arXiv subjects

O. Costin

Publications and source records attributed to O. Costin.

At least 19 recordsLinked to original sources

arXiv2022

Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are numerically efficient representations. For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays. We prove that relatively general functions, \'Ecalle resurgent ones, possess convergent dyadic expansions. These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).

math.CA↗

arXiv2020

On the domain of convergence of spherical harmonic expansions

Spherical harmonic expansions (SHEs) play an important role in most of the physical sciences, especially in physical geodesy. Despite many decades of investigation, the large order behavior of the SHE coefficients, and the precise domain of convergence for these expansions, have remained open questions. These questions are settled in the present paper for generic planets, whose shape (topography) may include many local peaks, but just one globally highest peak. We show that regardless of the smoothness of the density and topography, short of outright analyticity, the spherical harmonic expansion of the gravitational potential converges exactly in the closure of the exterior of the Brillouin sphere (The smallest sphere around the center of mass of the planet containing the planet in its interior), and convergence below the Brillouin sphere occurs with probability zero. More precisely, such over-convergence occurs on zero measure sets in the space of parameters. A related result is that, in a natural Banach space, SHE convergence of the potential below the Brillouin sphere occurs for potential functions in a subspace of infinite codimension (while any positive codimension already implies occurrence of probability zero). Provided a certain limit in Fourier space exists, we find the leading order asymptotic behavior of the coefficients of SHEs. We go further by finding a necessary and sufficient condition for convergence below the Brillouin sphere, which requires a form of analyticity at the highest peak, which would not hold for a realistic celestial body. Namely, a longitudinal average of the harmonic measure on the Brillouin sphere would have to be real-analytic at the point of contact with the boundary of the planet. It turns out that only a small neighborhood of the peak is involved in this condition.

math.CA↗

arXiv2020

Non-convergence of the spherical harmonic expansion of gravitational potential below the Brillouin sphere; the continuous case

For a singleton planet $P$ with gravitational potential $V$, we show that for each $\varepsilon > 0$ there exists a planet $P'$ with gravitational potential $V'$, with $(P',V')$ "$\varepsilon$-close" to $(P,V)$ (in an appropriate $C^0$-sense) for which the spherical harmonic expansion of $V'$ does not extend more than a distance $\varepsilon$ below the Brillouin sphere of $P'$.

math.FA↗

arXiv2016

A new type of factorial series expansions and applications

We construct a new type of convergent asymptotic representations, dyadic factorial expansions. Their convergence is geometric and the region of convergence can include Stokes rays, and often extends down to 0^+. For special functions such as Bessel, Airy, Ei, Erfc, Gamma and others, this region is C without an arbitrarily chosen ray effectively providing uniform convergent asymptotic expansions for special functions. We prove that relatively general functions, Ecalle resurgent ones possess convergent dyadic factorial expansions. We show that dyadic expansions are numerically efficient representations. The expansions translate into representations of the resolvent of self-adjoint operators in series in terms of the associated unitary evolution operator evaluated at some prescribed points (alternatively, in terms of the generated semigroup for positive operators).

math.CA↗

arXiv2015

Tronquée solutions of the Painlevé equation \P1

We analyze the one parameter family of tronquée solutions of the Painlevé equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in \cite{inprep} leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.

math.CA↗

arXiv2014

Decay estimates for One-dimensional wave equations with inverse power potentials

We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$ where $m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $t^{-m}$ for large $t$, which is consistent with existing mathematical and physical literature under slightly different assumptions (see e.g. Bizon, Chmaj, and Rostworowski, 2007; Donninger and Schlag, 2010; Schlag, 2007). Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being $const.x^{-α}$ where $α>2$ is a real number, as well as potentials of the form $const.x^{-m}+O(x^{-m-δ_1})$ with $δ_1>3$.

math.AP↗

arXiv2014

Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_I$

We show that the tritronquée solution of the Painlevé equation $\P1$, $ y"=6y^2+z$ which is analytic for large $z$ with $ \arg z \in (-\frac{3π}{5}, π)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in [-\frac{3π}{5}, π]}$ and the disk ${z: |z| < 37/20}$. This proves in particular the Dubrovin conjecture, an open problem in the theory of Painlevé transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero within less than 1/100 rigorous error bounds.

