Self-similar Method Research Articles

A new non-diffractive wave packet: The Scorer beam

We find the exact solution of the (1+1)-dimensional optical paraxial diffractive system by implementing the self-similarity method, leading to the discovery of a beam solution composed of Scorer functions, which we refer to as the Scorer beam. Through theoretical analysis and numerical simulation, we explore several characteristics of both infinite- and finite-energy Scorer beams, particularly focusing on their non-diffractive and self-accelerating properties. Additionally, we compare the Scorer beam with – and distinguish from – the related Airy beam. This investigation provides both theoretical and numerical insights that may guide the experimental realization of Scorer beams.

Diffusion Cascades and Mutually Coupled Diffusion Processes

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer’s function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay.

Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems

In this paper, a family of exact solutions to N-coupled nonlinear Schrödinger (N-CNLS) equations including bright and dark solitons, Akhmediev breathers, Kuznetsov-Ma breathers, and rogue waves are derived with the aid of a generic nonlinear wave assumption. As an application of the solutions, we present a scheme to realize amplitude coding in an identical waveguide system. Furthermore, by a general self-similar transformation method, exact controllable similaritons of N-coupled inhomogeneous NLS (N-CINLS) equations are constructed under relaxed compatibility conditions. Flexible modulations of similaritons are investigated in a typical inhomogeneous system. Based on the general self-similar transformation, we further study various and novel modulations of composite similariton of 2-CINLS regimes. The various similaritons presented here may be expected to find application in the control and transmission of nonlinear waves in realizable complex systems including those based on optical fibers and Bose-Einstein condensates.

A numerical study of heat and mass transfer characteristic of three-dimensional thermally radiated bi-directional slip flow over a permeable stretching surface

Within fluid mechanics, the flow of hybrid nanofluids over a stretching surface has been extensively researched due to their influence on the flow and heat transfer properties. Expanding on this concept by introducing porous media, the current study explore the flow and heat and mass transport characteristics of hybrid nanofluid. This investigation includes the effect of magnetohydrodynamic (MHD) with chemical reaction, thermal radiation, and slip effects. The nanoparticles, copper, and alumina are combined with water for the formation of a hybrid nanofluid. Using the self-similar method for the reduction of Partial differential equations (PDEs) to the system of Ordinary differential equations (ODEs). These nonlinear equation systems are solved numerically using the bvp4c (boundary value solver) technique. The effect of the different physical non-dimensional flow parameters on different flow profiles such as velocity, temperature, concentration, skin friction, Nusselt and mass transfer rate are depicted through graphs and tables. The velocity profiles diminish with the effect of magnetic and slip parameters. The temperature and concentration slip parameters reduce the temperature and concentration profile respectively. The higher values of magnetic factor lessened the skin friction coefficient for both slip and no-slip conditions. An elevation in the thermal slip parameter reduced the boundary layer thickness and the heat transfer from the surface to the fluid. The Nusselt number amplified with the climbing values of the radiation parameter. The mass transfer rate depressed with the solutal slip parameter. Comparison is made with the published work in the literature and there is excellent agreement between them.

Analysis of High-Order Bright–Dark Rogue Waves in (2+1)-D Variable-Coefficient Zakharov Equation via Self-Similar and Darboux Transformations

This paper conducts an in-depth study on the self-similar transformation, Darboux transformation, and the excitation and propagation characteristics of high-order bright–dark rogue wave solutions in the (2+1)-dimensional variable-coefficient Zakharov equation. The Zakharov equation is instrumental for studying complex nonlinear interactions in these areas, with specific implications for energy transfer processes in plasma and nonlinear wave propagation systems. By analyzing bright–dark rogue wave solutions—phenomena that are critical in understanding high-energy events in optical and fluid environments—this research elucidates the intricate dynamics of energy concentration and dissipation. Using the self-similar transformation method, we map the (2+1)-dimensional equation to a more tractable (1+1)-dimensional nonlinear Schrödinger equation form. Through the Lax pair and Darboux transformation, we successfully construct high-order solutions that reveal how variable coefficients influence rogue wave features, such as shape, amplitude, and dynamics. Numerical simulations demonstrate the evolution of these rogue waves, offering novel perspectives for predicting and mitigating extreme wave events in engineering applications.This paper crucially advances the practical understanding and manipulation of nonlinear wave phenomena in variable environments, providing significant insights for applications in optical fibers, atmospheric physics, and marine engineering.

