Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach

Let   Q ( x ) = q _{2 m} x ^{2 m} + q _{2 m −1} x ^{2 m −1}+⋯  be a polynomial of degree 2 m with   q _{2 m} > 0  , and let  {π _n ( x )} _{n ≥1}  be the sequence of monic polynomials orthogonal with respect to the weight   w ( x ) = e ^{− Q ( x )}  on     . Furthermore, let  α _n   and  β _n   denote the Mhaskar–Rakhmanov–Saff (MRS) numbers associated with Q ( x ). By using the Riemann–Hilbert approach, an asymptotic expansion is constructed for  π _n ( c_nz + d_n )  , which holds uniformly for all z bounded away from  (−∞, −1)  , where     and     .     

Viscous Critical-Layer Analysis of Vortex Normal Modes

The linear stability properties of an incompressible axisymmetrical vortex of axial velocity   W ₀( r )  and angular velocity  Ω₀( r )  are considered in the limit of large Reynolds number. Inviscid approximations and viscous WKBJ approximations for three-dimensional linear normal modes are first constructed. They are then shown to be singular at the critical points r_c satisfying  ω= m Ω₀( r_c ) + kW ₀( r_c )  , where ω is the frequency, k and m the axial and azimuthal wavenumbers. The goal of this paper is to resolve these singularities. We show that a viscous critical-layer analysis is analytically tractable. It leads to a single sixth-order equation for the perturbation pressure. This equation is identical to the one obtained in stratified shear flows for a Prandtl number equal to 1. Integral expressions for typical solutions of this equation are provided and matched to the outer inviscid and viscous approximations in the complex plane around r_c . As for planar flows, it is proved that the critical layer solution with a balanced behavior matches a non-viscous approximation in a  4π/3  sector of the complex-plane. As a consequence, the Frobenius expansions of a non-viscous mode on each side of a critical point r_c differ by a π phase jump.     

Author Index to Volumes 100 and 101: 1998

Through both analytical and numerical methods, we solve the eigenproblem u_zz >+(1/ z −λ−( z −1/ε)²) u =0 on the unbounded interval z ∈[−∞, ∞], where λ is the eigenvalue and u ( z )→0 as | z |→∞. This models an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere. It is the usual parabolic cylinder equation with Hermite functions as the eigenfunctions except for the addition of an extra term, which is a simple pole. The pole, which is on the interior of the interval, is interpreted as the limit δ→0 of 1/( z − i δ). The eigenfunction has a branch point of the form z  log( z ) at z =0, where the branch cut is on the upper imaginary axis. The eigenvalue is complex valued with an imaginary part, which we show, through matched asymptotics, to be approximately √ π exp(−1/ε²){1−2ε log ε+ε log 2+γε}. Because T ( λ ) is transcendentally small in the small parameter ε, it lies "beyond all orders" in the usual Rayleigh–Schrödinger power series in ε. Nonetheless, we develop special numerical algorithms that are effective in computing T ( λ ) for ε as small as 1/100.     

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs),   S_mn ( c , η)  , and their eigenvalues,  λ _mn   , for arbitrary complex size parameter c in the asymptotic regime of large  | c |  , m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary  arg( c )  , the PSWFs and their eigenvalues are accurately expressed by established prolate -type or oblate -type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points,   c ^mn _{○; r}   , in the complex c -plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m , n , and  arg( c )  . The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c , are derived for both the prolate - and oblate -type asymptotic expansions and for both  ( n − m )  even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of   c ^mn _{○; r}   . We document this ordering for the specific case of  arg( c ) =π/4  , which occurs for the diffusion equation in spheroidal coordinates. Some representative values of  λ _mn   and   S_mn ( c , η)  for large, complex c are also given.     

Conditional Lie Bäcklund Symmetries and Sign-Invariants to Quasi-Linear Diffusion Equations

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u _t =[ u ^m ( u _x ) ⁿ ] _x + P ( u ) u _x + Q ( u ) , where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η= u _xx + H ( u ) u ² _x + G ( u )( u _x )^{2− n} + F ( u ) u ^{1− n} _x and the Hamilton–Jacobi sign-invariant J = u _t + A ( u ) u ^{n +1} _x + B ( u ) u _x + C ( u ) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.     

Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

We prove that arbitrary (nonpolynomial) scalar evolution equations of order    m  ≥ 7  , that are integrable in the sense of admitting the canonical conserved densities   ρ⁽¹⁾, ρ⁽²⁾  , and   ρ⁽³⁾   introduced in [ 1 ], are polynomial in the derivatives    u _{m −  i}    for  i  = 0, 1, 2. We also introduce a grading in the algebra of polynomials in     u _k     with     k  ≥  m  − 2    over the ring of functions in     x ,  t ,  u , … ,  u _{m −3}    and show that integrable equations are scale homogeneous with respect to this grading .     

The Stokes and Krasovskii Conjectures for the Wave of Greatest Height

The integral equation:
φ_μ(s) = (1/3 π)∫^π ₀((sin φ_μ(t))/(μ ⁻¹+ ∫^t ₀sin φ_μ(u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ_μ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ_∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ_∞( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup _{s ∈[0,π]}φ_μ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer.     

典型實照函數的最小?妥钚∑

<正> 1.如果函數f(z)在包含實軸土某一區間的區域B中是正則的,f(z)在此實軸區間上取實值.在區域B的其餘地方f(z)與(z)同符號;即     

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

Asymptotic formulas, as  ɛ→ 0⁺  , are derived for the solutions of the nonlinear differential equation  ɛ u" + Q ( u ) = 0  with boundary conditions   u (-1) = u (1) = 0  or   u '(-1) = u '(1) = 0  . The nonlinear term Q ( u ) behaves like a cubic; it vanishes at   s _-, 0, s ₊  and nowhere else in  [ s _-, s ₊]  , where   s _- < 0 < s ₊  . Furthermore,   Q '( s _±) < 0, Q '(0) > 0  and the integral of Q on the interval [ s _-, s ₊] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.     

Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations   y ^{( n )}= F ( y ', y , x )  of arbitrary order   n ≥ 3  , where F is algebraic in   y , y '  and locally analytic in x . We prove that for   n > 3  these equations always admit movable branch points. In the case   n = 3  these equations admit movable branch points unless they are of the known class   y '= a ( x )( y ')²+ ( b ₂( x ) y ²+ b ₁( x ) y + b ₀( x )) y '+ ( c ₄( x ) y ⁴+ c ₃( x ) y ³+ c ₂( x ) y ²+ c ₁( x ) y + c ₀( x ))  , where   a ,  b_j ,  c_j   are locally analytic in x .     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

Rates of Convergence to Self-Similar Solutions of Burgers' Equation

We study the large-time behavior of solutions to Burgers' equation with localized initial conditions. Previous studies have demonstrated that these solutions converge to a self-similar asymptotic solution  Θ( x, t )  with an error whose   L_p   norm is of order   t ^{−1+1/2 p}   . Noting that the temporal and spatial translational invariance of the underlying equations leads to a two-parameter family of self-similar solutions  Θ( x − x _*, t + t _*)  , we demonstrate that the optimal choice of   x _*  and   t _*  reduces the   L_p   error to the order of   t ^{−2+1/2 p}   .     

Large Time Asymptotic Behaviors for Periodic Solutions of Generalized Burgers Equations With Spherical Symmetry or Linear Damping

Using the method of balancing arguments, large time asymptotic behaviors for the periodic solutions of generalized Burgers equations   u_t  +  u ³ u_x  +  ju /2 t  =δ/2 u_xx   and   u_t  +  u ³ u_x  +λ u  =δ/2 u_xx   subject to the periodic initial condition     and the vanishing boundary conditions   u (0,  t ) =  u ( l ,  t ) = 0,   t  ≥ 0   or    t ₀,  where   A ,  A ₁, δ, λ,  l ,  t ₀, ∈ R ⁺  and   j  = 1, 2  , are obtained.     

Reinhardt域上一类推广的Roper-Suffridge算子

研究一类推广的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)∑_(k=2)~nakz_k~pk,f′(z)1)(~1/p2)z_2,…,f′(z_1)~(1/pn)z_n)′,证明该算子在复欧氏空间中的Reinhardt域Ω_(n,p2,%…,pn)={z=(z_1,…,z_n)∈C~n:|z_|~2+∑_(k=2)~n|zk|~(pk)1,Pk∈N~+,k=2,…,n}上分别保持α次的殆β型螺形性,α次的β型螺形性及强β型螺形性.     

