American Institute of Mathematical Sciences

Advanced
2003, 2003(Special): 482-491. doi: 10.3934/proc.2003.2003.482

Application of weak turbulence theory to FPU model

Peter R. Kramer 1, , Joseph A. Biello 1, and  Yuri Lvov 1,

1. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 301 Amos Eaton Hall, 110 8th Street, Troy, NY 12180, United States, United States

Received  September 2002 Revised  April 2003 Published  April 2003

The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear dispersive system. A direct application of the standard kinetic equations would miss interesting dynamics of the energy transfer process starting from a large-scale excitation. This failure is traced to an enforcement of the exact resonance condition, whereas mathematically the resonance should be broadened due to the energy transfer happening on large but finite time scales. By allowing for the broadened resonance, a modified three-wave kinetic equation is derived for the FPU model. This kinetic equation produces some correct scaling predictions about the statistical dynamics of the FPU model, but does not model accurately the detailed evolution of the energy spectrum. The reason for the failure seems not to be one of the previously clarified reasons for breakdown in the weak turbulence theory.
Keywords: resonance broadening., Fermi-Pasta-Ulam model, weak turbulence.
Mathematics Subject Classification: Primary: 41A60,82C05,82C28, Secondary: 34E13, 37K6.
Citation: Peter R. Kramer, Joseph A. Biello, Yuri Lvov. Application of weak turbulence theory to FPU model. Conference Publications, 2003, 2003 (Special) : 482-491. doi: 10.3934/proc.2003.2003.482
[1]

Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710
[2]

Alexander Pankov, Vassilis M. Rothos. Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 835-849. doi: 10.3934/dcds.2011.30.835
[3]

Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991
[4]

Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135
[5]

Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371
[6]

Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719
[7]

W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111
[8]

Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781
[9]

Natalia Ptitsyna, Stephen P. Shipman. A lattice model for resonance in open periodic waveguides. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 989-1020. doi: 10.3934/dcdss.2012.5.989
[10]

Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations and Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021
[11]

T. Gallouët, J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2371-2391. doi: 10.3934/cpaa.2012.11.2371
[12]

Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255
[13]

François Baccelli, Augustin Chaintreau, Danny De Vleeschauwer, David R. McDonald. HTTP turbulence. Networks and Heterogeneous Media, 2006, 1 (1) : 1-40. doi: 10.3934/nhm.2006.1.1
[14]

Soizic Terrien, Christophe Vergez, Benoît Fabre, Patricio de la Cuadra. Emergence of quasiperiodic regimes in a neutral delay model of flute-like instruments: Influence of the detuning between resonance frequencies. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022011
[15]

Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090
[16]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[17]

Victor Zvyagin, Vladimir Orlov. Weak solvability of fractional Voigt model of viscoelasticity. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6327-6350. doi: 10.3934/dcds.2018270
[18]

Prasanta Kumar Barik, Ankik Kumar Giri. Weak solutions to the continuous coagulation model with collisional breakage. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6115-6133. doi: 10.3934/dcds.2020272
[19]

Gerhard Keller. Maximal equicontinuous generic factors and weak model sets. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6855-6875. doi: 10.3934/dcds.2020132
[20]

Eric Falcon. Laboratory experiments on wave turbulence. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 819-840. doi: 10.3934/dcdsb.2010.13.819

 Impact Factor: 

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy