Global Asymptotics for Meixner‐Pollaczek Polynomials with a Varying Parameter

In this paper, we study the uniform asymptotics of the Meixner‐Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann‐Hilbert problems first introduced by Deift and Zhou [1].     

The Resurgence Properties of the Incomplete Gamma Function II

In this paper, we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by Howls 1992. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function with large a and fixed positive λ, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptotic series of .     

Asymptotic Expansions for Second-Order Linear Difference Equations,II

Infinite asymptotic expansions are derived for the solutions to the second-order linear difference equation where p and q are integers, a(n) and b(n) have power series expansions of the form for large values of n, and a₀ ≠ 0, b₀ ≠ 0. Recurrence relations are also given for the coefficients in the asymptotic solutions. Our proof is based on the method of successive approximations. This paper is a continuation of an earlier one, in which only the special case p ≤ 0 and q = 0 is considered.     

Computation of the Coefficients Appearing in the Uniform Asymptotic Expansions of Integrals

The coefficients that appear in uniform asymptotic expansions for integrals are typically very complicated. In the existing literature, the majority of the work only give the first two coefficients. In a limited number of papers where more coefficients are given, the evaluation of the coefficients near the coalescence points is normally highly numerically unstable. In this paper, we illustrate how well‐known Cauchy‐type integral representations can be used to compute the coefficients in a very stable and efficient manner. We discuss the cases: (i) two coalescing saddles, (ii) two saddles coalesce with two branch points, and (iii) a saddle point near an endpoint of the interval of integration. As a special case of (ii), we give a new uniform asymptotic expansion for Jacobi polynomials in terms of Laguerre polynomials as that holds uniformly for z near 1. Several numerical illustrations are included.     

On the Multiplicity of Solutions Of a Differential Equation Arising in Chemical Reactor Theory

Consider the boundary value problem where β ? 0, τ ? 0. We are concerned with a mathematically rigorous numerical study of the number of solutions in any bounded portion of the positive quadrant (τ ? 0, β ? 0) of the τ, β plane. These correct computational results may then be matched with asymptotic (β→∞, τ ? 0) results developed earlier. These numerical results are based on the development of a posteriori error estimates for the numerical solution of an associated initial-value problem and a priori bounds on .     

The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, and , are given functions of t. In particular, we consider the case when the initial data have algebraic decay as , with as and as . The constant states and are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when and , while in initial‐value problem 2 we consider the case when and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to both initial‐value problems over all parameter values.     

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about ? |t| as t → ∞ have asymptotic expansions whose leading terms are where Ã₊, , Ã_?, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã₊, ) as functions of Ã_? and The nonlinear solutions with Ã_± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to Ã₊ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is ?W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.     

The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, while , are given continuous functions of t ( > 0). In particular, we consider the case when the initial data has algebraic decay as , with as and as . The constant states and are problem parameters. We focus attention on the case when (with ) and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to the initial‐value problem over all parameter values.     

On the Spectral Stability of Kinks in 2D Klein–Gordon Model with Parity‐Time‐Symmetric Perturbation

In a series of recent works by Demirkaya et al., stability analysis for the static kink solutions to the one‐dimensional continuous and discrete Klein–Gordon equations with a ‐symmetric perturbation has been performed. In the present paper, we study two‐dimensional (2D) quadratic operator pencil with a small localized perturbation. Such an operator pencil is motivated by the stability problem for the static kink in 2D Klein–Gordon field taking into account spatially localized ‐symmetric perturbation, which is in the form of viscous friction. Viscous regions with positive and negative viscosity coefficient are balanced. For the considered operator pencil, we show that its essential spectrum has certain critical points generating eigenvalues under the perturbation. Our main results are sufficient conditions ensuring the existence or absence of such eigenvalues as well as the asymptotic expansions for these eigenvalues if they exist.     

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.     

Variation of Parameters and the Renormalization Group Method

This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters. A regular perturbation procedure typically provides asymptotic solutions valid for bounded t values as a positive parameter ε tends to zero. One can eliminate secular terms by introducing a slowly‐varying amplitude obtained as a solution of an amplitude equation on intervals where is bounded. With sufficient stability hypotheses, the results may even hold for all . These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations.     

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

We present an approach for analyzing initial‐boundary value problems which are formulated on the finite interval (, where L is a positive constant) for integrable equation whose Lax pairs involve 3 × 3 matrices. Boundary value problems for integrable nonlinear evolution partial differential equations (PDEs) can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem (RHP). The relevant jump matrices are explicitly given in terms of the three matrix‐value spectral functions , and , which in turn are defined in terms of the initial values, boundary values at , and boundary values at , respectively. However, these spectral functions are not independent; they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half‐line.     

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G‐functions, or equivalently hypergeometric functions , also referred to as hyper‐Bessel functions. In the case it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general , but has not exhibited its reduction. After detailing the necessary working in the case , we consider the problem of reducing the 12 coupled differential equations in the case to a single differential equation for the resolvent. An explicit fourth‐order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third‐order nonlinear equation. The small and large s asymptotics of the fourth‐order equation are discussed, as is a possible relationship of the systems to so‐called four‐dimensional Painlevé‐type equations.     

Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points , N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for in the double scaling limit, namely, and , where and ; see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case is relatively simple (because it is very much like the case when b is fixed), the case is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and , and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.     

Quasi‐Uniform Spectral Schemes (QUSS), Part I: Constructing Generalized Ellipses for Graphical Grid Generation

Chebyshev and Legendre polynomial spectral methods are bedeviled by highly nonuniform grids. The separation between nearest neighbors of an N‐point grid at the center of the interval is larger than the spacing of a uniform grid with the same number of points. Quasi‐Uniform Spectral Schemes (QUSS) redistribute grid points and choose basis functions in order to recover this factor of as nearly as possible while retaining a high density of points near the endpoints to avoid the horrors of the Gibbs or Runge Phenomenon. Here, we introduce a systematic approach, dubbed “mapped cosine bases,” that embraces the widely used Kosloff/Tal‐Ezer functions as a special case. The mapped cosine approach uses grid points that are the images of a uniform grid under the coordinate mapping . Here, we show how to generalize the well‐known graphical construction of the Chebyshev grid using a circle to QUSS mappings using a generalized ellipse. This provides a way to visualize the maps and grids and the subtle differences between different mappings of the mapped cosine family. We illustrate and compare the Kosloff/Tal‐Ezer map with two new maps that use elliptic integrals and Jacobian theta functions, respectively. We show that the elliptic integral grid is an asymptotic approximation to the usual grid for prolate spheroidal functions. This suggests the conjecture that one can obtain the benefits of a prolate basis without the complications of prolate functions by using mapped polynomials instead.     

Large Time Asymptotics with Error Estimates to Solutions of a Forced Burgers Equation

This article deals with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on . Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole–Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to 0 for large time.     

A Solvable Nonlinear Wave Equation

It is pointed out that the nonlinear wave equation can be solved by quadratures. Here a and c are constants, A(y) and B(y) (arbitrary) functions; a t-dependence of all these quantities can also be accommodated. This wave equation can also be rewritten in the (purely differential) form via the substitutions .     

A Boundary Value Problem for Conjugate Conductivity Equations

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half‐plane with conductivity , . The representations are obtained via the so‐called unified transform method or Fokas method, involving a Riemann–Hilbert problem on the complex plane when p is even and on a two‐sheeted Riemann surface when p is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent p, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. The method used to derive our integral representations could be applied in any bounded simply connected domain of the right half‐plane with a smooth boundary.