Thermal Effects of Ambipolar Diffusion during the Gravitational Collapse of a Radiative Cooling Filament

In this study, we consider the effects of ambipolar diffusion during the gravitational collapse of a radiative cooling filamentary molecular cloud. Two separate configurations of magnetic field, i.e., axial and toroidal, are considered in the presence of the ambipolar diffusion for a radiative cooling filament. These configurations lead to two different formulations of the problem. The filament is radiatively cooled and heated by ambipolar diffusion in both cases of magnetic field configurations. The self-similar method is used to solve the obtained equations in each case. We found that the adiabatic exponent and ambipolar diffusivity play very important roles during the gravitational collapse of a cooling filament. The results show that the ambipolar heating significantly increases the temperature in the middle regions of a cooling filament. Furthermore, we found that the ambipolar diffusion has very important effects during the collapse, so that its heating effect is dominant over its dynamical effect in the middle regions of a cooling filament. The obtained results also address some regions where the rate of star formation is more or less compared to the observational reports.

Coupled models for propagation of explosive shock waves in cylindrical and spherical geometries

The propagation of shock waves in different geometries is crucial in engineering and scientific applications. A comprehensive model is developed to elucidate the hydrodynamic growth and decay of shock waves in cylindrical and spherical geometries by using the strong shock wave assumption. This model takes into consideration the conservation equations governing mass, momentum, and energy, thereby allowing for an accurate description of the coupled behavior between the piston and shock wave propagation. In contrast to the localized analysis employed in previous self-similar methods, this model incorporates the finite sound wave velocity to introduce the concept of retarded pressure on the piston surface. Consequently, the proposed model offers a multitude of advantages by providing a complete set of dynamic information concerning the trajectories, velocities, and accelerations of both the piston and shock wave. Furthermore, an asymptotic analytical solution is derived to describe the decay of shock waves in cylindrical and spherical geometries. To validate the theoretical analysis and illustrate the propagation characteristics of shock waves in these specific geometries, thorough comparisons are conducted. These findings contribute to the advancement of our understanding of shock wave dynamics in various physical systems, particularly in the field of plasma physics.

Classification and wall heat transfer of non-ideal CO2 isothermal laminar boundary layers

The heat transfer properties of non-ideal fluids are crucial in engineering applications. In order to investigate the wall heat transfer characteristics, the self-similar method has been employed to solve supercritical carbon dioxide (SCO2) isothermal laminar boundary layers. The temperature profile regimes are classified into 14 different categories based on their monotonicity and position relative to the pseudo-critical temperature, Tpc. For no trans-Tpc, the boundary layer exhibits subcritical liquid or supercritical vapor, being liquid-like or gas-like region. When free-stream temperature and wall temperature are both lower than Tpc at a large Mach number (Ma), the temperature profile intersects the Tpc twice. In this scenario, the boundary layer is composed of liquid layer, supercritical vapor layer and liquid layer adjacent to the wall, and the wall heat transfer is determined by the thickness of the two bottom layers. At once trans-Tpc, it is seen that the formation of supercritical vapor or liquid film due to pseudo-boiling/condensation plays a key role in the heat transfer deterioration or enhancement. Mainly related to the film thickness and the pseudo-boiling/condensation intensity, the influence of wall temperature and free-stream pressure on the wall heat transfer has been explored by introducing a new dimensionless number Nu*.

Soliton cluster solutions of nonlinear Schrödinger equations with variable coefficients in Bessel lattice

The goal of this article is to obtain soliton cluster solutions of (2 + 1)-dimensional nonlinear variable coefficient Schrödinger equations through computerized symbolic computation. By applying the self-similarity method, one soliton solution is constructed for the NLS equations with variable coefficient, which provide a model of the propagation of the soliton waves. Consequently, the solitonary cluster solution is achieved in different structures. Additionally, the propagation of the obtaining solitonary cluster solutions is analyzed and discussed. The results are useful to explain the soliton phenomena in nonlinear optics.

