Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed in this paper, including the classification and methods of constructing uniform asymptotics used to describe the structure of wave fields in these domains.
Abstract: A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed The fundamentals of the wave catastrophe theory are considered, including the classification and methods of constructing uniform asymptotics used to describe the structure of wave fields in these domains, together with an analysis of the structure of the field Classes of special functions used to construct uniform asymptotic expansions of wave fields are generally described together with the properties of these classes and the methods of computation
16 citations
••
TL;DR: In this article, the motion of a particle in a potential decreasing with time is considered, and different time and space rescaling are considered in order to obtain the asymptotic solutions.
Abstract: The motion of a particle in a potential decreasing with time as ‖X‖n is considered. Different time and space rescaling are considered in order to obtain the asymptotic solutions. The validity of adiabatic invariants is discussed. The classical critical case corresponds to the obtainment of self‐similar solutions for the quantum problem.
16 citations
•
TL;DR: After developing the tools required for application of the fixed point theory in the investigation, some general results about the long-time behavior of solutions of n-th order nonlinear differential equations are presented.
Abstract: We discuss a number of issues important for the asymptotic integration of ordinary differential equations. After developing the tools required for application of the fixed point theory in the investigation, we present some general results about the long-time behavior of solutions of n-th order nonlinear differential equations with an emphasis on the existence of polynomial-like solutions, the asymptotic representation for the derivatives and the effect of perturbations upon the asymptotic behavior of solutions.
16 citations
••
TL;DR: The Lagrangian analysis as mentioned in this paper is a special case of the Lagrangians, which was introduced by the W.K.B. method for solving the Dirac equation.
Abstract: H. Poincare defined asymptotic expansions. Their use by the W. K. B. method introduced a new kind of solution of linear differential equations. Maslov showed their singularities to be merely apparent. The clarification of those results leads to the introduction of "Lagrangian functions", of their scalar product and of "Lagrangian operators", which constitutes a new structure: the "Lagrangian analysis". The last step of its definition requires the choice of a constant. That constant has to be Planck's constant, when the equation is the Schrodinger or the Dirac equation describing the hydrogen atom-the study of atoms with several electrons is very incomplete. 1. Henri Poincare's main field, more precisely the one where the number of his publications is the highest, happens to be celestial mechanics. For instance, he tried to establish the convergence of the series by means of which the motion of the solar system is computed; it was a failure. He proved indeed the opposite: the divergence of those series, whose numerical values furnished the most impressive, precise and famous predictions in science during the last century! Henri Poincare explained that paradox: those series give a very good approximation of the wanted result, provided only their first terms, namely, a reasonable number of them, are taken into account. Of course, demanding mathematicians to be reasonable is dubious but Henri Poincare [4] made it clear by defining the asymptotic expansion ^mi0anx n of a function of x at the origin: it is a, formal series such that for each natural number N there exists a positive number cN such that I N I ƒ(•*) 2 <*n* < cN\x\ * for x near 0. (1.1) I "° I Thus an asymptotic expansion of ƒ is a formal series able to give a very good approximation of ƒ(*), when x is small, but unable to supply the exact value of/(x). 2. The W.K.B. method constructs asymptotic solutions of a linear differential equation n(x, ~ -j£ W *) 0 ( x G l = R ^ 6 i [0, oo[), (2.1) whose unknown is the function u and whose parameter v tends to /oo. Presented at the Symposium on the Mathematical Heritage of Henry Poincare in April, 1980; received by the editors June 15, 1980. 1980 Mathematics Subject Classification. Primary 47B99, 81C99; Secondary 35S99, 42B99.
16 citations