Small Parameter Research Articles

A generalized Allen–Cahn model with mass source and its Cahn–Hilliard limit

AbstractThe present paper is concerned with a fourth‐order Allen–Cahn model with logarithmic potential and mass source that describes the process of phase separation in two‐component systems accompanied by a flux of material. The existence of a global weak solution is obtained under appropriate hypotheses on the source term. Furthermore, we study its Cahn–Hilliard limit as a small parameter goes to zero. The main difficulty in the mathematical analysis of the model lies in the presence of the source term that leads to the nonconservation of mass, contrary to the original Cahn–Hilliard theory.

On Elliptic Lower-Dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems with Small Parameters

On Elliptic Lower-Dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems with Small Parameters

Uniqueness and non-degeneracy of solutions for nonlinear fractional Schrödinger equation with perturbation

This paper concerned a fractional Schrödinger equation in whole space, for ɛ > 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 12<s<1, N > 1 and 2<p<2NN−2s. We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions.

Approximate solutions for the Vlasov–Poisson system with boundary layers

We construct the approximate solutions to the Vlasov–Poisson system in a half-space, which arises in the study of the quasi-neutral limit problem in the presence of a sharp boundary layer, referred as to the plasma sheath in the context of plasma physics. The quasi-neutrality is an important characteristic of plasmas and its scale is characterized by a small parameter, called the Debye length. We present the approximate equations obtained by a formal expansion in the parameter and study the properties of the approximate solutions. Moreover, we present numerical experiments demonstrating that the approximate solutions converge to those of the Vlasov–Poisson system as the parameter goes to zero.

On a quenching problem of hyperbolic MEMS equation with a nonlocal term

We consider a nonlocal hyperbolic problem. It arises in the field of MEMS control. First, we review the stationary problem and state that there exists a unique and no solution for small and large parameter, respectively. Next, we introduce the results of global existence and dynamical properties of solution for the hyperbolic equation. Finally, we derive the result of the quenching behaviour. As proven in similar problems, we show that the quenching occurs for large parameter. This criterion is also expressed by means of the first eigenvalue of Laplace operator.

ЭКИНЧИ ТАРТИПТЕГИ ТУУНДУСУНА КАРАТА ЧЕЧИЛБЕГЕН ДИФФЕРЕНЦИАЛДЫК ТЕҢДЕМЕ ҮЧҮН ЧЕКТИК МАСЕЛЕНИН ЧЫГАРЫЛЫШЫН КОЛЛОКАЦИЯ-АСИМПТОТИКАЛЫК МЕТОД МЕНЕН ТАБУУ

Boundary value problems for ordinary differential equations is one of the sections of the general theory of differential equations. The difference between the boundary value problem and the Cauchy problem is that the solution of the differential equation must satisfy boundary conditions connecting the values of the desired function at more than one point. The most common boundary value problem refers to two-point boundary value problems for which boundary conditions are set at two points at the ends of the interval on which solutions are being sought. In addition, the special case of two-point boundary value problems also covers such an important section of the theory of differential equations as the theory of oscillations. In this article, the problem of constructing an asymptotic expansion in a small parameter for solving a boundary value problem for a second-order differential equation of an unresolved relative to the highest derivative of the desired solution is considered. The solution of the boundary value problem is found according to the algorithm of the collocation-asymptotic method. A numerical example of a boundary value problem is considered and an asymptotic decomposition is constructed for solving problems with a relatively small parameter, including the first two terms of the expansions.

Radiative corrections to aid the direct detection of the Higgsino-like neutralino dark matter: Spin-independent interactions

The lightest neutralino (χ˜10) is a good dark matter (DM) candidate in the R-parity conserving minimal supersymmetric Standard Model. In this work, we consider the light Higgsino-like neutralino as the lightest stable particle, thanks to a rather small Higgsino mass parameter μ. We then estimate the prominent radiative corrections to the neutralino-neutralino-Higgs boson vertices. We show that for Higgsino-like χ˜10, these corrections can significantly influence the spin-independent direct detection cross section, even contributing close to 100% in certain regions of the parameter space. These corrections, therefore, play an important role in deducing constraints on the mass of the Higgsino-like lightest neutralino DM, and thus the μ parameter. Published by the American Physical Society 2024

Dynamic Analysis of a Uniform Microbeam Resting on a Nonlinear Foundation Considering Its Curvature Subjected to a Mechanical Impact and Electromagnetic Actuation.

This study proposes an investigation into the nonlinear vibration of a simply supported, flexible, uniform microbeam associated with its curvature considering the mechanical impact, the electromagnetic actuation, the nonlinear Winkler-Pasternak foundation, and the longitudinal magnetic field. The governing differential equations and the boundary conditions are modeled within the framework of a Euler-Bernoulli beam considering an element of the length of the beam at rest and using the second-order approximation of the deflected beam and the Galerkin-Bubnov procedure. In this work, we present a novel characterization of the microbeam and a novel method to solve the nonlinear vibration of the microactuator. The resulting equation of this complex problem is studied using the Optimal Homotopy Asymptotic Method, employing some auxiliary functions derived from the terms that appear in the equation of motion. An explicit closed-form analytical solution is proposed, proving that our procedure is a powerful tool for solving a nonlinear problem without the presence of small or large parameters. The presence of some convergence-control parameters assures the rapid convergence of the solutions. These parameters are evaluated using some rigorous mathematical procedures. The present approach is very accurate and easy to implement, even for complicated nonlinear problems. The local stability near the primary resonance is studied.

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

A nearest-neighbor expansion of lepton flavor mixing in powers of the μ-τ permutation symmetry breaking effect

We point out that the observed pattern of lepton flavor mixing can be well described by a proper nearest-neighbor expansion of a constant 3×3 unitary matrix in powers of a small parameter characterizing the fine effect of μ-τ permutation symmetry breaking. We take an example of this kind for illustration, and provide complete discussions on the usefulness in the study of leptonic CP violation and unitarity triangles in matter.

