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Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter ( e ) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O ( e 2 ) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number ( R e ). Furthermore, at subcritical R e with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

Quantum tunnelling plays a significant role in many molecular reactions, yet it is difficult to describe for molecules with more than 10 atoms. Both commonly used approaches, conventional wavefunction methods and statistical methods, are either computationally intractable, or slowly-converging for such systems. In contrast, instanton theory can describe quantum tunnelling even in high-dimensional systems and without statistical errors. This is because instead of using the whole phase space, in instanton theory an asymptotic expansion is performed around an extremal-action trajectory called the instanton. In this work we extend instanton theory by deriving the first-order perturbative correction to the commonly used leading-order result. Furthermore, we provide mathematical background to the theory by introducing a Feynman-diagram formalism and showing equivalence of our theory to a Euclidean ϕ3 + ϕ4 theory. Physical chemists or other researchers can utilise this framework to incorporate findings from the vast field of quantum field theory (QFT) into the advancement of instanton theory. Lastly, we use the perturbatively corrected instanton theory to numerically calculate the tunnelling splitting of a proton tunnelling reaction occurring in malonaldehyde and reach good agreement with state of the art statistical methods, specifically Diffusion Monte Carlo (DMC) and Path integral molecular dynamics (PIMD). This good agreement shows promise as perturbatively corrected instanton theory requires far less computational resources than the previously mentioned methods and thus can likely be applied to high-dimensional systems, where those methods fail.

We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the definition of the shift operator for the free QW. Then we define an anisotropic Banach space as a modified Agmon-H\"{o}rmander's $\mathcal{B}^*$ space and we derive the asymptotic behavior at infinity of generalized eigenfunctions in these spaces. The scattering matrix appears in the asymptotic expansion of generalized eigenfunctions.

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

In the present paper, we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of orders $$\alpha ,1+\alpha ,2+\alpha ,3+\alpha $$ and $$4+\alpha $$ . The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to $\pm \infty $ . In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the ‘bold Feynman diagrammatic expansion’ involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In this paper, we approach the analytical investigation of the bold diagrammatic expansion in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature). Using non-perturbative methods, we rigorously construct the Luttinger–Ward formalism for the first time, and we prove that the bold diagrammatic series can be obtained directly via an asymptotic expansion of the Luttinger–Ward functional, circumventing the partial resummation technique. Moreover we prove that the Dyson equation can be derived as the Euler–Lagrange equation associated with a variational problem involving the Luttinger–Ward functional. We also establish a number of key facts about the Luttinger–Ward functional, such as its transformation rule, its form in the setting of the impurity problem, and its continuous extension to the boundary of the domain of physical Green’s functions.

Given an asymptotically Anti-de Sitter supergravity solution, one can obtain a microscopic interpretation by identifying the corresponding state in the holographically dual conformal field theory. This is of particular importance for heavy pure states that are candidate black hole microstates. Expectation values of light operators in such heavy CFT states are encoded in the asymptotic expansion of the dual bulk configuration. In the D1-D5 system, large families of heavy pure CFT states have been proposed to be holographically dual to smooth horizonless supergravity solutions. We derive the precision holographic dictionary in a new sector of light operators that are superdescendants of scalar chiral primaries of dimension (1,1). These operators involve the action of the supercharges of the chiral algebra, and they play a central role in the proposed holographic description of recently-constructed supergravity solutions known as "supercharged superstrata". We resolve the mixing of single-trace and multi-trace operators in the CFT to identify the combinations that are dual to single-particle states in the bulk. We identify the corresponding gauge-invariant combinations of supergravity fields. We use this expanded dictionary to probe the proposed holographic description of supercharged superstrata, finding precise agreement between gravity and CFT., 50 pages, v3: typos fixed

The main results of this paper are related to studying the dynamics of the Van-der-Pol oscillator, which is under the influence of a driving force and a hysteretic effect, formalized by the Bouc–Wen phenomenological model. The regularizing role of the hysteretic link in the reduction of chaotic regimes is established. The main results are obtained as part of the asymptotic expansion method (small parameter method). A system of two coupled Van-der-Pol oscillators is considered under the conditions of a driving force acting on one of them: the relationship between the amplitudes and frequencies of the natural oscillations of the oscillators is established in an analytical form. The paper also describes a system of cross-related Van-der-Pol oscillators, in which the driving force acting on one of them is determined by the output of a hysteretic transducer, to the input of which the mismatch between the oscillators’ velocities is applied. It is found that the coupling parameter between the oscillators and the parameter that determines the influence of the hysteretic link have a significant effect on the synchronization in the system.

