On non-local variational problems with lack of compactness related to non-linear optics

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.Comment: 30 page

Time evolution of localized solutions in 1-dimensional inhomogeneous FPU models

We study energy localization in a quartic FPU model with spatial inhomogeneity corresponding to a site-dependent number of interacting neighbors. Such lattices can have linear normal modes that are strongly localized in the regions of high connectivity and there is evidence that some of these localized modes persist in the weakly nonlinear regime. The present study shows examples where oscillations can remain localized for long times. Nonlinear normal modes are approximated by periodic orbits that belong to an invariant subspace of a Birkhoff normal form of the system that is spanned by spatially localized modes [F. Martínez-Farías et al., Eur. Phys. J. Special Topics 223, 2943 (2014), F. Martínez-Farías et al., Physica D 335, 10 (2016)]. The invariant subspace is suggested by the dispersion relation and also depends on the overlap between normal modes. Numerical integration from the approximate normal modes suggests that spatial localization persists over a long time in the weakly nonlinear regime and is especially robust in some disordered lattices, where it persists for large, (1), amplitude motions. Large amplitude localization in these examples is seen to be recurrent, i.e. energy flows back and forth between the initial localization region and its vicinity

Weakly nonlinear localization for a 1-D FPU chain with clustering zones

We study weakly nonlinear spatially localized solutions of a Fermi-Pasta-Ulam model describing a unidimensional chain of particles interacting with a number of neighbors that can vary from site to site. The interaction potential contains quadratic and quartic terms, and is derived from a nonlinear elastic network model proposed by Juanico et al. [1]. The FPU model can be also derived for arbitrary dimensions, under a small angular displacement assumption. The variable interaction range is a consequence of the spatial inhomogeneity in the equilibrium particle distribution. We here study some simple one-dimensional examples with only a few, well defined agglomeration regions. These agglomerations are seen to lead to spatially localized linear modes and gaps in the linear spectrum, which in turn imply a normal form that has spatially localized periodic orbits