math.CA↗

arXiv2013

A quasi-solution approach to nonlinear problems - the case of Blasius similarity solution

Using the simple case of Blasius similarity solution, we illustrate a recently developed general method that reduces a strongly nonlinear problem into a weakly nonlinear analysis. The basic idea is to find a quasi-solution $F_0$ that satisfies the nonlinear problem and boundary conditions to within small errors. Then, by decomposing the true solution $F=F_0+E$, a weakly nonlinear analysis of $E$, using contraction mapping theorem in a suitable space of functions provides the existence of solution as well as bounds on the error $E$. The quasi-solution construction relies on a combination of exponential asymptotics and standard orthogonal polynomial representations in finite domain.

math.CA↗

arXiv2013

Analytical approximation of Blasius' similarity solution with rigorous error bounds

We use a recently developed method \cite{Costinetal}, \cite{Dubrovin} to find accurate analytic approximations with rigorous error bounds for the classic similarity solution of Blasius of the boundary layer equation in fluid mechanics, the two point boundary value problem $f^{\prime \prime \prime} + f f^{\prime \prime} =0$ with $f(0)=f^\prime (0)=0$ and $\lim_{x \rightarrow \infty} f^\prime (x) =1$. The approximation is given in terms of a polynomial in $[0, \frac{5}{2}]$ and in terms of the error function in $[\frac{5}{2}, \infty)$. The two representations for the solution in different domains match at $x=\frac{5}{2}$ determining all free parameters in the problem, in particular $f^{\prime \prime} (0) =0.469600 \pm 0.000022 $ at the wall The method can in principle provide approximations to any desired accuracy for this or wide classes of linear or nonlinear differential equations with initial or boundary value conditions. The analysis relies on controlling the errors in the approximation through contraction mapping arguments, using energy bounds for the Green's function of the linearized problem.

math.CA↗

arXiv2012

The blockage problem

We investigate the totally asymmetric exclusion process on Z, with the jump rate at site i given by r_i=1 for i nonzero, r_0=r. It is easy to see that the maximal stationary current j(r) is nondecreasing in r and that j(r)=1/4 for r>=1; it is a long outstanding problem to determine whether or not the critical value r_c of r such that j(r)=1/4 for r>r_c is strictly less than 1. Here we present a heuristic argument, based on the analysis of the first sixteen terms in a formal power series expansion of j(r) obtained from finite volume systems, that r_c=1 and that for r less than 1 and near 1, j(r) behaves as 1/4-γ\exp[-{a/(1-r)}] with a approximately equal to 2. We also give some new exact results about this system; in particular we prove that j(r)=J_max(r), with J_max(r) the hydrodynamic maximal current defined by Seppalainen, and thus establish continuity of j(r). Finally we describe a related exactly solvable model, a semi-infinite system in which the site i=0 is always occupied. For that system, the critical r is 1/2 and the analogue j_s(r) of j(r) satisfies j_s(r)=r(1-r) for r<=1/2; j_s(r) is the limit of finite volume currents inside the curve |r(1-r)|=1/4 in the complex r plane and we suggest that analogous behavior may hold for the original system.

math-ph↗

arXiv2012

Global reconstruction of analytic functions from local expansions

A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros. There is a duality between the global analytic structure of the reconstructed function and the properties of the coefficients as a function of their index. Borel summability of a class of divergent series follow as a byproduct.

math.CV↗

arXiv2011

Global behavior of solutions of nonlinear ODEs in $\CC$: first order equations

We show that the solutions of first order nonlinear ODEs can be controlled globally in the complex domain, using a finite set of constants of motion defined in regions of $\CC$. These constants of motion enable us to obtain quantitative behaviors of the solutions far away from the origin, as well as to determine the position of singularities of the solution.

math.CA↗

arXiv2010

Boundary blow-up solutions in the unit ball : asymptotics, uniqueness and symmetry (v3)

We calculate the full asymptotic expansion of boundary blow-up solutions, for any nonlinearity f. Our approach enables us to state sharp qualitative results regarding uniqueness and ra-dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard nonlinearities f(u) = u^p, p > 1.