Asymptotic expansion method with respect to a small parameter for fractional differential equations with Riemann–Liouville derivate

In this paper, we proposed an asymptotic approach with respect to a small parameter for fractional differential equations, with the small parameter linked to the fractional order derivative. This approach allows the splitting of the field variable and consequently the Riemann–Liouville Integral as the sum of two contributions. One describes the unperturbed state and the other describes the behavior of the model in the perturbed state due to the presence of the fractional derivative. This approach allows mapping time-fractional differential equations into a system of two coupled partial differential equations with integer order. By solving these partial differential equations we are able to obtain solutions of the assigned fractional differential equations. To obtain solutions we determine the approximate Lie symmetries through which the system of two coupled partial differential equations is reduced to a system of two coupled ordinary differential equations. As an application, we consider the time fractional diffusion–reaction equation with a linear source term; we test the proposed procedure by comparing the profiles of the obtained approximate solutions with solutions found in the fractional context for values of the fractional parameter tending to the integer value.

The asymptotic solutions for the motion of a charged symmetric gyrostat in the irrational frequency case

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange’s scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body’s movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler–Poisson are solved through the utilization of Poincaré’s small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body’s position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems.

The relationship between two approximate symmetry frameworks

The two most popular and practically useful approaches to calculate approximate symmetries of differential equations with a small parameter are the Baikov–Gazizov–Ibragimov (BGI) and Fushchich–Shtelen (FS) frameworks. It is proven that BGI approximate symmetries form a subset of FS approximate symmetries. Examples are provided.

Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters: ɛ, representing the usual scale separation in vegetation-water models, and δ, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involving ɛ and δ, with and without a so-called superslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full partial differential equation model.

The homogenized dynamical model of a thermoelastic composite stitched with reinforcing filaments.

The dynamical problem of linear thermoelasticity for a body with incorporated thin rectilinear inclusions is studied. It is assumed that the inclusions (i.e. filaments and threads) are parallel to each other and the problem contains a small parameter [Formula: see text], which characterizes the distance between two neighbouring inclusions. Using the two-scale convergence approach, we find the limiting problem as [Formula: see text]. As a result, we get a well-posed homogenized model of an anisotropic inhomogeneous body with effective characteristics inheriting thermomechanical properties of inclusions.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

What is the physical origin of the gradient flow structure of variational fracture models?

We investigate a physical characterization of the gradient flow structure of variational fracture models for brittle materials: a Griffith-type fracture model and an irreversible fracture phase field model. We derive the Griffith-type fracture model by assuming that the fracture energy in Griffith's theory is an increasing function of the crack tip velocity. Such a velocity dependence of the fracture energy is typically observed in polymers. We also prove an energy dissipation identity of the Griffith-type fracture model, in other words, its gradient flow structure. On the other hand, the irreversible fracture phase field model is derived as a unidirectional gradient flow of a regularized total energy. We have considered the time relaxation parameter a mathematical approximation parameter, which we should choose as small as possible. In this research, however, we reveal the physical origin of the gradient flow structure of the fracture phase field model (F-PFM) and show that the small time relaxation parameter is characterized as the rate of velocity dependence of the fracture energy. It is verified by comparing the energy dissipation properties of those two models and by analysing a travelling wave solution of the irreversible F-PFM. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Approximations in Mean Square Analysis of Stochastically Forced Equilibria for Nonlinear Dynamical Systems

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement.

Nonlinear-Optical Analogies in Nuclear-Like Soliton Reactions: Selection Rules, Nonlinear Tunneling and Sub-Barrier Fusion–Fission

Abstract The main goal of our study is to reveal unexpected but intriguing analogies arising between optical solitons and nuclear physics, which still remain hidden from us. We consider the main cornerstones of the concept of nonlinear optics of nuclear reactions and the well-dressed repulsive-core solitons. On the base of this model, we reveal the most intriguing properties of the nonlinear tunneling of nucleus-like solitons and the soliton self-induced sub-barrier transparency effect. We describe novel interesting and stimulating analogies between the interaction of nucleus-like solitons on the repulsive barrier and nuclear sub-barrier reactions. The main finding of this study concerns the conservation of total number of nucleons (or the baryon number) in nuclear-like soliton reactions. We show that inelastic interactions among well-dressed repulsive-core solitons arise only when a “cloud” of “dressing” spectral side-bands appears in the frequency spectra of the solitons. This property of nucleus-like solitons is directly related to the nuclear density distribution described by the dimensionless small shape-squareness parameter. Thus the Fourier spectra of nucleus-like solitons are similar to the nuclear form factors. We show that the nuclear-like reactions between well-dressed solitons are realized by “exchange” between “particle-like” side bands in their spectra.

Multi-task neural networks by learned contextual inputs

This paper explores learned-context neural networks. It is a multi-task learning architecture based on a fully shared neural network and an augmented input vector containing trainable task parameters. The architecture is interesting due to its powerful task adaption mechanism, which facilitates a low-dimensional task parameter space. Theoretically, we show that a scalar task parameter is sufficient for universal approximation of all tasks, which is not necessarily the case for more common architectures. Empirically it is shown that, for homogeneous tasks, the dimension of the task parameter may vary with the complexity of the tasks, but a small task parameter space is generally viable. The task parameter space is found to be well-behaved, which simplifies workflows related to updating models as new data arrives, and learning new tasks with the shared parameters are frozen. Additionally, the architecture displays robustness towards datasets where tasks have few data points. The architecture’s performance is compared to similar neural network architectures on ten datasets, with competitive results.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.