The asymptotic behavior of weak time-periodic solutions to the Navier–Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

We construct an asymptotic expansion in a small parameter of a boundary layer solution of the boundary value problem for a system of two ordinary differential equations, one of which is a second-order equation and the other is a first-order equation with a small parameter at the derivatives in both equations. Such a system arises in chemical kinetics when modeling the stationary process in the case of fast reactions and in the absence of diffusion of one of the reacting substances. A significant feature of the studied problem is that one of the equations of the degenerate system has a triple root. This leads to a qualitative difference in the boundary layer component of the solution compared with the case of simple (single) roots of degenerate equations. The boundary layer becomes multizonal, and the standard algorithm for constructing the boundary layer series turns out to be unsuitable and is replaced with a new algorithm.

We analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical quantities are finite. We also consider a scaling limit, in which the problem tends to that with two Dirac $$\delta $$ δ ’s. In this limit the global Casimir energy diverges, but we obtain its asymptotic expansion, which turns out to be model dependent. On the other hand, outside singular supports of $$\delta $$ δ ’s the limit of energy density is a finite universal function (independent of the details of the nonsingular model before scaling). These facts confirm the conclusions obtained earlier for other systems within the approach adopted here: the form of the global Casimir force is usually dominated by the modification of the quantum state in the vicinity of macroscopic bodies.

International audience; The Petrowsky type equation $y_{tt}^\eps+\eps y_{xxxx}^\eps - y_{xx}^\eps=0$, $\eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\eps}$ occurring at the extremities, these boundary controls get singular as $\eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^\eps$ and derive a rigorous second order asymptotic expansion of the control of minimal weighted $L^2-$norm, with respect to the parameter $\eps$. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J-.L.Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any $\eps$ small enough. Numerical experiments support our study.

We consider the question of the possible existence of two-parameter envelope solitons in weakly dispersive media with an acoustic spectrum of linear waves. We propose an asymptotic procedure for finding such soliton solutions in the one-dimensional case and demonstrate the proposed method in the example of the modified Boussinesq equation. We study the question of nonlinear multidimensional localization of excitations in weakly dispersive media with an acoustic spectrum of linear waves.

Starting from a complete family for the unit sphere S n in the complex n-space C n (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on L 2 ( S n ).

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u ( t , x ) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u ( t , x ) and its asymptotic approximation ν ( t , x ). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.

An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where $$\gamma \ge 0$$ , and i, j, k are nonnegative integers. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from cartesian to spherical. The resulting expression for $$h\left( {s,{\mathbf{Q}}} \right)$$ includes a number of cases of elementary integrals, the most difficult of which is $$II\left( {n,\mu } \right) = \int_{0}^{\pi } {\cos^{n} \theta e^{ - \mu \cos \theta } } d\theta$$ , with a nonnegative integer n and positive μ. This integral can be formed by linearly combining modified Bessel functions of the first kind $$B(n,\mu ) = \frac{1}{\pi }\int\limits_{0}^{\pi } {e^{\mu \cos \theta } \cos \left( {n\theta } \right)d\theta }$$ , with a nonnegative integer n and negative μ. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for this expression for n = 0,1,2. The relative error of asymptotic function is in the order of 10−16 with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function of $$B\left( {n,\mu } \right)$$ based on the expression for $$e^{ - \mu } II\left( {n,\mu } \right)$$ when n is an integer and μ is real and large in absolute value.

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where \begin{document}$ {\delta} = {\delta}(x) $\end{document} is the delta function supported at the origin. In the \begin{document}$ L^2 $\end{document} supercritical setting \begin{document}$ p>3 $\end{document} , we construct self-similar blow-up solutions belonging to the energy space \begin{document}$ L_x^\infty \cap \dot H_x^1 $\end{document} . This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \begin{document}$ x = 0 $\end{document} imposed by the \begin{document}$ \delta $\end{document} term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \begin{document}$ 0 using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

A three-dimensional problem is considered for electromagnetic diffraction on a bound ideally conducting body containing a conical point. An asymptotic representation of the electromagnetic field is calculated in the vicinity of the conical point with the solution presented as the sum of singular and smooth parts.

