Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems
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 resonances. The contents of this part corresponds to the chapter titles listed below. Chapter 3. Resonances of reversible vector fields. Chapter 4. Analytic description of periodic orbits bifurcating from a pair of simple purely imaginary eigenvalues. Chapter 5. Constructive Floquet theory for periodic matrices near a constant one. Chapter 6. Inversion of affine equations around reversible homoclinic connections. Part III is devoted to the existence of homoclinic connections near resonances for reversible systems. Chapter 7 contains the study of the \(0^{2+}iw\) reversible resonance in \(\mathbb R^4\) for analytic vector fields. The author proves the existence of periodic solutions of arbitrary small size, the existence of reversible connections homoclinic to exponentially small periodic orbits and the generic nonexistence of homoclinic connections to \(0\). Chapter 8 extends the results of Chapter 7 to infinite dimensional analytic vector fields. It must be emphasized that this extension cannot be performed via a center manifold reduction since the reduction does not preserve the analyticity. Application to water waves is considered. Chapter 9 which is the final in the book is devoted to the \((i\omega _0)^2i\omega _1\) reversible resonances in \(\mathbb R^6\). It is proven the existence of periodic solutions of arbitrary small size, the existence of reversible connections homoclinic to exponentially small periodic orbits and the generic nonexistence of homoclinic connections to \(0\). In spite of the fact that the results of this chapter are very similar to the ones of Chapter 7, the interaction between frequencies \(\omega _0\) and \(\omega _1\) makes the study of this resonance far more intricate. Clearly and beautifully written, the book presents a rigorous and detailed study of intricate phenomena in reversible vector fields near resonances. The book corresponds to the current state of researches on resonances in finite and infinite dimensional reversible systems.
scientific article; zbMATH DE number 1503587