On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.     

On solutions with generalized power asymptotics to systems of differential equations

In the paper we study methods for constructing particular solutions with nonexponential asymptotic behavior to a system of ordinary differential equations with infinitely differentiable right-hand sides. We construct the corresponding formal solutions in the form of generalized power series whose first terms are particular solutions to the so-called truncated system. We prove that these series are asymptotic expansions of real solutions to the complete system. We discuss the complex nature of the functions that are represented by these series in the analytic case.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 851–861, December, 1995.     

Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

A classification of solutions of the first and second Painlevé equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the parameterization of the solutions is analyzed. It turns out that solutions of the Painlevé equations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of “truncated” solutions (intégrales tronquée) according to P. Boutroux’s classification. It is shown that all known special solutions of the first and second Painlevé equations belong to this class.     

Correlation asymptotics for non‐translation invariant lattice spin systems

We obtain asymptotic expressions for the Green kernels of certain non‐translation invariant transition matrices using methods of semiclassical and microlocal analysis. Combined with a result by Bach and Møller this yields asymptotic formulas for the truncated two‐point correlation functions of certain non‐translation invariant lattice models of real‐valued spins. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

On the asymptotic spectra of the bounded solutions of a nonlinear Volterra equation

We consider the asymptotic behavior of the bounded solutions of a nonlinear Volterra integrodifferential equation with a positive definite convolution kernel. Our main result states that (under appropriate assumptions) the asymptotic spectra of the solutions are contained in the set where the real part of the Fourier transform of the kernel vanishes. We also give a new asymptotic stability theorem, and present a new proof of a known result on the asymptotic behavior of the bounded solutions of a nonlinear, nondifferentiated Volterra equation.     

Nonlinear Transition Layers—The Second Painleve Transcendent

We investigate a large class of weakly nonlinear second-order ordinary differential equations with slowly varying coefficients. We show that the standard two-timing perturbation solution is not valid during the transition from oscillatory to exponentially decaying behavior. In all cases this difficulty is remedied by a nonlinear transition layer, whose leading-order character is described by one special nonlinear differential equation known as the second Painlevé transcendent (in essence a nonlinear Airy equation). The method of matched asymptotic expansions yields the desired connection formula. The second Painlevé transcendent also provides two other types of transitions: (1) between weakly nonlinear solutions (either oscillatory or exponentially decaying) and special fully nonlinear solutions, and (2) between two of these special nonlinear solutions. These special solutions are of three: different kinds: (a) slowly varying stable equilibrium solutions, (b) “exploding” solutions, and (c) solutions depending on both the fast and slow scales (which emerge from the unstable zero equilibrium solution).     

On the dynamics of Gowdy space‐times

We study the behavior near the singularity t = 0 of Gowdy metrics. We prove existence of an open dense set of boundary points near which the solution is smoothly “asymptotically velocity term dominated” (AVTD). We show that the set of AVTD solutions satisfying a uniformity condition is open in the set of all solutions. We analyze in detail the asymptotic behavior of “power law” solutions at the (hitherto unchartered) points at which the asymptotic velocity equals zero or one. Several other related results are established. © 2004 Wiley Periodicals, Inc.     

Distributions of Poles to Painlevé Transcendents via Padé Approximations

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.     

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross–Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces

In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 329–337, March, 2005.     

The question of the asymptotic behavior as |<Emphasis Type="Italic">t</Emphasis>| → ∞ of boundary value problem solutions for <Emphasis Type="Italic">p</Emphasis>-ADIC strings

Abatract  We obtain lower asymptotic estimates for tachyon fields of open and closed strings as |t| → ∞. They confirm the expressions in the first asymptotic term that were previously found as solutions of linearized equations; this is not confirmed in the second asymptotic term. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 325–333, December, 2008.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

Local existence for the one‐dimensional Vlasov–Poisson system with infinite mass

A collisionless plasma is modelled by the Vlasov–Poisson system in one dimension. We consider the situation in which mobile negative ions balance a fixed background of positive charge, which is independent of space and time, as ∣x∣ → ∞. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behaviour are shown to exist locally in time, and criteria for the continuation of these solutions are established. Copyright © 2006 John Wiley & Sons, Ltd.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Semicrystal with a singular potential in an accelerating electric field

We study the Schrödinger equation describing the one-dimensional motion of a quantum electron in a periodic crystal placed in an accelerating electric field. We describe the asymptotic behavior of equation solutions at large values of the argument. Analyzing the obtained asymptotic expressions, we present rather loose conditions on the potential under which the spectrum of the corresponding operator is purely absolutely continuous and spans the entire real axis.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.