Serge Aubry - Physics Department . Technion

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$Turbulence-degraded wave fronts as fractal surfaces - Physics ...$

VII Summary and concluding remarks The possible divergency of the second moment does not implies energy spreading as believed in early publications. Numerical observations (over times which are not too long for the above reasons) shows 1- There are initial conditions with practically no diffusion from the beginning (quasiperiodic?) and others which looks (initially) chaotic. 2- the participation number does not (clearly) diverge in any case. Moreover, it is rigorously proven it cannot diverge in DNLS models with norm conservation and large enough amplitude wave packet. 3- The same qualitative behavior for the spreading of an initially localized wavepacket in DNLS models with norm conservation and quartic KG models without norm conservation 4-Conjecture: Unless the wavepacket corresponds initialy to a quasiperiodic solution, it converges (slowly) to a limit profile which is a quasiperiodic and spatially localized solution (KAM torus). This profile may not have second moment or may have a second moment? (open question) Serge Aubry, Aubry, LLB, FRANCE
Conjectures:: There should be absence of diffusion for any initially localized wave packet in any nonlinear model at any dimension with purely discrete linear spectrum (Random, quasiperiodic or else) There is diffusion (at least partial in case of time periodic Discrete Breather formation) in all models for which the linear spectrum has an absolutely continous component. There is diffusion as soon the model is a at non vanishing temperature or if it submitted to a random noise (with broad spectrum) Numerical problem with random FPU model: The linear localization length diverges at low frequency. The long tail already exists in the linear case (diverging second moment) and cannot be considered as an effect of nonlinearity Serge Aubry, Aubry, LLB, FRANCE Final Comments
Page 1 and 2: Suppression of Energy Diffusion in
Page 3 and 4: I-Nonlinear spatially periodic syst
Page 5 and 6: Initial wavepacket Time periodic so
Page 7 and 8: II- Exact Time Periodic Solutions (
Page 9 and 10: Numerical Calculation of Localized
Page 11 and 12: RDNLS: a straightforward proof of e
Page 13 and 14: Anderson Representation of DNLS nor
Page 15: Quasiperiodic Periodic Solutions: B
Page 18 and 19: The probability to find an initial
Page 20 and 21: Assume the wavepacket spreads unifo
Page 22 and 23: Example: What is the behavior of a
Page 24 and 25: V Diffusion of a Wavepacket in nonl
Page 26 and 27: There are initial conditions with n
Page 28 and 29: VI-New Numerical Investigations G.
Page 30 and 31: Serge Aubry, Aubry, LLB, FRANCE t=1
Page 32 and 33: Participation number of norm distri
Page 34 and 35: Serge Aubry, Aubry, LLB, FRANCE Tim
Page 36 and 37: Scenario for the Absence of spreadi
Page 38 and 39: Warning about the problem of reliab

Serge Aubry - Physics Department . Technion

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