开拓劳宾生和戈鲁辛的几个定理

<正> 1.设 p 次对称函数(?)在单位圆|z|<1中是正则的单叶的,此种函数的全体成一函数族 S_p.当p=1时,简讯 S_1为 S.设ω=f(z)∈S_p 映照|z|<1于 W 面上时,其像关于原点成星形,此种 f(z)成 S_p 之一子族S_p.设 f(z)∈S_p,     

A Ray Approximation to a PdE that Arises in the Study of Tandem Queues

We use singular perturbation methods to analyze a diffusion equation that arose in studying two tandem queues. Denoting by p ( n ₁,  n ₂) the probability that there are n ₁ customers in the first queue and n ₂ customers in the second queue, we obtain the approximation p ( n ₁,  n ₂)∼ɛ² P ( X ,  Y )=ɛ² P (ɛ n ₁, ɛ n ₂), where ɛ is a small parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both normal and tangential derivatives. We analyze the boundary value problem using the ray method of geometrical optics and other singular perturbation techniques. This yields the asymptotic behavior of P ( X ,  Y ) for X and/or Y large.     

Multiwavelets: Regularity, Orthogonality, and Symmetry via Two–Scale Similarity Transform

An important object in wavelet theory is the scaling function φ( t ), satisfying a dilation equation φ( t )=∑ C_k φ(2 t − k ). Properties of a scaling function are closely related to the properties of the symbol or mask P (ω)=∑ C_ke ^{− i ω k} . The approximation order provided by φ( t ) is the number of zeros of P (ω) at ω=π, or in other words the number of factors (1+ e ^{− i ω}) in P (ω). In the case of multiwavelets P (ω) becomes a matrix trigonometric polynomial . The factors (1+ e ^{− i ω}) are replaced by a matrix factorization of P (ω), which defines the approximation order of the multiscaling function . This matrix factorization is based on the two-scale similarity transform (TST). In this article we study properties of the TST and show how it is connected with the theory of multiwavelets. This approach leads us to new results on regularity, symmetry, and orthogonality of multiscaling functions and opens an easy way to their construction.     

Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation u_t = uu_x + u_xxx , and the Kadomtsev–Petviashvili (KP) equation u_yy =( u_xxx + uu_x + u_t ) _x with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation with initial value u (0,  x )= u ₀( x ), we show that there is no solution holomorphic in any neighborhood of ( t ,  x )=(0, 0) in C² unless u ₀( x )= a ₀+ a ₁ x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y =0, u ( t ,  x , 0)= u ₀( x ,  t ), u_y ( t ,  x , 0)= u ₁( x ,  t ), where the Taylor coefficients of u ₀ and u ₁ around t =0, x =0 are assumed non-negative. We prove that there is no holomorphic solution around the origin in C³, unless u ₀ and u ₁ are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.     

Chebyshev expansions and rational approximations

Let A(z) = A_m(z) + a_mz^mB(z,m) where A_m(z) is a polynomial in z of degree m-1. Suppose A(z) and B(z,m) are approximated by main diagonal Padé approximations of order n and r respectively. Suppose that the number of operations needed to evaluate both sides of the above equations by means of the Padé approximations and polynomial noted are the same. Thus 4n = 3m + 4r. We address ourselves to the question of which procedure is more efficient? That is, which procedure produces the smallest error? A variant of this problem is the situation where A(z) and B(z,m) are approximated by their representations in infinite series of Chebyshev polynomials of the first kind truncated after n and r terms respectively. Here n = m + r.Let F(z) have two different series type representations in overlapping or completely disjoint regions of the complex z-plane. Suppose that for each representation there is a sequence of rational approximations of the same type, say of the Padé class, which converge for |arg z| < π except possibly for some finite set of points. Assume that the number of machine operations required to make evaluations using the noted approximations are the same. Again, we ask which procedure is best? Other variants are studied.General answers to the above questions are not known. Instead, we illustrate the ideas for a number of the rather common special functions.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.