Abstract
A generalized parametrically driven damped nonlinear Schrödinger equation is used to describe, close to the resonance, the dynamics of weakly dissipative systems, like a harmonically coupled pendula chain or an easy-plane magnetic wire. The combined effects of parametric forcing, spatial coupling, and dissipation allows for the existence of stable non-trivial uniform states as well as homogeneous pattern states. The latter can be regular or chaotic. A new family of localized states that connect asymptotically a non-trivial uniform state with a spatio-temporal chaotic pattern is numerically found. We discuss the parameter range, where these localized structures exist. This article is dedicated to Prof. Helmut R. Brand on the occasion of his 60th birthday.
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References
O. Descalzi, M. Clerc, S. Residori, G. Assanto (eds.), Localized States in Physics: Solitons and Patterns (Springer, Berlin, 2010)
Y.V. Kartashov, B.A. Malomed, L. Torner, Rev. Mod. Phys. 83, 247 (2011)
E. Knobloch, Nonlinearity 21, T45 (2008)
A. Ankiewics, N. Akhmediev, Dissipative Solitons: From Optics to Biology and Medicine (Springer, Berlin, 2008)
P. Coullet, Int. J. Bif. Chaos 12, 245 (2002)
W. van Saarloos, P.C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990)
G. Nicolis, I. Prigogine, Self-organization in Non Equilibrium systems (J. Wiley & sons, New York, 1977)
M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)
B.A. Malomed, A.A. Nepomnyashchy, Phys. Rev. A 42, 6009 (1990)
N. Verschueren, U. Bortolozzo, M.G. Clerc, S. Residori, Phys. Rev. Lett. 110, 104101 (2013)
P.D. Woods, A.R. Champneys, Physica D 129, 147 (1999)
P. Coullet, Phys. Rev. Lett. 84, 3069 (2000)
H.R. Brand, R.J. Deissler, Phys. Rev. Lett. 63, 2801 (1989)
H.R. Brand, R.J. Deissler, Physica A 204, 87 (1994)
R.J. Deissler, H.R. Brand, Phys. Rev. Lett. 72, 478 (1994)
R.J. Deissler, H.R. Brand, Phys. Rev. E 51, R852 (1995)
R.J. Deissler, H.R. Brand, Phys. Rev. Lett. 74, 4847 (1995)
N. Akhmediev, J.M. Soto-Crespo, G. Town, Phys. Rev. E 63, 056602 (2001)
A. Ankiewicz, N. Devine, N. Akhmediev, J.M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008)
N. Akhmediev, J.M. Soto-Crespo, Phys. Lett. A 317, 287 (2003)
N. Akhmediev, J.M. Soto-Crespo, Phys. Rev. E 70, 036613 (2004)
C. Cartes, J. Cisternas, O. Descalzi, H.R. Brand, Phys. Rev. Lett. 109, 178303 (2012)
O. Descalzi, H.R. Brand, Phys. Rev. E 82, 026203 (2010)
O. Descalzi, C. Cartes, J. Cisternas, H.R. Brand, Phys. Rev. E 83, 056214 (2011)
C. Cartes, O. Descalzi, H.R. Brand, Phys. Rev. E 85, 015205 (2012)
M. Faraday, Phil. Trans. R. Soc. 121, 299 (1831)
V.I. Arnold, Geometrical methods in the theory of ordinary differential equations (Springer-Verlag, New York, 1983)
J.K. Hale, H. Kocak, Dynamics and Bifurcations (Springer-Verlag, NY, 1991)
M. Clerc, P. Coullet, E. Tirapegui, Phys. Rev. Lett. 83, 3820 (1999)
J.W. Miles, J. Fluid Mech. 148, 451 (1984)
B. Denardo, et al., Phys. Rev. Lett. 68, 1730 (1992)
J.N. Kutz, et al., Opt. Lett. 18, 802 (1993)
S. Longhi, Phys. Rev. E 53, 5520 (1996)