math.AP↗

arXiv2010

Ionization of Coulomb systems in $\RR^3$ by time periodic forcings of arbitrary size

We analyze the long time behavior of solutions of the Schrödinger equation $iψ_t=(-Δ-b/r+V(t,x))ψ$, $x\in\RR^3$, $r=|x|$, describing a Coulomb system subjected to a spatially compactly supported time periodic potential $V(t,x)=V(t+2π/ω,x)$ with zero time average. We show that, for any $V(t,x)$ of the form $2Ω(r)\sin (ωt-θ)$, with $Ω(r)$ nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, {\em i.e.} $P(t,K)=\int_K|ψ(t,x)|^2dx\to 0$ as $t\to\infty$ for any compact set $K\subset\RR^3$. Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then $P(t,K)$ decays like $t^{-5/3}$ as $t \to \infty $. To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.

math.AP↗

arXiv2009

On the geometry of Julia sets

We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain $J$ from a smoother curve of lower Hausdorff dimension, by replacing pieces of the more regular curve by increasingly rescaled elementary "bricks" obtained from the transseries expression. Self-similarity of $J$, up to conformal transformation, is manifest in the formulas. The Hausdorff dimension of $J$ is estimated by the transseries formula. The analysis extends to polynomial maps.

math.DS↗

arXiv2009

Ionization in damped time-harmonic fields

We study the asymptotic behavior of the wave function in a simple one dimensional model of ionization by pulses, in which the time-dependent potential is of the form $V(x,t)=-2δ(x)(1-e^{-λt} \cosωt)$, where $δ$ is the Dirac distribution. We find the ionization probability in the limit $t\to\infty$ for all $λ$ and $ω$. The long pulse limit is very singular, and, for $ω=0$, the survival probability is $const λ^{1/3}$, much larger than $O(λ)$, the one in the abrupt transition counterpart, $V(x,t)=δ(x)\mathbf{1}_{\{t\ge 1/λ\}}$ where $\mathbf{1}$ is the Heaviside function.

math-ph↗

arXiv2008

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form $\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the boundary $i\RR$, under the assumption $g'\to\infty$ and further assumptions on $c_k$. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with $c_k=1$, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at $z=0$. When sufficient analyticity information on $g$ exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If $g(k)=k^b$, $c_k=1$, $b=n$ or $b=(n+1)/n$, $n\in\NN$, behavior near the boundary is roughly of the standard form $\Re(z)^{-b'}Q(x)$ where $Q(x)=1/q$ if $x=p/q\in\QQ$ and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form $x_{n+1}=λP(x_n)$, $|λ|<λ_0(P)$, turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map $P(x) =x-x^2$, $λ_0=1$, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

math.CV↗

arXiv2008

Integral formulation of 3-D Navier-Stokes and longer time existence of smooth solutions

We consider the 3-D Navier-Stokes initial value problem, $$ v_t - νΔv = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in \mathbb{T}^3 (*) $$ where $\mathcal{P}$ is the Hodge projection. We assume that the Fourier transform norms $ \| {\hat f} \|_{l^1 (\mathbb{Z}^3)}$ and $\| {\hat v}_0 \|_{l^{1} (\mathbb{Z}^3)}$ are finite. Using an inverse Laplace transform approach, we prove that an integral equation equivalent to (*) has a unique solution ${\hat U} (k, q)$, exponentially bounded for $q$ in a sector centered on $\RR^+$, where $q$ is the inverse Laplace dual to $1/t^n$ for $n \ge 1$. This implies in particular local existence of a classical solution to (*) for $t \in (0, T)$, where $T$ depends on $\| {\hat v}_0 \|_{l^{1}}$ and $\| {\hat f} \|_{l^1}$. Global existence of the solution to NS follows if $\| {\hat U} (\cdot, q) \|_{l^1}$ has subexponential bounds as $q\to\infty$. If $f=0$, then the converse is also true: if NS has global solution, then there exists $n \ge 1 $ for which $\| {\hat U} (\cdot, q) \|$ necessarily decays. We show the exponential growth rate bound of U, α, can be better estimated based on the values of ${\hat U}$ on a finite interval $[0,q_0]$. We also show how the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations suggest that this approach gives an existence time that substantially exceeds classical estimate.

math.AP↗