Oscillatory Integrals and Phenomena Beyond all Algebriac Orders: With Applications to Homoclinic Orbits in Reversible Systems

Eric Lombardi (auth.)3540677852, 9783540677857

During the last two decades, in several branches of science (water waves, crystal growth, travelling waves in one dimensional lattices, splitting of separatrices,…) different problems appeared in which the key point is the computation of exponentially small terms. This self-contained monograph gives new and rigorous mathematical tools which enable a systematic study of such problems. Starting with elementary illuminating examples, the book contains (i) new asymptotical tools for obtaining exponentially small equivalents of oscillatory integrals involving solutions of nonlinear differential equations; (ii) implementation of these tools for solving old open problems of bifurcation theory such as existence of homoclinic connections near resonances in reversible systems.

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Table of contents :
Introduction….Pages 1-19
“Exponential tools” for evaluating oscillatory integrals….Pages 22-76
Resonances of reversible vector fields….Pages 78-100
Analytic description of periodic orbits bifurcating from a pair of simple purely imaginary eigenvalues….Pages 101-122
Constructive floquet theory for periodic matrices near a constant one….Pages 123-134
Inversion of affine equations around reversible homoclinic connections….Pages 135-184
The 0 2+ iω resonance….Pages 186-325
The 0 2+ iω resonance in infinite dimensions. Application to water waves….Pages 327-357
The (iω 0 ) 2 iω 1 resonance….Pages 359-403