This paper is about solutions to PDEs in which the ratio between the largest and smallest values of the coefficient is very large. Such problems are referred to as high-contrast ones. A full asympt...

We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field \begin{document}$ {\mathbf{A}} $\end{document} and the potential field \begin{document}$ V $\end{document} . We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines \begin{document}$ {\rm{curl}}\ {\mathbf{A}} $\end{document} and \begin{document}$ V-\frac{1}{2}\nabla\cdot{\mathbf{A}} $\end{document} . We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error \begin{document}$ \mathcal{O}(\lambda^{-4}) $\end{document} recovers above two quantities with no additional information about \begin{document}$ {\mathbf{A}} $\end{document} and \begin{document}$ V $\end{document} . We also establish stability estimates for \begin{document}$ {\rm{curl}}\ {\mathbf{A}} $\end{document} and \begin{document}$ V-\frac{1}{2}\nabla\cdot{\mathbf{A}} $\end{document} .

In the paper, we obtain asymptotic expansion of the finite sum of some sequences $ S_{n} = \sum_{k = 1}^{n}\left(n^{2}+k\right) ^{-1} $ by using the Euler's standard one of the harmonic numbers.

Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class...

Under investigation in this work is the general three-component nonlinear Schrodinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.

In this paper, an integral transformation of the displacement is employed to determine the solution of the elastodynamics problem of two collinear Griffith cracks with constant velocity situated in a mid-plane of an infinite orthotropic strip where the boundaries are assumed to be stress-free. By use of the integral transformation of the displacement, the problem is reduced to a solution of the triple integral equations and the finite Hilbert transformation technique is used to solve it. The expressions of stresses are obtained asymptotically for large values of strip depth. An analytical expression of the stress intensity factor at the crack tip is obtained and represented graphically for two different orthotropic materials.

A study has been made of the pressure field in filtration of a weakly compressible fluid in a homogeneous isotropic compressible medium at high-amplitude perturbations. The equation used in formulating the problem is nonlinear: the densities of a porous medium and of a saturating fluid are assumed to be dependent on the function sought. Use is made of the fact that the dependences of the densities of the fluid and the skeleton material on pressure are approximated with a high degree of accuracy by the linear function. Consideration is given to one-dimensional plane horizontal filtration. The porosity and permeability of a porous medium, and also the viscosity of a filtering medium, are considered to be constant. A solution to the problem has been found with asymptotic expansions where a cofactor of the compressibility factor of the liquid acts as a small parameter. Analytical expressions have been found for the zero and first expansion factors. It has been shown that the zero expansion factor may be used to investigate the evolution of the pressure field of an incompressible liquid, whereas the expression for the first factor contains information on the contribution of nonlinearity due to the fluid’s compressibility. Values of the zero and first residual terms have been determined. It has been proved that the zero and first residual terms contain terms of only higher orders as far as the asymptotic-expansion parameter is concerned, i.e., the corresponding requirement of asymptoticity of expressions of the zero and first factors is met. On the basis of a computational experiment, the regularities of the dependence of the contribution of the nonlinearity under study on time and on the spatial coordinate have been established.

We consider the dimer model on a hexagonal lattice. This model can be represented as a “pile of cubes in a box.” The energy of a configuration is given by the volume of the pile. The partition function is computed by the classical MacMahon formula or as the determinant of the Kasteleyn matrix. We use the MacMahon formula to derive the scaling behavior of free energy in the limit as the lattice spacing goes to zero and temperature goes to infinity. We consider the case of a finite hexagonal domain, the case where one side of the hexagonal box is infinite, and the case of inhomogeneous Boltzmann weights. We obtain an asymptotic expansion of free energy, which is called finite-size corrections, and discuss the universality and physical meaning of the expansion coefficients.

In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications.

Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely., Comment: 15 pages, in press

Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus $$G_{13} $$ , while they decrease with an increase in the shear modulus $$G_{23} $$ . The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of $$G_{13}^{(2)} /G_{13}^{(1)} $$ increases, while they become smaller when the ratio of $$G_{23}^{(2)} /G_{23}^{(1)} $$ increases, and they increase with an increase in the mass density ratio for the bi-material V-notch.

