclic element in the model itself.

Integrability, and stability aspects for the non-autonomous perturbed Gardner KP equation: Solitons, breathers, [formula omitted]-type resonance and soliton interactions

This article examines the soliton-type solutions, their interactions, and the integrable properties of a non-autonomous perturbed Gardner KP (NPGKP) equation. For the NPGKP equation under consideration, a bilinear structure, and a Bäcklund transformation are designed explicitly, which claim the integrability of the system under some constraints. The stability of the obtained solutions is discussed using modulation instability. The bilinear form demonstrates the dynamic characteristics of multiple solitons, breathers, and their interactions in response to external impulses. Furthermore, it allows for determining the solitons’ amplitudes and velocities. The two-soliton solution yields a first-order breather solution. At the same time, the analytical investigation focuses on the interaction between a single breather and a single-soliton within the multi-soliton solution. This investigation also identifies the resonance of Y-shaped solitons and examines the dynamical characteristics of the interaction between resonant Y-shaped solitons and M-fissionable pulses. The multi-shock solutions and the collisions between shocks are analyzed in the presence of external influences. Graphical representations of the relationships between different sorts of achieved solutions are provided explicitly.

A Volume-of-Fluid method for multicomponent droplet evaporation with Robin boundary conditions

We propose a numerical method tailored to perform interface-resolved simulations of evaporating multicomponent two-phase flows. The novelty of the method lies in the use of Robin boundary conditions to couple the transport equations for the vaporized species in the gas phase and the transport equations of the same species in the liquid phase. The Robin boundary condition is implemented with the cost-effective procedure proposed by Chai et al. [1] and consists of two steps: (1) calculating the normal derivative of the mass fraction fields in cells adjacent to the interface through the reconstruction of a linear polynomial system, and (2) extrapolating the normal derivative and the ghost value in the normal direction using a linear partial differential equation. This methodology yields a second-order accurate solution for the Poisson equation with a Robin boundary condition and a first-order accurate solution for the Stefan problem. The overall methodology is implemented in an efficient two-fluid solver, which includes a Volume-of-Fluid (VoF) approach for the interface representation, a divergence-free extension of the liquid velocity field onto the entire domain to transport the VoF, and the temperature equation to include thermal effects. We demonstrate the convergence of the numerical method to the analytical solution for multicomponent isothermal evaporation and observe good overall computational performance for simulating non-isothermal evaporating two-fluid flows in two and three dimensions.

Psychometric assessment of the 10-item Thai version of the Experience in Close Relationship-Revised for Adolescents (ECR-R-10-AD)

The 18-item version of the Experiences in Close Relationships-revised (ECR-R-18) is a valid and reliable scale used among Thai adolescents. However, it revealed problematic items that impacted the scale’s internal consistency. The study aimed to achieve two objectives: (1) develop a new, shorter scale by retaining only highly loaded items equally between attachment anxiety and attachment avoidance, and (2) evaluate the psychometric properties of the shorter ECR-R version compared to the existing 18-item scale. Objective 1 was achieved through Study 1, involving 204 youths aged 16–18 years (64% female). All participants completed the 18-item ECR-R, and exploratory factor analysis was conducted to identify suitable items for the new ECR-R-AD. Objective 2 was fulfilled in Study 2, which included a total of 443 students in grades aged 15–18 years old (88% female) from Thai boarding schools in Northern Thailand. All participants completed both the 18-item ECR-R, and confirmatory factor analysis of both the existing 18-item and the new shorter scale was performed and compared. Additional measures including the Rosenberg Self-Esteem Scale, Perceived Stress Scale-10, and Relationship Questionnaire were completed alongside the ECR-R to assess convergent, discriminant, and criterion validity. The invariance test for the new ECR-R across genders was conducted using multigroup confirmatory factor analysis. For objective 1, Study 1 developed a new scale called "ECR-R-10-AD" with 10 items, comprising 5 for attachment anxiety and 5 for attachment avoidance. The McDonald’s omega values were 0.866 for avoidance and 0.823 for anxiety subscales. The corrected correlation between the ECR-R-18 and ECR-R-10-AD was significant. For objective 2, Study 2 found that the first-order two-factor solution model fit the data best for the ECR-R-10-AD. Convergent, discriminant, and criterion validity with other measurements and invariance tests based on sex were established for the ECR-R-AD. The ECR-R-10-AD provided sufficient psychometric properties among Thai adolescents. Factorial validity, convergent validity, and measurement invariance were established. As the ECR-R-10-AD is brief, it can be administered with less burden. Limitations and future research were discussed.

Error analysis of asymptotic solution of a heavy particle motion equation in fluid flows

For weakly inertial particles subjected to volumetric forces and Stokes drag force in fluid flows, we can solve the simplified particle motion equation using the perturbation method. This method allows us to obtain a recursive formula for the nth-order correction of the asymptotic solution of particle velocity. We verified the error of the asymptotic solution under two typical flow fields: a time-varying uniform flow field with a volumetric force field and a two-dimensional non-uniform cellular flow field. In the former, the relative error of the asymptotic solution of particle velocity and position increases with the Stokes number, and we provided a quantitative analysis of the results. In the latter, we verify and analyze the asymptotic solution from two perspectives: the behavior of a single particle and the collective behaviors of many particles. For asymptotic solutions with maximum velocity and position errors of less than 5%, we select the solution with the lowest order correction and designate it as the optimal asymptotic solution. The order of the optimal asymptotic solution increases with increasing Stokes numbers and motion durations. However, in most cases, for weakly inertial particles [St ∼ O(10−3)], and the time t* ∼ O(10), the first-order asymptotic solution can achieve accuracy, where both St and t* are defined using the flow field characteristic time, Tf = 4π s. The results validate the rationale behind utilizing first-order asymptotic solutions in the fast Eulerian method for turbulent dispersion of weakly inertial particles.

