Conclusions, Open Problems and Future Outlook

The final Chapter first summarizes and discusses the main conclusions described in the book. We then list a number of open problems, which we feel should be further pursued in continuation of what we have presented in earlier Chapters. We start with some recent results that extend the mathematical theory of integrability from the viewpoint of singularity analysis and continue with some directions that further develop the topics of nonlinear normal modes, localization, diffusion and the complex statistical properties of nonlinear lattices. Finally, regarding the future outlook of research in Hamiltonian dynamics, we briefly review three topics of great current interest that were not treated in the book, but are extremely important in view of their far-reaching experimental applications: (1) anomalous heat conduction and the discovery of mechanisms that control heat flow based on the dynamics of Hamiltonian lattices, (2) soliton dynamics in nonlinear photonic structures and (3) kinetic theory of Hamiltonian systems with applications to plasma physics.