Formal Asymptotic Expansion Research Articles

Nonlinear diffusion, boundary layers and nonsmoothness: Analysis of challenges in drift–diffusion semiconductor simulations

We study different discretizations of the van Roosbroeck system for charge transport in bulk semiconductor devices that can handle nonlinear diffusion. Three common challenges corrupting the precision of numerical solutions will be discussed: boundary layers, discontinuities in the doping profile, and corner singularities in L-shaped domains. We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as advanced Scharfetter–Gummel type finite-volume discretization schemes. The most problematic of these challenges are boundary layers in the quasi-Fermi potentials near ohmic contacts, which can have a drastic impact on the convergence order. Using a novel formal asymptotic expansion, our theoretical analysis reveals that these boundary layers are logarithmic and significantly shorter than the Debye length.

Formal oscillatory integrals and deformation quantization

Following [14] and [12], we formalize the notion of an oscillatory integral interpreted as a functional on the amplitudes supported near a fixed critical point $x_0$ of the phase function with zero critical value. We relate to an oscillatory integral two objects, a formal oscillatory integral kernel and the full formal asymptotic expansion at $x_0$. The formal asymptotic expansion is a formal distribution supported at $x_0$ which is applied to the amplitude. In [12] this distribution itself is called a formal oscillatory integral (FOI). We establish a correspondence between the formal oscillatory integral kernels and the FOIs based upon a number of axiomatic properties of a FOI expressed in terms of its formal integral kernel. Then we consider a family of polydifferential operators related to a star product with separation of variables on a pseudo-K\"ahler manifold. These operators evaluated at a point are FOIs. We completely identify their formal oscillatory kernels.

Validity of Formal Asymptotic Expansions for Singularly Perturbed Competition-Diffusion Systems

We consider a two-species competition-diffusion system involving a small parameter $\varepsilon>0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\varepsilon\approx0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen--Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\varepsilon\rightarrow0$. In this paper, we rigorously verify this latter ansatz for a large class of initial data. The proof relies on a rescaling argument, the super--subsolution method, and a Liouville type theorem for eternal solutions of parabolic systems. Roughly speaking, the Liouville type theorem states that any eternal solution that lies between two traveling waves is itself a traveling wave. The same Liouville type theorem was established for the scalar Allen--Cahn equation by Berestycki and Hamel. In view of their importance, we prove the Liouville type theorems in a rather general framework, not only for two-species competition-diffusion systems but also for $m$-species cooperation-diffusion systems possibly with time periodic or spatially periodic coefficients.

Singularly perturbed solutions of a class of non-Fourier temperature field distribution

Thermoelastic coupling model excited by laser is of great significance in engineering. To study the thermoelastic coupling model, the distribution of temperature field must be determined firstly. Because the laser excitation time is short (usually femtosecond), the traditional Fourier heat conduction law is no longer suitable. Therefore, it is necessary to establish the distribution of temperature field by using the non-Fourier heat conduction law. Previous studies on the temperature field model mostly use numerical analysis and computer simulation to discuss its numerical solution, but few can directly solve the analytical solution of the model. Up to now, there are few reports about using singularly perturbed analysis method to solve the asymptotic solution of temperature field model and determine the jumping position of heat conductivity coefficient. In this paper, a temperature field model is constructed by using the non-Fourier heat conduction law, i.e. a class of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The nonlinear singularly perturbed two-parameter hyperbolic equations with discontinuous coefficients are obtained when the heat conduction coefficients jump due to sharp temperature changes. By using the singularly perturbed biparametric expansion method, the asymptotic solution of the problem is obtained. First, the expansion of the problem is obtained by using singularly perturbed method. The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the internal and external solutions and the maximum modulus estimates of the time derivatives, and the formal asymptotic expansion of the solutions is obtained. Secondly, the singularly perturbed hyperbolic equation is corrected by the singular perturbation theory, and the derivative of the solution is estimated. The position expression of the jump of the thermal conductivity coefficient is determined by the Fourier transform, and the seam method is used to connect the seams of the two sides of the jump position of the thermal conductivity coefficient, thus the form asymptotic expansion of the solution is obtained. Finally, the uniform validity of the asymptotic solution is obtained by estimating the residual term, and the distribution of the temperature field with discontinuous heat conduction coefficient is obtained. In this paper, we have synthetically applied the knowledge of ordinary differential equations, partial differential equations, mathematical and physical equations, nonlinear acoustics, mathematical analysis, singular perturbation theory and so on, which enriched the study of non-Fourier temperature field model.

Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

非Fourier温度场分布的奇摄动解

A temperature field model for single-layer materials was constructed with the non-Fourier heat conduction law, i.e. a type of singularly perturbed hyperbolic equations with small parameters in an unbounded domain. The asymptotic solution to the problem was obtained with the singularly perturbed expansion method. Firstly, the singular perturbation method was used to obtain the external solution and boundary layer correction terms of the problem. Through estimation of the maximum norms of the internal solution and the external solution, and the maximum norms of the time derivative, and under the theory of linear parabolic equations, the existence and uniqueness of the internal and external solutions were obtained, and the formal asymptotic expansion of the solution was obtained. The L² estimator of the asymptotic solution was given with the remainder estimator. The uniform validity of the asymptotic solution and the distribution of the temperature field in the unbounded domain were got. Through singular perturbation analysis, the relationship between the non-Fourier temperature field and the Fourier temperature field was given, and the specific behaviors of the non-Fourier temperature field were described.

Selection of quasi-stationary states in the Navier–Stokes equation on the torus

The two dimensional incompressible Navier–Stokes equation on with , periodic boundary conditions, and viscosity is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if , while the dipole will be selected if . Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of δ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all , there is an initial time period in which the high modes decay at the rapid rate , while the low modes evolve at the slower rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.

Asymptotic behavior of optimal exercise strategy for a small number of executive stock options

Abstract We investigate the asymptotic behavior, as the number of options approaches zero, of the optimal exercise strategy of continuous-exercise perpetual executive stock options. Here the optimal strategy is described in terms of the free boundary of a degenerate variational inequality. To our knowledge, our method based on the convexity of the variational inequality at the initial time is new. Also, the unique formal asymptotic expansions of the free boundary and the solution of the variational inequality are derived.

On the infinite Prandtl number limit in two-dimensional magneto-convection

In this paper, the infinite limit of the Prandtl number is justified for the two-dimensional incompressible magneto-convection, which describes the nonlinear interaction between the Rayleigh–Bénard convection and an externally magnetic field. Both the convergence rates and the thickness of initial layer are obtained. Moreover, based on the method of formal asymptotic expansions, an effective dynamics is constructed to simulate the motion within the initial layer.

Modulational instability in a full‐dispersion shallow water model

AbstractWe propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.

Heat kernel asymptotics of the subordinator and subordinate Brownian motion

For a class of Laplace exponents, we consider the transition density of the subordinator and the heat kernel of the corresponding subordinate Brownian motion. We derive explicit approximate expressions for these objects in the form of asymptotic expansions: via the saddle point method for the subordinator’s transition density and via the Mellin transform for the subordinate heat kernel. The latter builds on ideas from index theory using zeta functions. In either case, we highlight the role played by the analyticity of the Laplace exponent for the qualitative properties of the asymptotics.

On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

We prove a relation between the scaling $h^\beta$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is $h^4$, as claimed in [AAE+12] using a formal asymptotic expansion. The proof involves calculating the $\Gamma$-limit for the elastic energies of small balls $B_h(p)$, scaled by $h^4$, and showing that the limit infimum energy is given by a square of a norm of the curvature at a point $p$. This $\Gamma$-limit proves asymptotics calculated in [AKM+16].

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

The solution of the three-dimensional nonlinear wave equation −U″TT + U″XX + U″YY + U″ZZ = f(eT, eX, eY, eZ, U) by means of the method of matched asymptotic expansions is considered. Here e is a small positive parameter and the right-hand side is a smoothly changing source term of the equation. A formal asymptotic expansion of the solution of the equation is constructed in terms of the inner scale near a typical butterfly catastrophe point. It is assumed that there exists a standard outer asymptotic expansion of this solution suitable outside a small neighborhood of the catastrophe point. We study a nonlinear second-order ordinary differential equation (ODE) for the leading term of the inner asymptotic expansion depending on three parameters: u″xx = u5 − tu3 − zu2 − yu − x. This equation describes the appearance of a step-like contrast structure near the catastrophe point. We briefly describe the procedure for deriving this ODE. For a bounded set of values of the parameters, we obtain a uniform asymptotics at infinity of a solution of the ODE that satisfies the matching conditions. We use numerical methods to show the possibility of locating a shock layer outside a neighborhood of zero in the inner scale. The integral curves found numerically are presented.