Noise masking of near-surface scattering (heterogeneities) on subsurface seismic reflectivity

Abstract Near-surface velocity variations are the main cause of seismic scattering in exploration seismology. Many studies create the near-surface heterogeneity as velocity models that have random velocity distribution, random objects, or irregular subsurface topography to study and mitigate the resultant scattering effects of the near-surface layer. Von Kármán (self-similar) method is a known method in the literatures for modeling heterogeneous earth in a statistical way. This research modifies the self-similar method, and throughout the work, it has proven that the self-similar provides a robust method for generating realistic near-surface velocity models with different spatial velocity distributions. This study creates four-velocity models with simple subsurface layering and structure, three of which include a near-surface layer in three different degrees of velocity heterogeneity. Synthetic acoustic seismic reflections are produced for the four-velocity models to investigate the resultant scattering effects of the near-surface velocity heterogeneity on the quality of seismic waveform coherency. Spectacular negative observations are witnessed of the near-surface layer involvement to the quality of seismic reflection coherency that increases as velocity dramatically varies. Subtracting the scattering noise, which is modeled using an exact heterogeneous model, enhances seismic reflection coherency for the subsurface layers, but waveforms that are affected by scattering must be reconstructed for true amplitude and seismic waveform analysis.

Dynamical behavior of ambipolar diffusion during the gravitational collapse of a radiative cooling filamentary cloud

This work aims to investigate the dynamic behavior of ambipolar diffusion during the gravitational collapse of a filamentary molecular cloud undergoing radiative cooling. The cloud undergoes collapse in the presence of the ambipolar diffusion effect while being cooled by radiation. According to the observations of CO emissions, the cooling rate is scaled as a function of density and temperature. Furthermore, in the presence of ambipolar diffusion, the profile of the toroidal magnetic field is dominated by the induction equation. The self-similar method is employed to solve the derived equations. The problem is highly sensitive to two parameters, namely the polytropic exponent γ and the ambipolar diffusivity. Accordingly, we consider the problem in three cases of the polytropic exponent. We have found that the presence of ambipolar diffusion results in a decrease in the flux-to-mass ratio at the outer regions of a cooling filament for all cases of γ. However, the temperature experiences a critical increase for cases where γ>0.8, leading to the cessation of self-similar collapse in the central regions of the cloud. The obtained results are fully consistent with previous studies regarding the star formation rate in the case of γ>0.8. Finally, based on the results of this study, it is possible to differentiate between hot and cold gravitational collapse in a radiative cooling filament.

A numerical algorithm for ground reaction curves of circular openings in strain-softening rock masses satisfying the generalized Hoek-Brown criterion

Based on the self-similarity method, a numerical algorithm for constructing ground reaction curves (GRC) of circular openings in elastic-strain softening rock mass that satisfies the generalized Hoek-Brown failure criterion is proposed. The algorithm is derived from the analysis of the axisymmetric plane strain model, where the far-field and excavation boundary are both axisymmetric stresses. Using the self-similar transformation, the plastic zone of the model was transformed into unit plane ring, which has unit radius at the boundary between the plastic zone and the elastic zone. The unit plane ring was dispersed into multiple concentric rings, for which the Runge-Kutta discrete equations of the stress and deformation were established based on corresponding self-similar governing equations. The equations were solved ring by ring in a successive manner, starting from the outer boundary to the excavation boundary. The comparison with the verified algorithm in literature shows that the results of the different algorithms are in good agreement and the relative errors of the plastic zone radius under various softening parameters are only 1.7%–2.0%. These results verified the correctness and reliability of the proposed new algorithm. Finally, the new algorithm is used to investigate the influence of the Hoek-Brown parameter a on GRC. A discussion of the potential capability of practical applications of this algorithm is also provided.

Study of the Wear Resistance of a Friction unit with a Non-Standard Support Profile and a Metal Coating

The theoretical calculations and the results of experiment for a friction unit consisting of a wedge-shaped support surface are presented in this research paper. The slider on the tread surface has a metal alloy. The lubricating fluid is represented by a lubricant with non-Newtonian characteristics; it has the laminar flow in this unit. The friction unit functions when the cavity between the slider and the support ring is completely filled. To solve the theoretical problem under given conditions, the equations of continuity, the rate of mechanical energy dissipation, the values of the expression for the displacement of a non-Newtonian fluid were used. In addition, equations were used to characterize the profile of a slider with a melt on the tread surface. The presented problem was solved by the self-similar variable method with asymptotic expansion in the smallest parameter. Upon calculation, it is given the equations for the first and zero pressure approximations and it is determined the velocity field are determined with and without the molten metal. Further, the friction force and load capacity were found using the non-Newtonian properties of the lubricating fluid and the above structural characteristics of non-standard surfaces for friction contact.

