Asymptotic Analysis Method Research Articles

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schrödinger equation (NLSE) and its variable–coefficient (vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities (especially the soliton accelerations and interaction forces); whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles, particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.

Asymptotic analysis of boundary layer of a thermo-dependent fluid flow

In this work, we applied the asymptotic analysis method based on a large Prandtl number for a boundary layer flow over a flat plate, of a thermo-dependent fluid obeying a power law. In order to make sense to the notion of boundary layer, we have to assume that the fluid is shear-thinning. The governing equations of the thermal boundary layer do not admit self-similar solutions but the asymptotic model derived in this work admits one. This approach was also employed to obtain a Nusselt number correlation as a function of the ratio K(T∞)/K(Tw), where K is the consistency index, T∞ and Tw are respectively the free stream and the wall temperature. In particular, we have shown that the shear-thinning nature of the fluid accentuates the effect of thermo-dependence and amplifies the heat transfer at the wall.

Multiscale analysis of stationary thermoelastic vibrations of a composite material.

The problem of description of stationary vibrations is studied for a planar thermoelastic body incorporating thin inclusions. This problem contains two small positive parameters [Formula: see text] and [Formula: see text], which describe the thickness of an individual inclusion and the distance between two neighbouring inclusions, respectively. Relying on the variational formulation of the problem, by means of the modern methods of asymptotic analysis, we investigate the behaviour of solutions as [Formula: see text] and [Formula: see text] tend to zero. As the result, we construct two models corresponding to the limit cases. At first, as [Formula: see text], by the version of the method of formal asymptotic expansions we derive a limit model in which inclusions are thin (of zero width). Then, from this limit model, as [Formula: see text], we derive a homogenized model, which describes effective behaviour on the macroscopic scale, i.e. on the scale where there is no need to take into account each individual inclusion. The limiting passage as [Formula: see text] is based on the use of the two-scale convergence theory. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Meso-scale method of asymptotic analysis of elastic vibrations in periodic and non-periodic multi-structures

Summary The method of meso-scale asymptotic approximations has proved to be very effective for the analysis of models of solids containing large clusters of defects, such as small inclusions or voids. Here, we present a new avenue where the method is extended to elastic multi-structures. Geometrically, a multi-structure makes a step up in the context of overall dimensions, compared to the dimensions of its individual constituents. The main mathematical challenge comes from the analysis of the junction regions assigned to the multi-structure itself. Attention is given to problems of vibration and on the coupling of vibration modes corresponding to displacements of different orientations. The method is demonstrated through the dynamic analysis of infinite or finite multi-scale asymmetric flexural systems consisting of a heavy beam connected to a non-periodic array of massless flexural resonators within some interval. In modelling the interaction between the beam and the resonators, we derive a vectorial system of partial differential equations through which the axial and flexural motions of the heavy beam are coupled. The solution of these equations is written explicitly in terms of Green’s functions having intensities determined from a linear algebraic system. The influence of the resonators on the heavy beam is investigated within the framework of scattering and eigenvalue problems. For large collections of resonators, dynamic homogenization approximations for the medium within the location of the resonant array are derived, leading to (i) the classical Rayleigh beam for symmetric systems and (ii) a generalized Rayleigh beam for asymmetric structures that support flexural–longitudinal wave coupling. Independent numerical simulations are also presented that demonstrate the accuracy of the analytical results.

Phase control of localization in the nonlinear two-mode system from harmonic mixing driving: Perturbative analysis and symmetry consideration

In this paper, we present a rigorous analysis of symmetry and underlying physics of the nonlinear two-mode system driven by a harmonic mixing field, by means of multiple scale asymptotic analysis method. The effective description in the framework of the second-order perturbative theory provides an accurate picture for understanding the Floquet eigenspectrum and dynamical features of the nonlinear two-mode system, showing full agreement with the prediction of symmetry considerations. We find that two types of symmetries play significant role in the dynamical features of this model, the mechanism behind which can be interpreted in terms of the effective description. The results are of relevance for the phase control of the atomic localization in Bose–Einstein condensates or switch of the optical signals in nonlinear mediums.

