Pinning and depinning of fronts bounding spatially localized structures in the forced complex Ginzburg-Landau equation describing the 1:1 resonance is studied in one spatial dimension, focusing on regimes in which the structure grows via roll insertion instead of roll nucleation at either edge. The motion of the fronts is nonlocal but can be analyzed quantitatively near the depinning transition.
REFERENCES
1.
Ackemann
, T.
, Firth
, W. J.
, and Oppo
, G.-L.
, “Fundamentals and applications of spatial dissipative solitons in photonic devices
,” in Advances in Atomic, Molecular and Optical Physics
, edited by P. R.
Berman
, E.
Arimondo
, and C. C.
Lin
(Academic Press
, 2009
), Vol. 57, pp. 323
–421
. 2.
Aranson
, I. S.
, Malomed
, B. A.
, Pismen
, L. M.
, and Tsimring
, L. S.
, “Crystallization kinetics and self-induced pinning in cellular patterns
,” Phys. Rev. E
62
, R5
–R8
(2000
). 3.
Barten
, W.
, Lücke
, M.
, and Kamps
, M.
, “Localized traveling-wave convection in binary-fluid mixtures
,” Phys. Rev. Lett.
66
, 2621
–2624
(1991
). 4.
Batiste
, O.
, Knobloch
, E.
, Alonso
, A.
, and Mercader
, I.
, “Spatially localized binary-fluid convection
,” J. Fluid Mech.
560
, 149
–158
(2006
). 5.
Beck
, M.
, and Wayne
, C. E.
, “Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity
,” SIAM Rev.
53
, 129
–153
(2011
). 6.
Bergeon
, A.
, and Knobloch
, E.
, “Spatially localized states in natural doubly diffusive convection
,” Phys. Fluids
20
, 034102
(2008
). 7.
Blanchflower
, S.
, “Magnetohydrodynamic convectons
,” Phys. Lett. A
261
, 74
–81
(1999
). 8.
Burke
, J.
, Houghton
, S. M.
, and Knobloch
, E.
, “Swift-Hohenberg equation with broken reflection symmetry
,” Phys. Rev. E
80
, 036202
(2009
). 9.
Burke
, J.
, and Knobloch
, E.
, “Localized states in the generalized Swift-Hohenberg equation
,” Phys. Rev. E
73
, 056211
(2006
). 10.
Burke
, J.
, and Knobloch
, E.
, “Homoclinic snaking: Structure and stability
,” Chaos
17
, 037102
(2007
). 11.
Champneys
, A. R.
, Knobloch
, E.
, Ma
, Y.-P.
, and Wagenknecht
, T.
, “Homoclinic snakes bounded by a saddle-centre periodic orbit
,” SIAM J. Appl. Dyn. Sys.
(submitted). 12.
Coullet
, P.
, and Emilsson
, K.
, “Strong resonances of spatially distributed oscillators: A laboratory to study patterns and defects
,” Physica D
61
, 119
–131
(1992
). 13.
Coullet
, P.
, Riera
, C.
, and Tresser
, C.
, “Stable static localized structures in one dimension
,” Phys. Rev. Lett.
84
, 3069
–3072
(2000
). 14.
Cox
, S. M.
, and Matthews
, P. C.
, “Exponential time differencing for stiff systems
,” J. Comput. Phys.
176
, 430
–455
(2002
). 15.
Dennin
, M.
, Ahlers
, G.
, and Cannell
, D. S.
, “Spatiotemporal chaos in electroconvection
,” Science
272
(5260
), 388
–390
(1996
). 16.
Houghton
, S. M.
, and Knobloch
, E.
, “The Swift-Hohenberg equation with broken cubic-quintic nonlinearity
,” Phys. Rev. E
84
, 016204
(2011
). 17.
Hoyle
, R.
, Pattern Formation
(Cambridge University Press
, Cambridge
, 2006
). 18.
Kolodner
, P.
, Bensimon
, D.
, and Surko
, C. M.
, “Traveling-wave convection in an annulus
,” Phys. Rev. Lett.
60
, 1723
–1726
(1988
). 19.
Krechetnikov
, R.
, and Marsden
, J. E.
, “Dissipation-induced instabilities in finite dimensions
,” Rev. Mod. Phys.
79
, 519
–553
(2007
). 20.
Lioubashevski
, O.
, Arbell
, H.
, and Fineberg
, J.
, “Dissipative solitary waves in driven surface waves
,” Phys. Rev. Lett.
76
, 3959
–3962
(1996
). 21.
Lombardi
, E.
, “Oscillatory integrals and phenomena beyond all algebraic orders: With applications to homoclinic orbits in reversible systems
,” in Lecture Notes in Mathematics
(Springer
, 2000
). 22.
Ma
, Y.-P.
, Burke
, J.
, and Knobloch
, E.
, “Defect-mediated snaking: A new growth mechanism for localized structures
,” Physica D
239
, 1867
–1883
(2010
). 23.
Ma
, Y.-P.
, and Knobloch
, E.
, “Localized states in the forced complex Ginzburg-Landau equation with 1:1 resonance
,” preprint
(2010
). 24.
Pomeau
, Y.
, “Front motion, metastability and subcritical bifurcations in hydrodynamics
,” Physica D
23
, 3
–11
(1986
). 25.
Rajchenbach
, J.
, Leroux
, A.
, and Clamond
, D.
, “New standing solitary waves in water
,” Phys. Rev. Lett.
107
, 024502
(2011
). 26.
Riecke
, H.
, and Granzow
, G. D.
, “Localization of waves without bistability: Worms in nematic electroconvection
,” Phys. Rev. Lett.
81
, 333
–336
(1998
). 27.
Sandstede
, B.
, “Stability of travelling waves
,” in Handbook of Dynamical Systems
(Elsevier Science
, 2002
), Vol. 2, pp. 983
–1055
. 28.
Sandstede
, B.
, and Scheel
, A.
, “Defects in oscillatory media: Toward a classification
,” SIAM J. Appl. Dyn. Syst.
3
, 1
–68
(2004
). 29.
Schneider
, T. M.
, Gibson
, J. F.
, and Burke
, J.
, “Snakes and ladders: Localized solutions of plane Couette flow
,” Phys. Rev. Lett.
104
, 104501
(2010
). 30.
Tobias
, S. M.
, Proctor
, M. R. E.
, and Knobloch
, E.
, “Convective and absolute instabilities of fluid flows in finite geometry
,” Physica D
113
, 43
–72
(1998
). 31.
Ueda
, K.-I.
, and Nishiura
, Y.
, “A mathematical mechanism for instabilities in stripe formation on growing domains
,” Physica D
241
, 37
–59
(2012
). 32.
Umbanhowar
, P. B.
, Melo
, F.
, and Swinney
, H. L.
, “Localized excitations in a vertically vibrated granular layer
,” Nature
382
, 793
–796
(1996
). 33.
van Saarloos
, W.
, “Front propagation into unstable states
,” Phys. Rep.
386
, 29
–222
(2003
). © 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
