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Motivated by the simplest case of tt*-Toda equations, we study the large and small x asymptotics for x > 0 of real solutions of the sinh-Godron Painlevé III( D 6 ) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann–Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions. [ABSTRACT FROM AUTHOR]

In a recent study, we showed that the large (x,t)\left(x,t) behavior of a class of physically relevant solutions of Boussinesq’s equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive xx-axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here, we provide a proof of this asymptotic formula.

We consider the linear elasticity system for a body coated with a thin elastic layer. The effect of the thin layer on that body can be described through an impedance operator linking the displacement and the traction on the interface. This operator cannot be in general explicitly computed for an arbitrary shape. In this paper, we consider the case of a linear elastic slab with Hooke's law. We compute by the Fourier transform the exact symbol of the impedance operator and prove that it is a pseudodifferential operator of order one. Then, we give a Padé-like approximation at order three (with respect to the thickness of the layer) for the impedance operator. The pattern of this approximation will be a helpful guide for approximating the impedance operator for a body whose boundary has nonzero curvature. [ABSTRACT FROM AUTHOR]

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order n that are uniformly valid for unbounded complex values of the argument z, including the real interval 0 ≤ z ≤ 1 in which the zeros in the right half plane are located: symmetry extends the results to the left half plane. The approximations are derived from the differential equation satisfied by these polynomials, and other independent solutions are also considered. For large n this equation is characterized by having a simple pole, and expansions valid at this singularity involve Bessel functions and slowly varying coefficient functions. The expansions for these functions are simpler than previous approximations, in particular being computable to a high degree of accuracy. Simple explicit error bounds are derived which only involve elementary functions, and thereby provide a simplification of previous expansions and error bounds associated with differential equations having a large parameter and simple pole. [ABSTRACT FROM AUTHOR]

We consider self-adjoint Dirac-type systems with rectangular matrix potentials on the interval [0, b), where 0 < b ≤ ∞. We present a new proof of the local Borg–Marchenko uniqueness theorem. The high-energy asymptotics of the Weyl–Titchmarsh functions and the local Borg–Marchenko uniqueness theorem are derived for locally smooth potentials at the right endpoint. [ABSTRACT FROM AUTHOR]

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form P (h) ≈ h 1 - m as h → 0 + with m > 1 , modeling repulsion between the liquid–gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any m > 1 and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behavior in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to π / 2 for all m > 1 and finite energy for all m < 3 . We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral data of two boundary value problems with separated boundary conditions. For this inverse problem, we solve the most fundamental question of the inverse spectral theory about the necessary and sufficient conditions of solvability. In addition, we prove the local solvability and stability of the inverse problem. Furthermore, we obtain very simple sufficient conditions of solvability in the self-adjoint case. The main results are proved by a constructive method that reduces the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. In the future, our results can be generalized to various classes of higher-order differential operators with integrable or distribution coefficients. [ABSTRACT FROM AUTHOR]

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The contribution of this paper is to show that, for the residual neural network model, the deep layer limit coincides with a parameter estimation problem for a nonlinear ordinary differential equation. In particular, whilst it is known that the residual neural network model is a discretisation of an ordinary differential equation, we show convergence in a variational sense. This implies that optimal parameters converge in the deep layer limit. This is a stronger statement than saying for a fixed parameter the residual neural network model converges (the latter does not in general imply the former). Our variational analysis provides a discrete-to-continuum Γ -convergence result for the objective function of the residual neural network training step to a variational problem constrained by a system of ordinary differential equations; this rigorously connects the discrete setting to a continuum problem. [ABSTRACT FROM AUTHOR]

We develop in this paper a Susceptible Exposed Infectious Hospitalized and Recovered (SEIHR), spread model. In the model studied, we introduce a recruitment constant, to take into account the fact that newborns can transmit disease. The disease-free and endemic equilibrium points are computed and analyzed. The basic reproduction number 𝒭0 is acquired, when 𝒭0 ≤ 1, the disease dies out and persists in the community whenever 𝒭0 > 1. From numerical simulation, we illustrate our theoretical analysis.

