Self-similar Approach Research Articles

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

Beyond stiffness: Multiscale viscoelastic features as biomechanical markers for assessing cell types and states

Cell mechanics are pivotal in regulating cellular activities, diseases progression, and cancer development. However, the understanding of how cellular viscoelastic properties vary in physiological and pathological stimuli remains scarce. Here, we develop a hybrid self-similar hierarchical theory-microrheology approach to accurately and efficiently characterize cellular viscoelasticity. Focusing on two key cell types associated with livers fibrosis—the capillarized liver sinusoidal endothelial cells and activated hepatic stellate cells—we uncover a universal two-stage power-law rheology characterized by two distinct exponents, αshort and αlong. The mechanical profiles derived from both exponents exhibit significant potential for discriminating among diverse cells. This finding suggests a potential common dynamic creep characteristic across biological systems, extending our earlier observations in soft tissues. Using a tailored hierarchical model for cellular mechanical structures, we discern significant variations in the viscoelastic properties and their distribution profiles across different cell types and states from the cytoplasm (elastic stiffness E1 and viscosity η), to a single cytoskeleton fiber (elastic stiffness E2), and then to the cell level (transverse expansion stiffness E3). Importantly, we construct a logistic-regression-based machine-learning model using the dynamic parameters that outperforms conventional cell-stiffness-based classifiers in assessing cell states, achieving an area under the curve of 97% vs. 78%. Our findings not only advance a robust framework for monitoring intricate cell dynamics but also highlight the crucial role of cellular viscoelasticity in discerning cell states across a spectrum of liver diseases and prognosis, offering new avenues for developing diagnostic and therapeutic strategies based on cellular viscoelasticity.

Prescaling Relaxation to Nonthermal Attractors.

We study how isotropic and homogeneous far-from-equilibrium quantum systems relax to nonthermal attractors, which are of interest for cold atoms and nuclear collisions. We demonstrate that a first-order ordinary differential equation governs the self-similar approach to nonthermal attractors, i.e., the prescaling. We also show that certain natural scaling-breaking terms induce logarithmically slow corrections that prevent the scaling exponents from reaching the constant values during the system's lifetime. We propose that, analogously to hydrodynamic attractors, the appropriate mathematical structure to describe such dynamics is the transseries. We verify our analytic predictions with state-of-the-art 2PI simulations of the large-N vector model and QCD kinetic theory.

Tectonic Duets: Self-Similar Approach to Investigate Eastern Anatolian Fault’s Recent Seismicity, with Special Emphasis on the 6 February 2023 Earthquake Doublet

Abstract Earthquake clusters possess profound potential for discerning antecedent seismic cues. This study examines the self-similarity of earthquakes to characterize recent seismic patterns in the prolonged quiescent Eastern Anatolian fault zone (EAFZ). We thoroughly investigate the correlation fractal dimension (Dc) formulated upon the scale-invariant relative clustering in earthquake pattern identification. We provide a comprehensive examination of pre- and postseismicity patterns of the Mw 7.7 Nurdağı-Pazarcık, Mw 7.6 Ekinözü, and Mw 6.7 Elazığ earthquakes, each shattering different segments of the EAFZ. The spatiotemporal fluctuations of Dc suggest the commencement of the preparatory process observed around October 2021 to February 2022 for the 2023 dual catastrophe, whereas in the case of the Mw 6.7 Elazığ, this was witnessed from November 2018. Prior to the 2023 events, low-moderate Dc regions predominated on the Pazarcık segment and Cardak fault. The Pürtürge segment that ruptured with an Mw 6.7 event was within a low Dc area. We identified a consistent relationship between stress levels and Dc for the 2023 twin events and the Mw 6.7 earthquake, with low Dc indicative of high stress. Intriguingly, mainshocks and a substantial proportion of their aftershocks have occurred within areas characterized by low to moderate Dc. Various fault zones like Malatya, Amanos, and Adiyaman are situated in areas with low Dc. The southwestern area of the Amanos segment exhibits clustering, elevated stress levels, and low Dc, followed by the Mw 7.7. Therefore, it is imperative to maintain vigilant monitoring of this region to prevent another disaster.

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.