Laminar flame speed of H2/CH4/air mixtures with CO2 and N2 dilution

The laminar flame speeds of H2/CH4/air mixtures with CO2 and N2 dilution were systematic investigated experimentally and numerically over a wide range of H2 blending ratios (0–75 vol%) with CO2 (0–67 vol%) and N2 (0–67 vol%) dilution in the fuels. The experimental measurements were conducted via the Bunsen flame method incorporating the Schlieren technique under the condition of equivalence ratios from 0.8 to 2.0. To gain an insightful understanding of the experimental observations, detailed numerical simulation was carried out using Chemkin-Pro with GRI3.0-Mech. The experimental measurements were also used to validate the corresponding performance of a semiempirical correlation derived through asymptotic analysis method coupled with the reduced chemistry mechanism. The results showed that at lower H2 fraction (xH2 ≤ 0.5), the laminar flame speeds of H2/CH4/air mixtures displayed great linearly increase with the growth of H2 fractions. The combustion of mixtures with low H2 contents was dominated by CH4 conversion which was mainly controlled by the increasing OH radicals produced from the key oxidation reactions of H + O2 = O + OH. With the further increase of H2 fractions, the methane-dominated combustion gradually transformed into the methane-inhibited hydrogen combustion, resulting to the growth of laminar flame speeds was dramatical and non-linear. Due to the larger heat capacity and chemical kinetic effect, CO2 presented a stronger dilution effect on reducing the laminar flame speeds than N2. With the addition of CO2, the increasing stronger competition for H radical through CO + OH = CO2 + H with H + O2 = O + OH due to the significant reduction of H mole fractions, leading to the larger decrease of laminar flame speeds of mixtures. Besides, although the contribution of thermal effect of CO2 decreased near the equivalence ratio, the thermal effect of CO2 still preformed the dominated contribution to the total dilution effect. A comparison between the experimental data and estimated results using the semiempirical correlation showed that, the correlation using new modified coefficients provided the satisfactorily accuracy predictions on the laminar flame speeds of diluted H2/CH4/air mixtures at lower xH2 (xH2 ≤ 0.5) and lower xdilution (xdilution = 0.25). The estimated results were generally located within a deviation range of ±20% errors except for two unsatisfactory eases occurred at conditions of xH2 = 0.75 and xCO2 = 0.67. The considerably poor predictions were attributed to the significantly variation of the chemical kinetics under high H2 content and large CO2 dilution conditions.

Solitons, breathers and rogue waves in the coupled nonlocal reverse-time nonlinear Schrödinger equations

In this paper, we obtain the N-bright-bright soliton, N-dark-bright soliton, general N-breather and Nth-order semirational rogue wave solutions of the coupled nonlocal reverse-time nonlinear Schrödinger (NLS) equations by using the Darboux transformation and limit technique. It is demonstrated that these solutions can be globally bounded or collapsed with singularities. The elastic collisions of the bounded two-bright-bright solitons and two-dark-bright solitons are discussed through the asymptotic analysis method. The degenerate bounded or collapsing Akhmediev breather and Kuznetsov-Ma breather, and the bounded breather-soliton and breather-breather mixed waves are graphically shown. In particular, there exist degenerate bounded or collapsing rogue waves, bounded or collapsing rogue waves coexisting and interacting with dark-bright solitons or breathers in the coupled nonlocal reverse-time integrable models, most of which have no counterparts in the coupled local NLS equations. The modulation instability analysis concerning the plane-wave background is performed to illustrate the existence of rogue waves.

Linear Filtering with Fractional Noises: Large Time and Small Noise Asymptotics

The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman--Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman--Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asymptotic analysis of the integro-differential filtering equation arising in this case. We establish existence of the steady-state error limit and find its asymptotic scaling in the high signal-to-noise regime. Closed form expressions are derived in a number of important cases.

Vector multi-pole solutions in the [formula omitted]-coupled Hirota equation

In this paper, we study the vector multi-pole (MP) solutions of the r-coupled Hirota equation (RCHE). First of all, based on the Darboux transformation and the N-soliton solution, we derive the explicit formulas of the arbitrary-order vector MP solutions. Then, through the balance between exponential and algebraic terms, we give all asymptotic solitons in the vector double- and triple-pole solutions via an asymptotic analysis method. Furthermore, we analyze the soliton interactions in vector MP solutions. By comparing with the scalar Hirota equation, we find that the RCHE has richer collision properties: the position shift of each component soliton grows logarithmically with r, the asymptotic solitons have the same shape in each component, and their amplitudes are proportional among the components.

Numerical simulation of front dynamics in a nonlinear singularly perturbed reaction–diffusion problem

A new approach to the numerical simulation of the reaction front motion in a nonlinear singularly perturbed partial differential equation of the reaction–diffusion type is proposed. It is shown how the methods of asymptotic analysis allow reducing the statement of the original problem to a problem of a smaller dimension, the numerical solving of which uses methods of diagnostics of the solution’s blow-up. Numerical experiments demonstrate the effectiveness of the proposed approach.

Asymptotic modal analysis application to a cavity-plate system to predict sound transmission loss

The asymptotic modal analysis (AMA) method has previously been developed to predict the high-frequency dynamic response of structural and acoustical systems [Sung et al., Asymptotic Modal Analysis of Structural and Acoustical Systems (Morgan & Claypool Pub., 2021)]. In the AMA method, the dynamic response is obtained as the asymptotic limit of the classical modal analysis (CMA) solution as the averaged response in large frequency bands. In previous work, the AMA method has been applied to uncoupled individual structural and acoustical systems. In the asymptotic limit, the AMA results for individual structural and acoustical systems have been shown to approach the corresponding statistical energy analysis (SEA) results. The AMA method has also previously been applied to a coupled cavity-plate system to predict the high frequency response for piston-like excitation of the plate. In this paper, the AMA method is applied to a cavity–plate system subjected to random excitation of the plate to predict the sound transmission loss into the cavity. The AMA results in the asymptotic limit for the cavity-plate system are shown to approach the corresponding CMA and SEA results. In addition, the AMA method is able to predict the spatial distribution of the plate and cavity responses.

