By Christophe Besse, Jean-Claude Garreau
Focusing at the interface among arithmetic and physics, this booklet deals an creation to the physics, the math, and the numerical simulation of nonlinear structures in optics and atomic physics. The textual content covers a large spectrum of present examine at the topic, that's a very energetic box in physics and mathematical physics, with a really huge diversity of implications, either for primary technological know-how and technological purposes: mild propagation in microstructured optical fibers, Bose-Einstein condensates, disordered platforms, and the newly rising box of nonlinear quantum mechanics.
Accessible to PhD scholars, this ebook may also be of curiosity to post-doctoral researchers and pro academics.
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Additional resources for Nonlinear Optical and Atomic Systems: At the Interface of Physics and Mathematics
But if we skip them, we return to the linear wave equation, restore integrability, and recover Anderson localization. So then, for that enlarged wave packet, we can again add trapping hard walls, but keep the nonlinear terms, and ask the question whether the dynamics inside the wave packet remains regular, or will be chaotic at large enough times. Again the experience of molecular dynamics tells that the dynamics will stay chaotic with high probability, but the decoherence times increase. Therefore the possible flaw in the argument when dropping the nonlinear terms is the time scale.
T/. n2 n1 / ; (58) 42 S. Flach Fig. 17 Under a kick strength of k D 5, measures for ˇ D 0:3 (blue) and ˇ D 10 (red), for both quasiperiodic sequences set by D 1 (solid line), and for random sequences (dashed line, see  for details). Upper row: Mean logarithms for energy < log10 E >. The clouds around the quasiperiodic sequences correspond to one standard deviation error. Lower row: finite-difference derivative of the above. Grey horizontal lines correspond to exponents for weak and strong chaos regimes.
The predicted subdiffusive exponents are controlled only by the lattice dimension, and the power of nonlinearity. So far we discussed the resulting nonlinear diffusion for uncorrelated random potentials l . For linear wave equations, a number of other correlated potentials are known to result in wave localization for a corresponding linear wave equation. 1 Subdiffusive Destruction of Aubry-Andre Localization Let us replace the uncorrelated disorder potential in Sect. 2 ˛AA l C Â/ : (54) For the linear wave equation ˇ D 0 and any irrational choice of ˛AA this results in the well-known Aubry-Andre localization .