This talk will survey each of these efforts and discuss some of our successes and failures. Although the small amplitude waves do not influence the nonlinear field, they are modulated by its presence. The trapping phenomenon is associated with a kind of integrability of the coupled system consisting of the nonlinear field and the modulated linear field. With the help of this integrability, a generalized transform method will be presented for solving the general initial-value problem for the modulated linear waves. Perturbation theory for nearly integrable couplings will be presented, and numerical simulations will be used to illustrate the scattering and resonance effects that are present far from integrability.
Applications range from planar waveguide optics to wave propagation in molecular chains.
This talk summarizes joint work with N. N. Akhmediev, J. A. Besley, P. L. Christiansen, S. R. Clarke, A. Soffer, and M. I. Weinstein. In this talk I will present his proof. We discuss what is known about this conjecture, and in particular that it is approximately true for large dense graphs. The nonconvexity of their underlying energy potential is a challenge for mathematical analysis, since convexity plays an important role in the classical theories of existence and regularity. In the last years one main point of interest was to develop techniques to circumvent these difficulties.