The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this ‐transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d‐orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose ‐transform is a d‐orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.     

Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight

We study the Hankel determinants associated with the weight where , , , is analytic in a domain containing [ ? 1, 1] and for . In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as and . We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.     

Conservation Laws for the Fully Nonlinear Long Wave Equations

The fully nonlinear long wave equations describe the motion over a flat bottom of a two-dimensional inviscid fluid with a free surface in a gravitational field in the long wave approximation. These equations are shown to possess an infinite number of conservation laws (in two space dimensions) in the form The conserved densities T and the fluxes ?X and ?Y are polynomials in the height h and the horizontal and vertical components of velocity, u and v, and also in integrals of powers of u. The method of proof is a modification of the method recently devised by D. J. Benney to prove that these same equations possess an infinite number of conservation laws (in one space dimension) in the form where T and X are polynomials in the height h and integrals of powers of u. Conservation laws which explicitly contain x and t are also given.     

Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment,from Finite n to Double Scaling

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by , , and . We find that each one satisfies a second‐order differential equation. We show that after a double scaling, the large second‐order differential equation in the variable a with n as parameter satisfied by can be reduced to the Jimbo–Miwa–Okamoto σ form of the Painlevé V equation.     

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.     

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.     

The Kidder Equation:

The Kidder problem is with and where . This looks challenging because of the square root singularity. We prove, however, that for all . Other very simple but very accurate curve fits and bounds are given in the text; . Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are when the coefficients . An initial‐value problem is obtained if is known; the slope Chebyshev series has only a fourth‐order rate of convergence until a simple change‐of‐coordinate restores a geometric rate of convergence, empirically proportional to . Kidder's perturbation theory (in powers of α) is much inferior to a delta‐expansion given here for the first time. A quadratic‐over‐quadratic Padé approximant in the exponentially mapped coordinate predicts the slope at the origin very accurately up to about . Finally, it is shown that the singular case can be expressed in terms of the solution to the Blasius equation.     

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.     

Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation

Complex analytical structure of Stokes wave for two‐dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave prop agating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with radians angle on the crest. A conformal map is used that maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half‐plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half‐plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase in the numerical precision and subsequent increase in the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one square‐root branch point per horizontal spatial period λ of Stokes wave located at the distance from the real line. The increase in the scaled wave height from the linear limit to the critical value marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called the Stokes wave of the greatest height). Here, H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10^?26. The number of poles in tables increases from a few for near‐linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with      

Nonlocality and the Inverse Scattering Transform for the Pavlov Equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so‐called Pavlov equation , in this paper we establish the following. 1. The nonlocal term arising from its evolutionary form corresponds to the asymmetric integral . 2. Smooth and well‐localized initial data evolve in time developing, for , the constraint , where . 3. Because no smooth and well‐localized initial data can satisfy such constraint at , the initial () dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.     

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.     

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 .     

Multiple Homoclinic Solutions for Second‐Order Perturbed Hamiltonian Systems

In this paper, we study the second‐order perturbed Hamiltonian systems where is a parameter, is positive definite for all but unnecessarily uniformly positive definite for , and W is either asymptotically quadratic or superquadratic in x as . Based on variational methods, we prove the existence of at least two nontrivial homoclinic solutions for the above system when small enough.     

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.     

The Evolution of Finite Amplitude Solitary Rossby Waves on a Weak Shear

The evolution equation is derived for finite amplitude, long Rossby waves on a weak shear generalizing an earlier version given by Benney [1].     

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the neighborhood of , the uniform asymptotic expansion is associated with Bessel function of order β as . The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.     

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.     

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.