Spiral shocks induced in a galactic gaseous disk: Hydrodynamic understanding of observational properties of spiral galaxies

Context. We investigate the properties of spiral shocks in a steady, adiabatic, non-axisymmetric, self-gravitating, mass-outflowing accretion disk around a compact object. Aims. We obtained the accretion-ejection solutions in a galactic disk and applied them to spiral galaxies in order to investigate the possible physical connections between some observational quantities of galaxies. Methods. We considered the self-gravitating disk potential to examine the properties of the galactic gaseous disk. We obtained spiral shock-induced accretion-ejection solutions following the point-wise self-similar approach. Results. We observed that the self-gravitating disk profoundly affects the dynamics of the spiral structure of the disk and the properties of the spiral shocks. We find that the observational dispersion between the pitch angle and shear rate and between the pitch angle and star formation rate in spiral galaxies contains some important physical information. Conclusions. There are large differences among the star formation rates of galaxies with similar pitch angles. These differences may be explained by the different star formation efficiencies caused by distinct galactic ambient conditions.

Market index price prediction using Deep Neural Networks with a Self-Similarity approach

Stock indexes are of vital importance to understand financial and economic markets of sectors and nations. Given the importance of market indexes, researchers, investors, and policy makers are continuously working to improve models to forecast market movements; small improvements in modeling have the potential for large financial gains. To this end, machine learning approaches have gained in popularity as both software and hardware performance have increased. In this work, we apply machine learning applications to the S&P 500, DAX, AEX and the SMI indexes to improve forecasting performance. In particular, we take advantage of the fractal and Self-Similarity behaviors that exist in the time series of these indexes using simple Recurrent Neural Network, Multilayer Perceptron, and Long-Short Term Memory architectures. We apply the architecture using 60-, 30-, and 15-min windows of daily series for each approach and index. The results indicate that for the S&P 500 the proposed self-similarity models outperform all base approaches. However, results are not uniform across all models and indexes. Overall, performance differences across models and indexes are presented. To check the difference between the forecasts of the self-similarity and base models, Model Confidence Set was used.

A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet

Nano-fluids have considerable importance in the field of thermal development that relates to several industrial systems. There are some key applications in recent construction systems flow, as well as microscale cooling gadgets and microstructure electric gadgets for thermal migration. The current investigation concludes the study of electrically conducting nano-fluid flow and heat transfer analysis in two-dimensional boundary layer flow over a curved extending surface in the coexisting of magnetic field, heat generation and thermal radiation. The small sized particles of copper (Cu) are taken as nanoparticles and water is assumed to be the base fluid. We used quasi-linearization and central difference approximation to numerically solve the system of coupled equations obtained from the partial differential equations (PDEs) by incorporating the concept of similarity. The impacts of non-dimensional parameters on velocity, concentration and thermal profiles have been discussed with the help of suitable graphs and tables. It has been noticed that the velocity decelerated with the effect of the magnetic field interaction parameter. Thermal radiation caused an increase in temperature.

Self-Similarity and Power-Law Spectra of Polymer Melts and Solutions.

Both the Rouse and Doi-Edwards models can be expressed by the relaxation spectra, in the form of power-law functions. The concept of self-similarity has offered a simple solution to many problems in polymer physics. Since the solutions derived from self-similarity are power-law functions, it is essential to check whether the relaxation spectrum of polymeric fluids can be derived by self-similarity. In this study, the power-law spectrum of an unentangled polymer solution is derived by using the self-similarity approach, which does not work for entangled polymeric fluids. Although Baumgaertel et al. (Rheol. Acta 29, 400–408 (1990)) showed that the power-law spectrum can quantitatively describe the linear viscoelasticity of monodisperse polymer melts, regardless of molecular weight, they did not find the universality of the exponent of the spectrum because they found different exponents for different polymers. Under the consideration existing the universality of linear viscoelasticity of polymer melts, this paper deals with the universality of the exponent by employing a new regression algorithm and confirms that the exponent is independent of the type of polymer.

Quasisimple Wave Solutions of Euler’s System of Equations for Ideal Gas

Quasisimple wave solutions of Euler’s system of equations for ideal gas are investigated under the assumption of spherical and cylindrical symmetries. These solutions are proved to be stabilized into sound wave solutions and cavitation. It is proved that if initial conditions from outside the invariant region approach to transitional solution, then reciprocal of the self-similar parameter goes to infinity. However, when initial conditions stabilize into sound waves or cavitation, then reciprocal of self-similar parameter approaches finite value. Further, it is proved that initial conditions can be parametrized so that some of the initial conditions stabilize into sound wave solutions. The rest of the initial conditions are proved to be stabilized into cavitation. This extends the work of G. I. Taylor to the case of cavitation. It is proved that quasisimple wave solutions exist for the balance laws comprised of Euler’s system of equations in the case of cylindrically and spherically symmetric cases. The description applies to the motion of cylindrical and spherical piston in real life. In particular, self-similar description of appearance of vacuum in the motion of cylindrical and spherical piston is given.