A study of the dynamics of nonlinear waves at far field has been presented for real gases using quintic B-spline collocation method. To examine the dynamics at far field, “fast variable” in the governing system of equations has been introduced. Using asymptotic expansion of the flow variables in new fast variables, an evolution equation has been obtained for the behavior of waves at far from source. The evolution equation is modified Burger’s equation, which is known to have exact solution for planar flow. However, the solution for the corresponding cylindrical and spherical symmetric flow for the equation is unknown. The obtained numerical results have been compared with the exact solution for the planar flow. The error norms of numerically computed values are found very small when compared to corresponding exact solution. A Von Neumann stability analysis of the scheme is presented. The scheme is found unconditionally stable. It shows that the method is quite efficient to capture the dynamics of flow at far field. Results corresponding to other two geometries have been presented at different time. Further, a study on the effect of the real medium on the flow is presented. Schematic shows that with the increase in non-ideal property of the medium, the formation of shock becomes faster.

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball $$\left\{\begin{array}[]{llll}\phantom{\varepsilon^{3}}\dot{x}=y,&x,\,y\in \mathbb{R}^{2},\quad u\in\mathbb{R}^{2},\\ \varepsilon^{3}\dot{y}=Jy+u,&\,\|u\|\leq 1,\quad 0

We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter e > 0) whose limit (when e tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter e) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

Purpose The purpose of this study is sensitivity analysis of the L2-norm and H1-seminorm of the solution of a diffusive–convective–reactive problem to topological changes of the underlying material. Design/methodology/approach The topological derivative method is used to devise a simple and efficient topology design algorithm based on a level-set domain representation method. Findings Remarkably simple analytical expressions for the sensitivities are derived, which are useful for practical applications including heat exchange topology design and membrane eigenvalue maximization. Originality/value The topological asymptotic expansion associated with a diffusive–convective–reactive equation is rigorously derived, which is not available in the literature yet.

Within the field-theory approach, the critical behavior of disordered spin systems with frozen point impurities and additional dipole–dipole interactions is investigated. The system is described by the Heisenberg model. The computation is performed in a two-loop approximation in three-dimensional space. The Pade–Borel method is used to sum asymptotic series. It is shown that a threshold value of the dipole–dipole interaction intensity exists such that the influence of point frozen impurities becomes important once the value exceeds this threshold. For small values of the dipole–dipole interaction intensity, the dependence of critical exponents on it is obtained.

The Sommerfeld integral for an impedance plane has played an important role in many electromagnetic applications. A direct numerical computation may not be efficient and cannot provide a robust and accurate result especially for highly conductive media and/or antennas located near the surface. Two complete asymptotic series are formulated for the integral, which is a power series of the reciprocal of the distance between the source and observation point. The asymptotic series are nonuniform, so the application of the series is limited and is not accurate for the aforementioned cases. Hence, in this article, a new uniform asymptotic expansion of the integral is derived, which is represented in terms of repeated integrals of the complementary error function. The representation is very similar to the conventional uniform expansion of the integral. However, the formulated expansion is a complete power series of the reciprocal of the distance, while the conventional series contains only the first-order term. For a far-field region, the proposed expansion is mathematically shown to reduce the known complete nonuniform expansion for the first few terms of the expansion. For some scenarios that include near-earth propagation and/or highly conductive media, the proposed formulation is numerically verified and its mathematical properties are examined.

In this paper, we investigate the coupled cubic-quintic nonlinear Schrodinger equation, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. The breather wave solutions and novel bright–dark rogue wave solutions can be constructed by using the Darboux-dressing transformation and asymptotic expansion. These solutions show the spatiotemporal patterns of novel bright–dark rogue waves. It is demonstrated that the coupled or multi-component systems contain more interesting rogue wave phenomena than single component systems. Our results can be of importance in understanding and predicting the rogue waves of coupled cubic-quintic nonlinear Schrodinger equation.