Influence of the CNT addition on the mechanical performance of three-phase composite plates

This work has analyzed the bending, free vibration and buckling performance of a three-phase (CNT/Polymer/Carbon fiber) composite plate as a function of the carbon fiber volume fraction and the CNT volume fraction, under several values of the plate aspect ratio (a/b) and the plate slenderness parameter (h/b), and considering the presence of defects due to the poor dispersion on the CNTs. Based on a two-steps homogenization technique, the mechanical properties of the carbon fibers reinforced –CNT enriched– polymeric matrix are computed. Then, the mechanical response of the plate is computed according to the first-order shear deformation plate theory solutions presented in Reddy’s books. The numerical results have revealed the presence, in three-phase (CNT/Polymer/Carbon fiber) composite plates, of a CNT volume fraction threshold above which the bending, free vibration and buckling performance of the plate decay with the carbon fiber volume fraction.

Viscoelastic Hertzian Impact

The problem of normal impact of a rigid sphere on a Maxwell viscoelastic solid half-space is considered. The first-order asymptotic solution is constructed in the framework of Hunter’s model of viscoelastic impact. In particular, simple analytical approximations have been derived for the maximum contact force and the time to achieve it. A linear regression method is suggested for evaluating the instantaneous elastic modulus and the mean relaxation time from a set of experimental data collected for different spherical impactors and impact velocities.

Challenging beyond-the-standard-model solutions to the fine-structure anomaly in heavy muonic atoms

The leading-order contribution of a new boson to the muonic fine-structure anomaly, which refers to a discrepancy between the predicted transition energies and spectroscopic measurements of μ−90Zr, μ−120Sn, and μ−208Pb, is investigated. We consider bosons of scalar, vector, pseudoscalar, and pseudovector type. Spin-dependent couplings sourced by pseudoscalars or pseudovectors are disfavoured as solutions to the anomaly due to the nuclei in question having vanishing angular momentum. Spin-independent interactions resulting from scalar or vector exchange are also disfavoured because no parameter space exists to simultaneously fit different atomic states of the same nucleus. Therefore, we conclude that a ‘Beyond-the-Standard-Model’ resolution of the muonic fine-structure anomaly is generally disfavoured, and the first-order solution by a single new boson is excluded.

Baseband modulational instability and interacting localized mixed waves in coherently coupled optical media

We consider a coherently coupled nonlinear Schrödinger equations (model equations) that describe interacting localized waves in coherently coupled optical media. The baseband modulational instability in such regimes is investigated, leading the result that the modulational instability does not necessary imply the baseband modulational instability. By means of a set of linear transformations, the model equations are converted into two independent scalar nonlinear Schrödinger equations. Based on the generalized perturbation (n,N−n)–fold Darboux transformation, we report and discuss analytical mixed localized wave solutions of these scalar equations. A first-order rogue wave solution of the scalar nonlinear Schrödinger equations is also presented. We then show that linear superpositions of different found analytical mixed localized wave solutions of the derived scalar equations results into several kinds of coherent nonlinear mixed structures such as, coexisting bright/dark soliton-breather, rogue wave-breather-soliton, interacting rogue waves, bright/dark breathers, and bright/dark solitons.

Psychometric properties of the Italian version of the Occupational Fatigue Exhaustion/Recovery (OFER) scale among healthcare professionals

ABSTRACT Objective The purpose of this study was to analyze the psychometric properties of the Italian version of the Occupational Fatigue Exhaustion/Recovery (OFER). Methods A cross-sectional study among 434 healthcare workers was undertaken. The factorial validity of the OFER, was compared using standard confirmatory factor analysis (CFA) with exploratory structural equation modeling (ESEM). Internal consistency, convergent validity, and factorial invariance (across gender) were assessed. Results A three-factor first-order ESEM solution provided the best and most parsimonious representation of the data: χ2(63) = 212.81, p < .001, RMSEA = 0.074 (90% CI 0.063–0.085), CFI = 0.949, TLI = 0.915. The three-factors were chronic fatigue, acute fatigue, and intershift recovery. Chronic and acute factors showed excellent internal consistency (ω > 0.80), however intershift recovery showed less internal consistency (ω < 0.70). OFER scores were found to be positively correlated with FAS scores, supporting its convergent validity. Conclusions The OFER was factorially invariant across gender. Overall, our findings provided evidence for the satisfactory psychometric properties of the OFER among Italian workers, confirming its value for use in both practical occupational and academic research settings.

A multiple-timing analysis of temporal ratcheting

We develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and αω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\omega $$\\end{document}, where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a rational number. If α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a ratio of odd and even integers (e.g., 21,32,43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{2}{1},\\,\ frac{3}{2},\\,\ frac{4}{3}$$\\end{document}), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document}. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for α=32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =\ frac{3}{2}$$\\end{document} and 43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{4}{3}$$\\end{document} appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems.Graphic abstract

Riemann–Hilbert approach for the inhomogeneous discrete nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable κ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.