Ratio of kaon-to-pion production cross-sections in BeBe collisions as a function of $$\sqrt{s}$$

The inclusive spectra of pions and kaons produced in $Be Be$ collisions as functions of their transverse momentum $p_t$ at mid-rapidity are calculated within the self-similarity approach. A satisfactory description of NA61/SHINE data on these spectra and the ratio of $K^\pm$ to $\pi^\pm$ meson production cross sections in $Be Be$ collisions as a function of $\sqrt{s}$ are presented. The similarity of these observables to the ones for $pp$ collisions at mid-rapidity and in the wide range of initial energies is illustrated.

A Self-similar Solution of Hot Accretion Flow: The Role of the Kinematic Viscosity Coefficient

Abstract We investigate the dependency of the inflow-wind structure of a hot accretion flow on the kinematic viscosity coefficient. In this regard, we propose a model for the kinematic viscosity coefficient to mimic the behavior of the magnetorotational instability that would be maximal at the rotation axis. Then, we compare our model with two other prescriptions from numerical simulations of the accretion flow. We solve two-dimensional hydrodynamic equations of hot accretion flows in the presence of thermal conduction. The self-similar approach is also adopted in the radial direction. We calculate the properties of the inflow and the wind such as velocity, density, and angular momentum for three models of the kinematic viscosity prescription. On inspection, we find that in our suggested model the wind is less efficient at extracting the angular momentum outward where the self-similar solutions are applied than it is in two other models. The solutions obtained in this paper might be applicable to hydrodynamical numerical simulations of hot accretion flows.

Existence, uniqueness and positivity on a free-boundary high order diffusion cooperative system

We study existence, uniqueness and positivity conditions for a cooperative system formulated with a high order diffusion. In addition, we show and characterize the instabilities due to the high order diffusion for which a positivity and a comparison principle hold after rescaling. Instabilities shall be understood as the oscillatory behaviour of solutions acting as an inherent feature driven by complex exponential bundles of solutions. Firstly, a shooting method approach is used to show the existence of such instabilities. Afterwards, the existence of solutions is assessed in the self-similar approach and characterized by analytical and numerical evidences for a single equation in and for the complete system. Finally, a positive kernel is shown to exist and a sharp estimation is obtained.

A self-similar solution for the water hammer equation used in an infinite long pipeline

The present study uses the self-similar approach, developed by Pistiner (2020), for solving the water hammer equation during the process of valve closure in infinite long pipeline. The obtained solution shows that the transient flow in the pipeline is characterized by three zones: the “near zone” located near the closing valve and is characterized by a positive flow; the “far zone” in the pipeline lying at an infinite distance from the valve where the flow is also positive and the “intermediate backflow zone”, which is lying among the above-mentioned zones and is characterized by reverse flow. The results obtained in this study may be useful to give some insights into the understanding of transient flow process in pipes as well as provide approximate estimation for transient waves in pipes and finally, the applicability of the model to predict head distribution in the pipeline has been shown.

Ratio of cross-sections of kaons to pions produced in pp collisions as a function of $$\sqrt{s}$$

A calculation of the inclusive spectra of pions and kaons produced in $pp$ collisions as functions of their transverse momentum $p_t$ at mid-rapidity is presented within the self-similarity approach. A satisfactory description of the data within a wide range of initial energies is presented. We focus mainly on the ratio of cross-sections of $K^\pm$ to $\pi^\pm$ mesons produced in $pp$ collisions as a function of $\sqrt{s}$. A fast rise of this ratio, when the initial energy increases starting from the kaon production threshold up to $\sqrt{s}\simeq$ 20-30 Gev, is revealed together with its very slow increase up to LHC energies. The energy dependence of this ratio is due to the conservation laws of four-momenta and quantum numbers of the initial and produced hadrons, and to the Regge behavior of cross-sections at large energies. The more or less satisfactory agreement of these ratios with NA61-SHINE, RHIC and LHC data is demonstrated.