We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded and that the complex limit set of each solution exists and is compact and connected.

One of directions in the development of Lomov's regularization method is the approach related to holomorphic regularization of singularly perturbed problems, which allows one to construct solutions to such problems in the form of series in powers of a small parameter that converge in the usual sense. For boundary-value problems, the problem of pseudo-holomorphic continuation of solutions is very urgent. In this paper, we examine a boundary-value problem for a Tikhonov system and give conditions for the existence of its pseudo-holomorphic solution. [ABSTRACT FROM AUTHOR]

Lomov's regularization method is generalized to resonant, weakly nonlinear, singularly perturbed systems in the case of intersecting roots of the characteristic equation of the limit operator. For constructing asymptotic solutions, the regularization of the original problem by using normal forms developed by the authors is performed. In the absence of resonance, the regularizing normal form is linear, whereas in the presence of resonances, it is nonlinear. In this paper, the resonant case of a weakly nonlinear problem is considered. By using an algorithm of normal forms, we construct an asymptotic solution of any order (with respect to a parameter) and justify this algorithm. [ABSTRACT FROM AUTHOR]

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author's knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance. [ABSTRACT FROM AUTHOR]

In this paper, we investigate so-called slowly traveling wave solutions for a quasilinear parabolic equation in detail. Over the past three decades, the motion of the plane curve by the power of its curvature with positive exponent α has been intensively investigated. For this motion, blow-up phenomena of curvature on cusp singularity in the plane curve with self-crossing points have been studied by several authors. In their analysis, particularly in estimating the blow-up rate, the slowly traveling wave solutions played a significantly important role. In this paper, aiming to clarify the blow-up phenomena, we derive an asymptotic expansion of the slowly traveling wave solutions with respect to the parameter κ , which is proportional to the maximum of the curvature of the curve, as κ goes to infinity. We discovered that the result depends discontinuously on the parameter δ = 1 + 1 / α . It suggests that the blow-up phenomenon may also drastically change according to parameter δ . [ABSTRACT FROM AUTHOR]

We are interested in a coupled system of two abstract Cauchy problems; one of them is posed in small time interval. The aim is to obtain a condition that replaces the effect of this small interval. This condition is constructed from an approximation of Steklov–Poincaré operator. In this paper, we analyze the construction and the approximation of this operator. Finally, the results obtained will be applied to a problem of diffraction of an electromagnetic wave by a perfectly conducting planar obstacle coated with thin layer of dielectric material. [ABSTRACT FROM AUTHOR]

In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature. [ABSTRACT FROM AUTHOR]

We consider a nonlinear ordinary differential equation of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the origin. The methods created in power geometry in recent years make it possible to compute formal solutions to that equation in the form of Dulac series. We describe the corresponding algorithm and prove a sufficient convergence condition for such formal solutions. [ABSTRACT FROM AUTHOR]

Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$ -linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort. [ABSTRACT FROM AUTHOR]

We consider the asymptotic expansion of the one-parameter Mittag-Leffler function E a (- x) for x → + ∞ as the parameter a → 1 . The dominant expansion when 0 < a < 1 consists of an algebraic expansion of O (x - 1) (which vanishes when a = 1 ), together with an exponentially small contribution that approaches e - x as a → 1 . Here we concentrate on the form of this exponentially small expansion when a approaches the value 1. Numerical examples are presented to illustrate the accuracy of the expansion so obtained. [ABSTRACT FROM AUTHOR]