I.V. Barashenkov, M.M. Bogdan, V.I. Korobov, Europhys. Lett. 15, 113 (1991)
M.G. Clerc, S. Coulibaly, D. Laroze, EPL 90, 38005 (2010)
D. Urzagasti, D. Laroze, M.G. Clerc, S. Coulibaly, H. Pleiner, J. Appl. Phys. 111, 07D111 (2012)
N.V. Alexeeva, I.V. Barashenkov, G.P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000)
I.V. Barashenkov, E.V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999)
V.S. Shchesnovich, I.V. Barashenkov, Physica D 164, 83 (2002)
E.V. Zemlyanaya, N.V. Alexeeva, Theor. Math. Phys. 159, 870 (2009)
I.V. Barashenkov, E.V. Zemlyanaya, T.C. van Heerden, Phys. Rev. E 83, 056609 (2011)
I.V. Barashenkov, E.V. Zemlyanaya, Phys. Rev. E 83, 056610 (2011)
M.G. Clerc, S. Coulibaly, D. Laroze, Phys. Rev. E 77, 056209 (2008)
M.G. Clerc, S. Coulibaly, D. Laroze, Int. J. Bif. Chaos 19, 2717 (2009)
M.G. Clerc, S. Coulibaly, D. Laroze, Int. J. Bif. Chaos 19, 3525 (2009)
M.G. Clerc, S. Coulibaly, D. Laroze, Physica D 239, 72 (2010)
M.G. Clerc, S. Coulibaly, D. Laroze, EPL 97, 30006 (2012)
I.V. Barashenkov, S. Cross, B.A. Malomed, Phys. Rev. E 68, 056605 (2003)
J. Burke, A. Yochelis, E. Knobloch, SIAM J. Appl. Dyn. Sys. 7, 651 (2008)
Y.-P. Ma, J. Burke, E. Knobloch, Physica D 239, 1867 (2010)
E. Kenig, B.A. Malomed, M.C. Cross, R. Lifshitz, Phys. Rev. E 80, 046202 (2009)
P. Coullet, T. Firsch, G. Sonnino, Phys. Rev. E 49, 2087 (1994)
C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, NY, 1999)
D. Laroze, J. Bragard, O.J. Suarez, H. Pleiner, IEEE Trans. Mag. 47, 3032 (2011)
J. Bragard, H. Pleiner, O.J. Suarez, P. Vargas, J.A.C. Gallas, D. Laroze, Phys. Rev. E 84, 037202 (2011)
D. Laroze, D. Becerra-Alonso, J.A.C. Gallas, H. Pleiner, IEEE Trans. Mag. 48, 3567 (2012)
C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, G. Iooss, Physica D 29, 95 (1987)
O. Batiste, E. Knobloch, A. Alonso, I. Mercader, J. Fluid Mech. 560, 149 (2006)
J. Burke, E. Knobloch, Phys. Rev. E 73, 056211 (2006)
J. Burke, E. Knobloch Chaos 17, 037102 (2007)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D, 16, 285 (1985)
J.P. Eckmann, D. Ruelle, S. Ciliberto, Phys. Rev. A, 34, 4971 (1986)
K. Geist, U. Parlitz, W. Lauterborn, Prog. Theo. Phys. 83, 875 (1990)
J.D. Scheel, M.C. Cross, Phys. Rev. E 74, 066301 (2006)
A. Karimi, M.R. Paul Phys. Rev. E 85, 046201 (2012)
D. Laroze, P.G. Siddheshwar, H. Pleiner, Commun. Nonlinear Sci. Numer. Simulat. 18, 2436 (2013)
C. Bonatto, J.A.C. Gallas, Phys. Rev. Lett. 101, 054101 (2008)
G.M. Ramirez-Avila, J.A.C. Gallas, Phys. Lett. A 375, 143 (2010)
J.A.C. Gallas, Int. J. Bif. Chaos 20, 197 (2010); and references therein
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery. Numerical Recipes in FORTRAN (Cambridge University Press, UK, 1992)
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Urzagasti, D., Laroze, D. & Pleiner, H. Localized chaotic patterns in weakly dissipative systems. Eur. Phys. J. Spec. Top. 223, 141–154 (2014). https://doi.org/10.1140/epjst/e2014-02089-x
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DOI: https://doi.org/10.1140/epjst/e2014-02089-x