By Christophe Besse, Jean-Claude Garreau

Focusing at the interface among arithmetic and physics, this e-book bargains an advent to the physics, the maths, and the numerical simulation of nonlinear platforms in optics and atomic physics. The textual content covers a large spectrum of present examine at the topic, that is a very lively box in physics and mathematical physics, with a truly vast variety of implications, either for basic technological know-how and technological purposes: gentle propagation in microstructured optical fibers, Bose-Einstein condensates, disordered structures, and the newly rising box of nonlinear quantum mechanics.

Accessible to PhD scholars, this e-book can also be of curiosity to post-doctoral researchers and pro academics.

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**Example text**

7). A packet launched in the weak chaos regime stays in this regime, while one launched in the strong chaos regime spreads to the point that it eventually crosses over into the asymptotic regime of weak chaos at later times. Fig. 7 Left plot: parametric space of disorder, W, vs. the frequency shift induced by nonlinearity, ı, for the DNLS model. Three spreading regimes are shown for dynamics dictated by: (1) weak chaos (pale blue), (2) strong chaos (green), and (3) the onset of self-trapping (pale red).

The number of disorder realizations was as large as 400, and integration times extended up to t D 108 . Initial states were wave packets occupying a typical localization volume V 30 of the linear wave equation. In Fig. 11 the results for D 2 are shown. The weak chaos exponent measures as ˛ 0:21 which is very close to the theoretical prediction ˛ D 0:2. Extensions to D 1:5; 1:3 in the weak chaos regime and to D 0:7; 0:5 in the strong chaos regime show very good agreement between the numerically observed exponents, and the theoretical predictions in Fig.

T/ increases up to 1=3 and stays at this value for later times. t/ first rises up to 1=2, keeps this value for one decade, and then drops down, as predicted. t/. Additionally, we also mention numerics for W 2 f1; 2; 6g with respective initial packet widths of L D V 2 f361; 91; 11g [51]. Results are qualitatively similar to those shown in Fig. 8, and thus omitted for graphical clarity. The duration of the strong chaos regime with ˛ D 1=2 (and thus the location of the crossover) is largely dependent on how deep in the strong chaos regime the state is initially.