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.     

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 .     

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.     

On Mean Convergence of Fourier-Bessel Series of Negative Order

The expansion of f ∈ L^p(0, 1) Fourier series of Bessel functions of order converges to f in L^p whenever Let be the space of p-integrable functions with respect to the measure t dt and where {s_n}, n = 1, 2, …, is the set of positive zeros of J_v. Then, the expansion of in a Fourier series of functions ψ_n, ?1 < ν < ?½, converges to in whenever      

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.     

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 Distributional Hankel Transform of Marcel Riesz's Ultrahyperbolic Kernel

In this article we give a sense to the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel. First we evaluate Rα(u) in α = ?2k and α = 2k for the cases μ even and ν odd, μ even and ν even, and μ odd and ν odd, μ odd and ν even, where and Finally in Section 4 we obtain the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel.     

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.     

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetrie decreasing re arrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing ?. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.     

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].     

On the Global Behavior of Solutions of a Coupled System of Nonlinear Schrödinger Equation

We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the H¹‐norm of the solution globally bounded in the Manakov case, when ?” We found that in the Manakov case, and when , the solution stays in , and also that the H¹‐norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.     

On the Eigenvalue Problem for a Certain

This paper studies the spectral properties of the partial differential operator over a finite region Ω. This operator, which arises in the analysis of nonaxisymmetric, rapidly rotating compressible flows, is treated as a perturbation of the operator which is generated by the terms Using the fact that , when defined on a suitable domain, is closed and self-adjoint, it is shown that [when acting on elements of ] is an operator with compact resolvent whose generalized eigenvectors are complete in ?² (Ω).     

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 .     

Exact N-Wave Solutions of Generalized Burgers Equations

Exact N-Wave solutions for the generalized Burgers equation where j, α, β, and γ are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.     

Euler-Maclaurin Summation of Trigonometric Fourier Series

Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the series in which α and β(>0) are rational numbers satisfying 0<β/α≤1, where λ is an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials in λ, are obtained as the sum of a logarithic term and an infinite series in powers of λ, whose expansion is valid when 0<λ≤(2π/α) and is exact. When the Fourier series is expressible as a polynomial in λ, the method identifies that polynomial.     

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.     

Using Symmetries à Rebours

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .     

On the Generalizations of a Theorem of Beppo Levi

In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where φ(x) is a function that verifies φ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and φ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: where φ_n(x) is a sequence of functions and Bn(a) designates the n-dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of φ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval.     

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