The present research article presents the investigation of diffraction phenomenon of EM-plane wave by a non-symmetric plate of finite length under the influence of cold plasma. The Wiener–Hopf equation is formulated by boundary value problem related to this model and Fourier transform. The standard way of Wiener–Hopf procedure is used to tackle the resulting equation. Asymptotic expansion and modified stationary phase method are used to find the result for the diffracted wave by finite plate under the assumption of Neumann boundary conditions in an anisotropic medium. The case of an isotropic medium has been obtained by assigning the particular values to elements of permittivity tensor. The high-frequency signal can be assumed only when very large operating frequency is taken into account as compared to the cyclotron frequency. Various physical parameters for isotropic and anisotropic medium are discussed graphically.

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries (KdV) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation (DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. High-accuracy numerical solutions are then obtained through the fast Fourier transform (FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly, periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction. Keywords: Particle trajectories, Linear shear current, Solitary waves, Direct numerical simulation

Inclusion of dispersion effects in density-functional calculations is now standard practice in computational chemistry. In many dispersion models, the dispersion energy is written as a sum of pairwise atomic interactions consisting of a damped asymptotic expansion from perturbation theory. There has been much recent attention drawn to the importance of "many-body" dispersion effects, which by their name imply limitations with a pairwise atomic expansion. In this perspective, we clarify what is meant by many-body dispersion, as this term has previously referred to two very different physical phenomena, here classified as electronic and atomic many-body effects. Atomic many-body effects refer to the terms in the perturbation-theory expansion of the dispersion energy involving more than two atoms, the leading contribution being the Axilrod-Teller-Muto three-body term. Conversely, electronic many-body effects refer to changes in the dispersion coefficients of the pairwise terms induced by the atomic environment. Regardless of their nature, many-body effects cause pairwise non-additivity in the dispersion energy, such that the dispersion energy of a system does not equal the sum of the dispersion energies of its atomic pairs taken in isolation. A series of examples using the exchange-hole dipole moment (XDM) method are presented to assess the relative importance of electronic and atomic many-body effects on the dispersion energy. Electronic many-body effects can result in variation in the leading-order C6 dispersion coefficients by as much as 50%; hence, their inclusion is critical for good performance of a pairwise asymptotic dispersion correction. Conversely, atomic many-body effects represent less than 1% of the total dispersion energy and are much less significant than higher-order (C8 and C10) pairwise terms. Their importance has been previously overestimated through empirical fitting, where they can offset underlying errors stemming either from neglect of higher-order pairwise terms or from the base density functional.

An asymptotic solution of ray equations is used for numerical simulation of the electromagnetic wave propagation in a smoothly non-uniform Earth-ionosphere duct. Being an extension of the adiabatic invariant approach, our method allows one to easily calculate and visualize all descriptive and energetic characteristics of global HF radio waves, including transition between multi-hop and ricochet propagation modes, antipodal focusing, geometric divergence and ohmic absorption. A quasi-parabolic model of F2 layer with the parameters taken from the international model IRI-2012 is used in numerical examples. From the mathematical point of view, it is an example of two-scale asymptotic expansion in a small parameter $\nu \sim {H/L} \sim L/\pi R$ where ${H}$ is a characteristic height of the ionospheric layer, ${L}$ is the scale of its horizontal non-uniformity, ${R}$ is the Earth radius.

We generalize the results on the existence of an over-stable solution of singularly perturbed differential equations to the equations of the form \begin{document}$ \varepsilon\ddot{x}-F(x,t,\dot{x},k(t), \varepsilon) = 0 $\end{document} . In this equation, the time dependence prevents from returning to the well known case of an equation of the form \begin{document}$ \varepsilon dy/dx = F(x,y,a, \varepsilon) $\end{document} where \begin{document}$ a $\end{document} is a parameter. This can have important physiological applications. Indeed, the coupling between the cardiac and the respiratory activity can be modeled with two coupled van der Pol equations. But this coupling vanishes during the sleep or the anesthesia. Thus, in a perspective of an application to optimal awake, we are led to consider a non autonomous differential equation.

In this work, stress and displacement fields were computed around a crack tip in the case of nearly incompressible and isotropic neo-Hookean material. The constitutive equation was linearized, so that the Cauchy stress tensor could be written as a sum of two components: the linear response in term of elastic Hooke’s law and the nonlinear one. Based on this decomposition, an asymptotic analysis has been developed, the fields of linear elastic fracture mechanics (LEFM-theory) are the zero-order terms of the asymptotic expansion. The validity of the proposed theory has been checked in the case of a mode-I crack problem. A numerical model was constructed using a finite element method. It has shown that the computed fields arising from this theory are qualitatively in agreement with those of the finite element simulations.