In many industrial applications, reactive filters, which remove toxic substances using chemical reactions, prove very efficient and versatile. One such example is the removal of sulphur dioxide from flue gas by converting it into liquid sulphuric acid. In this paper, we extend a model developed in Kiradjiev et al. (SIAM J Appl Math 82:591–619, 2021) for the operation of a reactive filter to consider the long-time behaviour and determine how the filter becomes clogged with liquid and becomes inefficient. We explore how the parameters that govern the operation of the filter affect its efficiency and discuss optimal filtration strategies. We also consider a distinguished asymptotic limit in terms of the key dimensionless parameters and obtain an explicit solution that compares well with the numerical solution to the full problem. Finally, we compare the results of our model with those from a simplified model for a reactive filter considered in Breward and Kiradjiev (J Eng Math, 129(14):1–28, 2021). [ABSTRACT FROM AUTHOR]

Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 푛-th order accurate; 푛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed. [ABSTRACT FROM AUTHOR]

A scalar nonlinear integro-differential equation with time-variable and bounded delays and generalized Caputo proportional fractional derivative is considered. The main goal of this paper is to study the stability properties of the zero solution. Results are given concerning stability, exponential stability, asymptotic stability, and boundedness of solutions. The investigations are based on an application of a quadratic Lyapunov function, its generalized Caputo proportional derivative, and a modification of the Razumikhin approach. Some auxiliary properties of the generalized Caputo proportional derivative are proved. Five illustrative examples are included. [ABSTRACT FROM AUTHOR]

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the 'entire part' of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which can be considered as a generalization of the differential equation for the zero-order cylinder functions. Since the asymptotics of our entire part of the Hankel function contains logarithmic terms, this new function can be used to derive asymptotic expansions of some other functions whose asymptotics contain logarithmic terms too. In the Fourier image it can be shown that these expansions are not merely asymptotic but actually convergent. [ABSTRACT FROM AUTHOR]

We consider a family of linearly viscoelastic shells with thickness 2 ε , all having the same middle surface S = θ (ω ¯) ⊂ I R 3 , where ω ⊂ I R 2 is a bounded and connected open set with a Lipschitz-continuous boundary γ and θ ∈ C 3 (ω ¯ ; I R 3) . The shells are clamped on a portion of their lateral face, whose middle line is θ (γ 0) , where γ 0 is a non-empty portion of γ . The aim of this work is to show that the viscoelastic Koiter's model is the most accurate two-dimensional approach in order to solve the displacements problem of a viscoelastic shell. Furthermore, the solution of the Koiter's model, ξ K ε = (ξ K , i ε) , is in H 1 (0 , T ; V K (ω)) , with ξ K , i ε : [ 0 , T ] × ω ¯ → R the covariant components of the displacements field ξ K , i ε a i of the points of the middle surface S and where V K (ω) : = { η = (η i) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω) ; η i = ∂ ν η 3 = 0 in γ 0 } , with ∂ ν denoting the outer normal derivative along γ . Under the same assumptions as for the viscoelastic elliptic membranes problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic elliptic membrane) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; L 2 (ω)) for the normal component, as ε → 0 . Under the same assumptions as in the viscoelastic flexural shell problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic flexural shell) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; H 2 (ω)) for the normal component, as ε → 0 . Also, we obtain analogous results assuming the same assumptions as in the viscoelastic generalized membranes problem. Therefore, we justify the two-dimensional viscoelastic model of Koiter for all kind of viscoelastic shells. [ABSTRACT FROM AUTHOR]

We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system. [ABSTRACT FROM AUTHOR]

Using the so-called direct scheme method, an asymptotic expansion of n-th order to the solution of a class of singularly perturbed linear-quadratic optimal problem with integral constraint on control is constructed. The expansion contains three type functions. Two of them are boundary layer functions in the vicinities of two fixed end-points, and the remain is regular function. A numerical example is represented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach's fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space. [ABSTRACT FROM AUTHOR]

This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

We study the dynamics of a two-level crossing model with a polynomial modification of the linear Landau–Zener tunneling. For a cubic modification, we express the non-adiabatic transition amplitudes analytically via the bi-confluent Heun functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations in a large part of the parameter space. The analytical approximations are validated by comparison with numerical results. Finally, we discuss the application to fast high fidelity quantum gates in quantum computing. [ABSTRACT FROM AUTHOR]

