Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems
type: Item
instance of: scholarly article
title: Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems
zbMATH Open document ID: 0959.34002
author: Eric Lombardi
published in: Lecture Notes in Mathematics
publication date: 2000-09-10T00:00:00Z
review text: Many physical problems are governed by differential equations which solutions can be represented in terms of power series of a small parameter \(\varepsilon\). In some cases these power series diverge, and this divergence may reflect that the system has a solution for which such an expansion misses some exponentially small term like \(\exp (-1/\varepsilon ^2)Z(t)\) lying beyond any algebraic orders. This kind of phenomena typically occurs for physical problems governed by reversible vector fields near resonances and are known in many branches of science (dendritic crystals growth, quantum tunneling, KAM theory, water waves theory and others). Many works dealing with this subject for a large class of problems provide no rigorous proof. The book presents a systematic and unified study of these ``nonlinear problems beyond any algebraic orders'' in finite and infinite dimensions. The crucial point of the analysis is the description of the analytic continuation of solutions which enables to catch exponentially small terms which are ``hidden beyond all orders on the real axis'', so that the main difficulty is to determine the complex singularities of solutions to nonlinear differential equations. The book begins by studying a simple reversible model system in \(\mathbb R^3\) which leads to the problem of computing oscillatory integrals of the form \[ \int_{-\infty}^{+\infty}\text{e}^{i\omega t/\varepsilon}f(X(t))dt, \tag{1} \] where \(f\) is an analytic function and \(X(t)\) is a particular solution to the system. This problem in its turn leads to the problem of computing the nature and the position of singularities of solutions to complex differential equations. Some examples of oscillatory integrals hidden in dynamical systems arising in different physical problems are outlined. Part I proposes a complete set of ``exponential tools'' for evaluating oscillatory integrals of the form (1). Part II gives all necessary prerequisites for studying reversible vector fields near…
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 76B25
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34M30
Mathematics Subject Classification ID: 37K55
zbMATH DE Number: 1503587
zbMATH Keywords: dynamical system
zbMATH Keywords: reversible vector fields
zbMATH Keywords: resonances
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: oscillatory integrals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0104102
OpenAlex ID: W4246672319
DOI: 10.1007/BFB0104102
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description: scientific article; zbMATH DE number 1503587