We consider the self-diffusiophoresis of axisymmetric particles using a continuum description where the interfacial chemical reaction is modeled by first-order kinetics with a prescribed axisymmetric distribution of rate-constant magnitude. We employ the standard macroscale framework where the interaction of solute molecules with the particle boundary is represented by diffusio-osmotic slip. The dimensionless problem governing the solute transport involves two parameters (the particle slenderness ϵ and the Damkohler number Da) as well as two arbitrary functions which describe the axial distributions of the particle shape and rate-constant magnitude. The resulting particle speed is determined throughout the solution of the accompanying problem governing the flow about the force-free particle. Motivated by experimental configurations, we employ slender-body theory to investigate the asymptotic limit ϵ ≪ 1. In doing so, we seek algebraically accurate approximations where the asymptotic error is smaller than a positive power of ϵ. The resulting approximations are thus significantly more useful than those obtained in the conventional manner, where the asymptotic expansion is carried out in inverse powers of ln ϵ. The price for that utility is that two linear integral equations need to be solved: one governing the axial solute-sink distribution and the other governing the axial distribution of Stokeslets. When restricting the analysis to spheroidal particles, no need arises to solve for the Stokeslet distribution. The integral equation governing the solute-sink distribution is then solved using a numerical finite-difference scheme. This solution is supplemented by a large-Da asymptotic analysis, wherein a subtle nonuniformity necessitates a careful treatment of the regions near the particle ends. The simple approximations thereby obtained are in excellent agreement with the numerical solution.

Distributed scattering centre (DSC) is a particular type of scattering centres which has the strongest scattering amplitude among all scattering centres. Furthermore, their distributed signatures in radar images have directly shown geometries of targets. Therefore, DSCs play an important role in target recognition. DSCs are currently described by the attributed scattering centre (ASC) model, which is developed out of the asymptotic expansion solutions to canonical geometries. However, rather than canonical geometries, real radar targets have various geometric structures, which show different signatures in radar images. In order to characterise these various signatures, the parametric model is modified. Scattered fields from planar and single-curved surfaces with arbitrary shapes are studied in this study and motivated by which, the parametric models of DSCs for more general structures are presented. To validate these models, the scattered waves and the post-imaging results simulated by these models are compared with those obtained by a rigorous full-wave numerical method, and those obtained by the conventional ASC model. The comparison results demonstrate these models have a higher accuracy over the conventional model in simulations of scattered waves, as well as inverse synthetic aperture radar image signatures of different geometric structures.

The results of mathematical modeling of stationary physical processes in the electron-hole plasma of the active region (i-region) of integral p-i-n-structures are presented. The mathematical model is written in the framework of the hydrodynamic thermal approximation, taking into account the phenomenological data on the effect on the dynamic characteristics of charge carriers of heating of the electron-hole plasma as a result of the release of Joule heat in the volume of the i-th region and the release of recombination energy. The model is based on a nonlinear boundary value problem on a given spatial domain with curvilinear sections of the boundary for the system of equations for the continuity of the current of charge carriers, Poisson, and thermal conductivity. The statement of the problem contains a naturally formed small parameter, which made it possible to use asymptotic methods for its analytical-numerical solution. A model nonlinear boundary value problem with a small parameter is reduced to a sequence of linear boundary value problems by the methods of perturbation theory, and the physical domain of the problem with curvilinear sections of the boundary is reduced to the canonical form by the method of conformal mappings. Stationary distributions of charge carrier concentrations and the corresponding temperature field in the active region of p-i-n-structures are obtained in the form of asymptotic series in powers of a small parameter. The process of refining solutions is iterative, with the alternate fixation of unknown tasks at different stages of the iterative process. The asymptotic series describing the behavior of the plasma concentration and potential in the region under study, in contrast to the classical ones, contain boundary layer corrections. It was found that boundary functions play a key role in describing the electrostatic plasma field. The proposed approach to solving the corresponding nonlinear problem can significantly save computing resources

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(\tau)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \pi i \tau}$ approaches a root of unity, i.e. as $\widetilde \tau \equiv m \tau + n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(\tau)$ has terms of the form $\widetilde \tau^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde \tau \to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index., Comment: v3: minor corrections and clarifications added