Calculation of the Free Energy of the Ising Model on a Cayley Tree via the Self-Similarity Method

In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy of the Ising model having the prolonged next nearest and nearest neighbor interactions and external field on a two-order Cayley tree using the self-similarity of the semi-infinite Cayley tree. The phase transition problem for the Ising system is investigated under the given conditions.

Self-similar diffuse boundary method for phase boundary driven flow

Interactions between an evolving solid and inviscid flow can result in substantial computational complexity, particularly in circumstances involving varied boundary conditions between the solid and fluid phases. Examples of such interactions include melting, sublimation, and deflagration, all of which exhibit bidirectional coupling, mass/heat transfer, and topological change of the solid–fluid interface. The diffuse interface method is a powerful technique that has been used to describe a wide range of solid-phase interface-driven phenomena. The implicit treatment of the interface eliminates the need for cumbersome interface tracking, and advances in adaptive mesh refinement have provided a way to sufficiently resolve diffuse interfaces without excessive computational cost. However, the general scale-invariant coupling of these techniques to flow solvers has been relatively unexplored. In this work, a robust method is presented for treating diffuse solid–fluid interfaces with arbitrary boundary conditions. Source terms defined over the diffuse region mimic boundary conditions at the solid–fluid interface, and it is demonstrated that the diffuse length scale has no adverse effects. To show the efficacy of the method, a one-dimensional implementation is introduced and tested for three types of boundaries: mass flux through the boundary, a moving boundary, and passive interaction of the boundary with an incident acoustic wave. Two-dimensional results are presented as well these demonstrate expected behavior in all cases. Convergence analysis is also performed and compared against the sharp-interface solution, and linear convergence is observed. This method lays the groundwork for the extension to viscous flow and the solution of problems involving time-varying mass-flux boundaries.

Multi-Spacecraft Observations of an Interplanetary Coronal Mass Ejection Interacting with Two Solar-Wind Regimes Observed by the Ulysses and Twin-STEREO Spacecraft

We present a combined study of a coronal mass ejection (CME), revealed in a unique orbital configuration that permits the analysis of remote-sensing observations on 27 June 2007 from the twin Solar Terrestrial Relations Observatory (STEREO)-A and -B spacecraft and of its subsequent in situ counterpart outside the ecliptic plane, the interplanetary coronal mass ejection (ICME) observed on 04 July 2007 by Ulysses at 1.5 AU and heliographic-Earth-ecliptic coordinates system (HEE) 33° latitude and 49° longitude. We apply a triangulation method to the STEREO Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) COR2 coronagraph images of the CME, and a self-similar expansion fitting method to STEREO/SECCHI Heliospheric Imager (HI)-B. At Ulysses we observe: a preceding forward shock, followed by a sheath region, a magnetic cloud, a rear forward shock, followed by a compression region due to a succeeding high-speed stream (HSS) interacting with the ICME. From a minimum variance analysis (MVA) and a length-scale analysis we infer that the magnetic cloud at Ulysses, with a duration of 24 h, has a west-north-east configuration, length scale of ≈0.2 AU, and mean expansion speed of 14.2 km s−1. The relatively small size of this ICME is likely to be a result of its interaction with the succeeding HSS. This ICME differs from the previously known over-expanding types observed by Ulysses, in that it straddles a region between the slow and fast solar wind that in itself drives the rear shock. We describe the agreements and limitations of these observations in comparison with 3D magneto-hydrodynamic (MHD) heliospheric simulations of the ICME in the context of a complex solar-wind environment.

Controllable optical rogue waves in inhomogeneous media

This paper investigates new rogue wave solutions of the nonlinear Schrödinger equation with variable coefficients, utilizing the self-similar transformation method. A new rogue wave family is introduced, which displays different wave structures. When one chooses appropriate variable coefficients, a series of first-order, second-order, and third-order rogue waves are obtained. We explore the generation and interaction of rational polynomial rogue waves in detail, and discuss some characteristics of their structures. The paper provides theoretical basis for studying the dynamic behavior of optical rogue wave in inhomogeneous media, and presents potential application value for control of rogue waves in fibers, plasmas and other fields of physics.