Radiation drive temperature measurements in aluminum via radiation-driven shock waves: Modeling using self-similar solutions

We study the phenomena of radiative-driven shock waves using a semi-analytic model based on self-similar solutions of the radiative hydrodynamic problem. The relation between the Hohlraum drive temperature TRad and the resulting ablative shock DS is a well-known method for the estimation of the drive temperature. However, the various studies yield different scaling relations between TRad and DS based on different simulations. In T. Shussman and S. I. Heizler, Phys. Plasmas 22, 082109 (2015), we have derived full analytic solutions for the subsonic heat wave, which include both the ablation and the shock wave regions. Using this self-similar approach, we derive here the TRad(DS) relation for aluminum, using the detailed Hugoniot relations and including transport effects. By our semi-analytic model, we find a spread of ≈40 eV in the TRad(DS) curve as a function of the temperature profile's duration and its temporal profile. Our model agrees with the various experiments and the simulations data, explaining the difference between the various scaling relations that appear in the literature.

Soliton evolution and associated sonic horizon formation dynamics in two-dimensional Bose-Einstein condensate with quintic-order nonlinearity.

This study explored the two-dimensional Bose-Einstein condensate with an inter-particle nonlinear interaction up to quintic order. Based on the two-dimensional Gross-Pitaevskii equation model with quintic-order nonlinearity, we first derived the bright soliton solution for the system based on the self-similar approach and the modified variational method. We identified that the kinematic quantities derived from the two methods agreed very well. The two-dimensional sonic horizon formation dynamics was then calculated based on the bright soliton solution that was obtained. The periodic formation and evolution patterns of the two-dimensional sonic horizon were quantitatively analyzed and pictorially illustrated. The results can be used to guide sonic black hole related phenomenon observations in a two-dimensional Bose-Einstein condensate with quintic-order nonlinearity.

The Importance of Hall Effect in the Self-similar Collapse of a Filamentary Cloud

Abstract Thanks to the observational and simulation works, the importance of the nonideal magnetohydrodynamic (MHD) effects, i.e., Hall effect, ohmic resistivity, and ambipolar diffusion, have been well established at various stages of cloud evolution. To get a comparison between the Hall effect with other effects, we aim to model the time evolution of a rotating filamentary molecular cloud during the isothermal/polytropic collapse phase in the presence of the Hall drift. Three components of the velocity vector are investigated when the angular momentum is fully coupled with the magnetic field at large radii of a filament. For this purpose, the nonideal MHD equations in the self-similar formalism are considered at large radii of a molecular cloud where the magnetic field evolution is affected by the Hall drift. Then, the connection between the self-similar approach with the observational data from the filamentary clouds is examined to get a realistic model. Due to the existence of Hall drift, the significant changes on the rotation of the cloud can be seen when the cloud switches from the isothermal collapse phase to the polytropic collapse phase. Also, the results of this model are useful in the study of the multiple star formation process as well as the initial conditions for driving the outflows during the collapse of the filamentary clouds. Finally, we found that there are some conditions for the comparability of the Hall effect with the ambipolar diffusion in the outer regions of the clouds.

New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence

We use the Lie symmetry theory to derive new scaling laws for passive scalar dynamics under the influence of a constant mean gradient of the scalar in decaying homogeneous isotropic turbulence. For this purpose, we apply symmetry analysis to the equations for two-point correlation of the scalar and velocity fluctuations. It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis. In the context of the current analysis also, scaling laws for one-point quantities of the scalar variance , the transverse scalar flux and the variance of the turbulent velocity fluctuations are established, which are essentially related to the scaling symmetries. A key step to derive the scaling laws is the symmetry breaking induced by the constant mean scalar gradient. We use the results of a highly resolved direct numerical simulation of Gauding et al. (Comput. Fluids, vol. 180, 2019, pp. 206–217) to verify the scaling laws and the self-similarity of the two-point correlation functions. It is shown that the general symmetry solutions obtained from symmetry results provide a very good similarity to these functions.

Soliton dynamics for quantum systems with higher-order dispersion and nonlinear interaction

We investigated three-dimensional quantum systems with higher-order dispersion and nonlinear effects. The systems’ soliton dynamics is studied based on the (3+1)-dimensional higher-order nonlinear Schrödinger equation (NLSE). Based on the self-similar approach and the bright soliton-type solution of the (1+1)-dimensional NLSE, we derived the analytical bright soliton solution for the (3+1)-dimensional NLSE with higher-order dispersion and nonlinear effects, with the typical soliton feature pictorially demonstrated. Our study illustrates that a higher-dimensional medium with higher-order dispersion and nonlinear effects supports soliton behavior. This demonstrates the applicability of the theoretical treatment presented in this work.