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2
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85036324212
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C. Szwaj (private communication)
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C. Szwaj (private communication);
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4
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5344260477
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A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
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Phys. Rev. A
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, pp. 877
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Gatti, A.1
Wiedemann, H.2
Lugiato, L.A.3
Marzoli, I.4
Oppo, G.L.5
Barnett, S.M.6
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6
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0011189154
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see also Magnétisme, Tome 1: Les Fondements (University of Grenoble Press, Grenoble, 2001)
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R. E. Rosensweig, Sci. Am. 247, 135 (1982);see also Magnétisme, Tome 1: Les Fondements (University of Grenoble Press, Grenoble, 2001).
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(1982)
Sci. Am.
, vol.247
, pp. 135
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Rosensweig, R.E.1
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7
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0000037738
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Kyoug J. Lee, W. D. McCormick, Qi Ouyang, and H. L. Swinney, Science 261, 192 (1993).
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(1993)
Science
, vol.261
, pp. 192
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Lee, K.J.1
McCormick, W.D.2
Ouyang, Q.3
Swinney, H.L.4
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10
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85036222819
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S. Metens, Ph.D thesis, University of Brussels, Belgium, 1998
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S. Metens, Ph.D thesis, University of Brussels, Belgium, 1998.
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15
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85036276864
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Eqs. (5) and (6), the damping coefficients (Formula presented) have been introduced to prevent divergence of the solutions. A weakly nonlinear analysis was performed for (Formula presented) by using the expansion (1) with (Formula presented) for w, and a similar expansion for u (with wave numbers varying between 0 and 2, due to the quadratic coupling). Using the same notation as in Refs. 1 18 the coupling coefficient between the two active and passive modes is (Formula presented) where (Formula presented) is the Lorentzian (Formula presented) and (Formula presented) and the self-interaction coefficient is (Formula presented) The condition required 1 for a turbulent crystal [i.e., that the coupling function β(θ) is flat and very small in the whole domain (0, π)] is fulfilled only for (Formula presented) very small (Formula presented) and (Formula presented) close to 3/2; but this case leads to a negative (Formula presented) A positive (Formula presented) is reachable only for peaked (nonflat) β(θ), so that it would lead to the usual N-gon structures, with (Formula presented) 2, 3
-
In Eqs. (5) and (6), the damping coefficients (Formula presented) have been introduced to prevent divergence of the solutions. A weakly nonlinear analysis was performed for (Formula presented) by using the expansion (1) with (Formula presented) for w, and a similar expansion for u (with wave numbers varying between 0 and 2, due to the quadratic coupling). Using the same notation as in Refs. 1, 18 the coupling coefficient between the two active and passive modes is (Formula presented) where (Formula presented) is the Lorentzian (Formula presented) and (Formula presented) and the self-interaction coefficient is (Formula presented) The condition required 1 for a turbulent crystal [i.e., that the coupling function β(θ) is flat and very small in the whole domain (0, π)] is fulfilled only for (Formula presented) very small (Formula presented) and (Formula presented) close to 3/2; but this case leads to a negative (Formula presented) A positive (Formula presented) is reachable only for peaked (nonflat) β(θ), so that it would lead to the usual N-gon structures, with (Formula presented) 2, 3.
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16
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0028386874
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The model in Eqs. (9) and (10) gives comparable results with the mean-field model introduced by G. L. Oppo, M. Brambilla, and L. Lugiato, Phys. Rev. A 49, 2028 (1994)
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(1994)
Phys. Rev. A
, vol.49
, pp. 2028
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Oppo, G.L.1
Brambilla, M.2
Lugiato, L.3
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17
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0012842292
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Phys. Rev. Afor small values of (Formula presented) as shown by M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, 61, 043806 (2000).
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(2000)
, vol.61
, pp. 43806
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Tlidi, M.1
Le Berre, M.2
Ressayre, E.3
Tallet, A.4
Di Menza, L.5
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19
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0000426270
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A. C. Newell, T. Passot, C. Bowman, N. Ercolani, and R. Indik, Physica D 97, 185 (1996).
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(1996)
Physica D
, vol.97
, pp. 185
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Newell, A.C.1
Passot, T.2
Bowman, C.3
Ercolani, N.4
Indik, R.5
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21
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85036269545
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P. Manneville, Structures Dissipatives Chaos et Turbulene (Aléa, Saclay, France, 1990)
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P. Manneville, Structures Dissipatives Chaos et Turbulene (Aléa, Saclay, France, 1990).
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24
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85036205075
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For nonanalytic functions, or for analytic functions (Formula presented) without any pole in the complex k plane, the result could be different. For example, in the case of a square shaped structure factor (Formula presented) for (Formula presented) the asymptotic behavior (Formula presented) leads to the power law behavior (Formula presented)
-
For nonanalytic functions, or for analytic functions (Formula presented) without any pole in the complex k plane, the result could be different. For example, in the case of a square shaped structure factor (Formula presented) for (Formula presented) the asymptotic behavior (Formula presented) leads to the power law behavior (Formula presented)
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25
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85036347425
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This procedure was implemented by P. Manneville in the case of the SH equation (private communication)
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This procedure was implemented by P. Manneville in the case of the SH equation (private communication).
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26
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85036296829
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A weakly nonlinear analysis proves that the mean-field model, and therefore also Eqs. (9) and (10), reduce to the SH equation (2), in the limit of small negative mistunings 25 (Formula presented) But our study of the DOPO concerns the case (Formula presented) outside the validity range of the approximation of a DOPO by the SH equation
-
A weakly nonlinear analysis proves that the mean-field model, and therefore also Eqs. (9) and (10), reduce to the SH equation (2), in the limit of small negative mistunings 25 (Formula presented) But our study of the DOPO concerns the case (Formula presented) outside the validity range of the approximation of a DOPO by the SH equation.
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27
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85036181734
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L. Di Menza, M. Le Berre, Y. Pomeau, E. Ressayre, and A. Tallet, Rencontre du Non-Linéaire (IHP, Paris, 2001)
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L. Di Menza, M. Le Berre, Y. Pomeau, E. Ressayre, and A. Tallet, Rencontre du Non-Linéaire (IHP, Paris, 2001).
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28
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0030213616
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G. J. de Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
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(1996)
Phys. Rev. A
, vol.54
, pp. 1609
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de Valcarcel, G.J.1
Staliunas, K.2
Roldan, E.3
Sanchez-Morcillo, V.J.4
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30
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85036281523
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Phys. Rev. ED. BoyerJ. Vinals65, 046119 (2002).
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(2002)
, vol.65
, pp. 46119
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Boyer, D.1
Vinals, J.2
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