Water surge impingement onto a vertical wall: A new self-similarity solution for impact pressure

The impingement process of water surge onto a vertical wall and the impact pressure are studied analytically in this work. We propose a new initial-boundary value problem particularly for the fluid motion near the corner of the horizontal bed and the vertical wall. The explicit solutions of the velocity and the pressure fields are analytically obtained using the self-similarity method under some verifiable physical assumptions. The impact pressure is found to be proportional to the product of the squared incident surge front velocity and the density of water, with a constant coefficient of around 0.867. We compare the analytical solution of the impact pressure with some existing laboratory data. The analytical solution agrees with the median value of the stochastic data of impact pressure from laboratory experiments. Subsequently, the velocity and the pressure fields from the analytical model are compared to the numerical simulation results based on OpenFOAM. The comparisons validate the physical assumptions made in the analytical derivation, demonstrating fair consistency. The analytical model successfully describes the early stage of the contact process between the surge front and the wall and provides a theoretical basis for the physics of water surge impingement.

МЕТОД ВЫЯВЛЕНИЯ АНОМАЛИЙ В СЕТЕВОМ ТРАФИКЕ

Introduction. Computer networks (CN) are highly developed systems with a multi-level hierarchical structure. The use of information and communication technologies in the CN to collect information allows an attacker to influence networks through cyber-attacks. This is facilitated by the massive use of outdated operating systems, ineffective protection mechanisms and the presence of multiple vulnerabilities in unsecured network protocols. Such vulnerabilities help a potential attacker to change the settings of network devices, listen and redirect traffic, block network interaction and gain unauthorized access to the internal components of the CN. The impact of cyber-attacks leads to the appearance of abnormal traffic activity in the CN. For its constant monitoring and detection in the CN, it is necessary to take into account the presence of a large number of network routes, on which sharp fluctuations in data transmission delays and large packet losses periodically occur, new properties of network traffic appear, which requires ensuring high quality of application service. All this served as an incentive to search for new methods of detecting and predicting cyber-attacks fractal analysis can also be attributed to them. The aim of the work is to develop a conceptual method for detecting anomalies caused by cyber-attacks in network traffic through the use of fractal analysis. Methods used. The main provisions of the fractal theory and the use of self-similarity assessment methods proposed by this theory, such as the extended Dickey-Fuller test, R/S analysis and the DFA method, are applied. When testing fractal methods that allow conducting studies of long-term dependencies in network traffic. The scientific novelty lies in the fact that the proposed method correctly identifies anomalies caused by the impact of cyber-attacks, and also allows you to predict and detect both known and unknown computer attacks at an early stage of their manifestation. Practical significance. The presented methodology can be used as an early detection system for cyber-attacks, based on the detection of anomalies in network traffic and the adoption of effective measures to protect the network.

Propagation of cylindrical shock waves in rotational axisymmetric dusty gas with magnetic field: Isothermal flow

In this article, the effect of the dust particles is studied on the propagation of a cylindrical shock wave in rotational axisymmetric ideal gas under isothermal flow conditions with the magnetic field. Here, magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are supposed to vary according to a power law in the undisturbed medium. With the help of Sakurai's technique, we obtain approximate solutions analytically by expanding the flow parameters in the form of a power series in ϕ=(CV)2. The power series method is used to derive the zeroth and the first-order approximations. The solutions for the zeroth-order approximation are constructed in analytical form. Distributions of the hydrodynamical quantities are analyzed graphically behind the shock front. Also, the effects of shock Cowling number (co), mass fraction of the solid particles in the mixture (kp), adiabatic exponent (γ), and rotational parameter (L) on the flow variables are discussed. It is investigated that the density and pressure near the line of symmetry in the case of isothermal flow become zero, and hence a vacuum is formed at the axis of symmetry when the flow is isothermal. The present work may be used to verify the correctness of the solution obtained by self-similarity and numerical methods. Furthermore, the results obtained in the present work are found to be in good agreement with those obtained from the study by Nath and Singh [Can. J. 98, 1077 (2020)].