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in (Ann Inst H Poincaré Anal Non Linéaire, 4(1):1–47 1987). We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work. [ABSTRACT FROM AUTHOR]

The paper investigates the interaction between the notions of expansiveness and admissibility. We consider a polynomially bounded discrete evolution family and define an admissibility notion via the solvability of an associated difference equation. Using the admissibility of weighted Lebesgue spaces of sequences, we give a characterization of discrete evolution families which are polynomially expansive and also those which are expansive in the ordinary sense. Thereafter, we apply the main results in order to infer continuous-time characterizations for the notions of expansiveness through the solvability of an associated integral equation. [ABSTRACT FROM AUTHOR]

We consider a class of power-logarithmic germs. We call them parabolic Dulac germs, as they appear as Dulac germs (first-return germs) of hyperbolic polycycles. In view of formal or analytic characterization of such a germ f by fractal properties of several of its orbits, we study the tubular ε -neighborhoods of orbits of f with initial points x 0 . We denote by A f (x 0 , ε) the length of such a tubular ε -neighborhood. We show that, even if f is an analytic germ, the function ε ↦ A f (x 0 , ε) does not have a full asymptotic expansion in ε in the scale of powers and (iterated) logarithms. Hence, this partial asymptotic expansion cannot contain necessary information for analytic classification. In order to overcome this problem, we introduce a new notion: the continuous time length of the ε -neighborhood A f c (x 0 , ε) . We show that this function has a full transasymptotic expansion in ε in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length ε ↦ A f (x 0 , ε) . Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f. [ABSTRACT FROM AUTHOR]

High accuracy analytic functions have been determined for the ground state eigenvalue of the quantum anharmonic potential x4+λx6. The procedure here used is a new application of the multipoint quasi-rational approximation method (MPQA) to quantum mechanics. Previous applications of the technique were performed in the case where the eigenvalues of the potential were already known for the particular case of λ=0. The extension of the method, where this condition is not accomplished, is presented here. As a first step, power and asymptotic expansions have been now also determined. This is followed by the search of a bridge function connecting both expansions, and later the determination of the parameters. Accuracies smaller than 0.1% have been found with only nine parameters.

For the second-order nonlinear ordinary differential equation u xx ' ' = u 5 − tu 3 − x , we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5− tu3− x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2→∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients. [ABSTRACT FROM AUTHOR]

We analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients. [ABSTRACT FROM AUTHOR]

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class of positive integer-valued distributions with power tails. [ABSTRACT FROM AUTHOR]

Linear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z). [ABSTRACT FROM AUTHOR]

The Painlevé-III equation with parameters Θ 0 = n + m and Θ ∞ = m - n + 1 has a unique rational solution u (x) = u n (x ; m) with u n (∞ ; m) = 1 whenever n ∈ Z . Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of u n (x ; m) in the limit n → + ∞ with m ∈ C held fixed. We isolate an eye-shaped domain E in the y = n - 1 x plane that asymptotically confines the poles and zeros of u n (x ; m) for all values of the second parameter m. We then show that unless m is a half-integer, the interior of E is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of E but blows up near the origin, which is the only fixed singularity of the Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ for u n (x ; m) that we compare with u n (x ; m) itself for finite values of n to illustrate their accuracy. We also consider the exceptional cases where m is a half-integer, showing that the poles and zeros of u n (x ; m) now accumulate along only one or the other of two "eyebrows," i.e., exterior boundary arcs of E. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak asymptotic in the C -plane of some wave functions resulting from the WKB-techniques applied to a Schrödinger equation with quartic oscillator and having some boundary condition. As a first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions. Especially, we investigate the properties of the Cauchy transform of the root counting measure of re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form C 2 (z) + r (z) C (z) + s (z) = 0 , where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential - p (z) d z 2 where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential. [ABSTRACT FROM AUTHOR]

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression. [ABSTRACT FROM AUTHOR]