Bifurcations of Invariant Tori in Second-Order Quasilinear Evolution Equations in Hilbert Spaces and Scenarios of Transition to Turbulence

In this paper, we consider second-order quasilinear differential equations in a separable Hilbert space for which the well-known Landau–Hopf scenario of transition to turbulence can be realized. We prove increasing of the control parameter leads to the consecutive appearance of invariant tori of increasing dimensions. In this case, the invariant torus of the largest possible dimension appears to be attractive. The results are obtained by using methods of the qualitative theory of dynamical systems with an infinite-dimensional space of initial conditions: the method of integral manifolds, the theory of normal forms, and also asymptotic methods of analysis of dynamical systems.

Theoretical analysis of time-dependent jetting on the surface of a thin moving liquid layer

Unsteady two-dimensional problem of a thin liquid layer with prescribed time-dependent influx into the layer, position of the influx section, and the thickness of the liquid at this section is studied by methods of asymptotic analysis. The ratio of the rate of the liquid thickness variation at the influx section to the influx velocity plays a role of a small parameter of the problem. The influx parameters are such that the flow in the thin layer is inertia dominated, with gravity, surface tension, and liquid viscosity being approximately negligible. Such flows were studied with respect to several applications, some of which are listed in the Introduction. One of the applications concerns with splashing during droplet impact onto a rigid substrate and related kinematic discontinuity propagating along the spray sheet, which is produced by the spreading droplet. This type of splashing was studied by Yarin and Weiss [“Impact of drops on solid surfaces: Self-similar capillary waves, and splashing as a new type of kinematic discontinuity,” J. Fluid Mech. 283, 141–173 (1995)] within a quasi-one-dimensional approach averaging the flow velocity over the layer thickness. We also start with the one-dimensional thin-layer approximation assuming the influx flow is accelerated. Such influx conditions lead to unbounded growth of the thickness of the liquid layer at a certain location and at a certain time instant within the one-dimensional approach. The present study recovers for the first time the structure of the flow close to this singularity using methods of asymptotic analysis. To this aim, the second-order outer solution, which is valid outside the region of the unbounded flow, is derived. The second-order outer solution is used to find proper stretched inner variables and the equations governing the inner flow at the leading order. It is shown that the inner free-surface flow in the stretched variables is two-dimensional, potential, non-linear, and independent of any parameters of the original problem.

Dark-dark solitons, soliton molecules and elastic collisions in the mixed three-level coupled Maxwell–Bloch equations

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n-fold Darboux transformation is constructed on basis of a linear 3×3 matrix eigenvalue problem. As an application, the n-dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n-dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method.

Resource Retrial Queue with Two Orbits and Negative Customers

In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measures.

Soliton, breather-like and dark-soliton-breather-like solutions for the coupled long-wave–short-wave system

In this paper, we will obtain the exact N-soliton solution of the coupled long-wave–short-wave system via the developed Hirota bilinear method. Through manipulating the relevant parameters, we will construct different types of solutions which include breather-like solutions and dark-soliton-breather-like solutions. Moreover, we will demonstrate that the interactions of two-soliton and two-breather-like solutions are all elastic through asymptotic analysis method. Finally, we will display the interactions through illustrations.

A Laplace asymptotic integral-based reliability analysis method combined with artificial neural network

Reliability analysis aims to estimate the exceedance probability of structural response over a prescribed threshold value, which is essentially a multi-dimensional integral problem. In this paper, a novel and efficient reliability analysis method combining Laplace asymptotic integral and artificial neural network is proposed. Laplace asymptotic integral is employed to approximate the multi-dimensional integral to calculate the failure probability, and an active artificial neural network is taken as the surrogate model. Different from the existing surrogate model-based reliability methods, the proposed method focuses on approximating the limit state function in the vicinity of the target design point, instead of that in the whole sample space. No candidate sample population for active learning is needed, which significantly reduces the requirement for computer memory and improves the calculation efficiency. A novel learning function based on an optimization formulation is proposed in this paper to select the most informative samples for local approximation. Some numerical examples as well as a practical engineering issue are studied, and the superiority of the proposed method over other surrogate model-based reliability analysis methods in terms of efficiency and accuracy are verified.

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

We consider the linear water-wave problem in a periodic channel Pi ^h subset {{mathbb {R}}}^3, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough h is proven.

Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions

This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

Development of Methods of Asymptotic Analysis of Transition Layers in Reaction\u2013Diffusion\u2013Advection Equations: Theory and Applications

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.