Optimal control of linear systems with balanced reduced-order models: Perturbation approximations

In this article we study balanced model reduction of linear systems for feedback control problems. Specifically, we focus on linear quadratic regulators with collocated inputs and outputs, and we consider perturbative approximations of the dynamics in the case that the Hankel singular values corresponding to the hardly controllable and observable states go to zero. To this end, we consider different perturbative scenarios that depend on how the negligible states scale with the small Hankel singular values, and derive the corresponding limit systems as well as approximate expressions for the optimal feedback controls. Our approach that is based on a formal asymptotic expansion of an algebraic Riccati equations associated with the Pontryagin maximum principle and that is validated numerically shows that model reduction based on open-loop balancing can also give good closed-loop performance.

The Asymptotically Self-Similar Regime for the Einstein Vacuum Equations

We develop a local theory for the construction of singular spacetimes in all spacetime dimensions which become asymptotically self-similar as the singularity is approached. The techniques developed also allow us to construct and classify exact self-similar solutions which correspond to the formal asymptotic expansions of Fefferman–Graham’s ambient metric.

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.

The tunneling effect for a class of difference operators

We analyze a general class of self-adjoint difference operators [Formula: see text] on [Formula: see text], where [Formula: see text] is a multi-well potential and [Formula: see text] is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30–35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs.The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian [Formula: see text] is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance [Formula: see text] induced by [Formula: see text], we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first [Formula: see text] eigenvalues of [Formula: see text] converge to the first [Formula: see text] eigenvalues of the direct sum of harmonic oscillators on [Formula: see text] located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions or quasimodes for the operator [Formula: see text], acting on [Formula: see text], via restriction to the lattice [Formula: see text].Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted [Formula: see text]-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

The sharp interface limit for the stochastic Cahn–Hilliard equation

Nous etudions l’equation de Cahn–Hilliard stochastique dependante en $\varepsilon$, posee en dimensions deux et trois, dans la limite de l’interface nette $\varepsilon \rightarrow$ 0. Le parametre $\varepsilon$ est positif et mesure la largeur de couches de transition generees pendant la separation de phase. Nous couplons aussi la puissance de bruit a ce parametre. Nous determinons la limite a l’aide de series asymptotiques formelles. Dans l’echelle appropriee, nos resultats indiquent que l’ equation de Cahn–Hilliard stochastique converge vers un probleme de Hele-Shaw avec un forcage stochastique dans l’equation de la courbure. Dans le cas d’un bruit suffisamment petit, nous prouvons rigoureusement que la limite est un probleme Hele-Shaw deterministe. Finalement, nous discutons des estimations necessaires afin d’etendre le resultat rigoureux en presence de bruit d’une intensite plus grande.

An asymptotic expansion approach for variable refractive index optical fibers (in weak guidance case)

To study the propagation of guided modes in an optical fiber under weak guidance assumptions, we usually use a scalar approximation of the Maxwell’s system which generally leads to the Helmholtz equations. Most of works in the literature treat optical fibers where the refractive index is constant in the core. In this work, we give more detailed calculations and expend the study to the case where the index of refraction is variable in the core of the fiber. The technique of calculation we propose is based on a formal asymptotic expansion that allows us to derive the equation and the transmission conditions of the scalar eigenvalue problem from Maxwell’s equations in the vectorial case. In the case where the refractive index is variable, which is the main contribution of this work, we carry out asymptotic expansions of lower order and higher order leading to a scalar eigenvalue problem.

Nonlinear difference equations for the generalized little q-Laguerre polynomials

In this paper, a study is made on polynomials orthogonal with respect to the generalized little q-Laguerre weight, defined by , . Here , and . This weight is supported on the exponential lattice . Let the subleading coefficient of of the monic polynomials be From the q-analogue of the ladder operators, the associated supplementary conditions and a ‘sum rule’, we deduce a system of difference equations satisfied by This system is used to obtain the first few terms in a formal asymptotic expansion of . We express the recurrence coefficients in terms of this subleading coefficient and show that the first few terms in the formal expansions in powers of agree with the first few terms for the corresponding expansions of the recurrence coefficients in the classical case. Moreover, we find certain non-linear difference equations for the recurrence coefficients of the monic polynomials, auxiliary functions in the ladder operators and for . We also observe the phenomenon of singularity confinement, related to that observed in the q-discrete Painlevé equations. Furthermore, we give a generalization of the weight function, characterized by for on the exponential lattice. In this situation we find another system of difference equations satisfied by and study its behaviour for large n. The paper ends with the discussion on a deformation of the generalized little q-Laguerre weight.