Oscillatory pattern formation with a conserved quantity

Nonlinearity

Volumn 18, Issue 3, 2005, Pages 1031-1056

  Winterbottom, D M ^a   Matthews, P C ^a   Cox, S M ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b UNIVERSITY OF ADELAIDE   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29744458394     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/18/3/006     Document Type: Article

References (47)

1. Aranson I S and Kramer L 2002 The world of the complex Ginzburg-Landau equations Rev. Mod. Phys. 74 99-143
2. Balmforth N J and Casti A 1998 Convection in a slowly diffusing, weakly stratified salt field Phys. Lett. A 238 35-42
3. Benjamin T B and Feir J E 1967 The disintegration of wave trains on deep water J. Fluid Mech. 27 417-30
4. Brusch L, Zimmermann M G, van Hecke M, Bär M and Torcini A 2000 Modulated amplitude waves and the transition from phase to defect chaos Phys. Rev. Lett. 85 86-9
5. Coullet P, Fauve S and Tirapegui E 1985 Large scale instability of nonlinear standing waves J. Lett. Phys. 46 L787-91
6. Coullet P and Iooss G 1990 Instabilities of one-dimensional cellular patterns Phys. Rev. Lett. 64 866-9
7. Cox S M and Matthews P C 2001 New instabilities in two-dimensional rotating convection and magnetoconvection Physica D 149 210-29
8. Cox S M and Matthews P C 2002 Exponential time differencing for stiff systems J. Comput. Phys. 176 430-55
9. Cox S M and Matthews P C 2003 One-dimensional pattern formation in systems with a conserved quantity Bifurcations, Symmetry and Patterns (Basel: Birkhäuser) pp 135-40
10. Cox S M, Matthews P C and Pollicott S L 2004 Swift-Hohenberg model for magnetoconvection Phys. Rev. E 69 066314
11. Cross M C and Hohenberg P C 1993 Pattern formation outside of equilibrium Rev. Mod. Phys. 65 851-1112
12. Doelman A, Hek G and Valkhoff N 2004 Stabilization by slow diffusion in a real Ginzburg-Landau system J. Nonlinear Sci. 14 237-78
13. Eckhaus W 1965 Studies in Non-Linear Stability Theory (Berlin: Springer)
14. Frisch U, She S S and Thual O 1986 Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model J. Fluid Mech. 168 221-40
15. Hernández-Garcia E and Montagne R 2000 Localized structures in coupled Ginzburg-Landau equations Phys. Lett. A 273 239-44
16. Herrero H and Riecke H 1994 Front structures in a real Ginzburg-Landau equation coupled to a mean field J. Int. Bifurc. Chaos 4 1343-6
17. Herrero H and Riecke H 1995 Sound pairs of fronts in a real Ginzburg-Landau equation coupled to a mean field Physica D 85 79-92
18. Herrero H and Riecke H 1997 Holes and chaotic pulses of traveling waves coupled to a long-wave mode Phys. Lett. A 235 493-8
19. Ipsen M and Sørensen P G 2000 Finite wavelength instabilities in a slow mode coupled complex Ginzburg-Landau equation Phys. Rev. Lett. 84 2389-92
20. Janiaud B, Pumir A, Bensimon D, Croquette V, Richter H and Kramer L 1992 The Eckhaus instability for travelling waves Physica D 55 269-86
21. Knobloch E 1992 Nonlocal amplitude equations Pattern Formation in Complex Dissipative Systems vol 65, ed S Kai (Singapore: World Scientific) pp 263-74
22. Knobloch E 1995 Remarks on the use and misuse of the Ginzburg-Landau equation Nonlinear Dynamics and Pattern Formation in the Natural Environment vol 335, ed A Doelman and A van Harten (Harlow: Longman) pp 130-46
23. Knobloch E and De Luca J 1990 Amplitude equations for travelling wave convection Nonlinearity 3 975-80
24. Kolodner P 1994 Stable, unstable, and defected confined states of traveling-wave convection Phys. Rev. E 50 2731-55
25. Komarova N L and Newell A C 2000 Nonlinear dynamics of sand banks and sand waves J. Fluid Mech. 415 285-321
26. Matkowsky B J and Volpert V 1993 Stability of plane wave solutions of complex Ginzburg-Landau equations Q. Appl. Math. 51 265-81
27. Matthews P C and Rucklidge A M 1993 Travelling and standing waves in magnetoconvection Proc. R. Soc. Lond. A 441 649-58
28. Matthews P C and Cox S M 2000 Pattern formation with a conservation law Nonlinearity 13 1293-320
29. Melo F, Umbanhowar P and Swinney H L 1994 Transition to parametric wave patterns in a vertically oscillated granular layer Phys. Rev. Lett. 72 172-5
30. Newell A C 1974 Envelope equations Lect. Appl. Math. 15 157-63
31. Proctor M R E 2001 Finite amplitude behaviour of the Matthews-Cox instability Phys. Lett. A 292 181-7
32. Riecke H 1992 Self-trapping of traveling-wave pulses in binary mixture convection Phys. Rev. Lett. 68 301-4
33. Riecke H 1992 Ginzburg-Landau equation coupled to a concentration field in binary-mixture convection Physica D 61253-9
34. Riecke H 1996 Solitary waves under the influence of a long-wave mode Physica D 92 69-94
35. Riecke H and Granzow G D 1998 Localization of waves without testability: worms in nematic electroconvection Phys. Rev. Lett. 81 333-6
36. Riecke H 1999 Localized structures in pattern-forming systems IMA Volume in Mathematical Applications vol 115 (New York: Springer) pp 215-29
37. Riecke H and Kramer L 2000 The stability of standing waves with small group velocity Physica D 137 124-42
38. Sakaguchi H 1993 Phase dynamics and localized solutions to the Ginzburg-Landau type amplitude equations Prog. Theor. Phys. 89 1123-46
39. Sakaguchi H 1995 Phase dynamics and the coupled complex Ginzburg-Landau equations Prog. Theor. Phys. 93 491-502
40. Sakaguchi H 1996 Localized solutions to the coupled complex Ginzburg-Landau equations Prog. Theor. Phys. 95 823-7
41. Sakaguchi H 1996 Travelling wave-standing wave transition in the coupled complex Ginzburg-Landau equations Phys. Scr. T 67 148-50
42. Schöpf W 1993 Convection in binary fluids: amplitude equations, codimension-2 bifurcation, and thermal fluctuations Phys. Rev. E 47 1739-64
43. Tsimring L S and Aranson I S 1997 Localized and cellular patterns in a vibrated granular layer Phys. Rev. Lett. 79 213-16
44. Umbanhowar P, Melo F and Swinney H 1996 Localized excitations in a vertically vibrated granular layer Nature 38 2793-6
45. van Hecke M, Storm C and van Saarloos W 1999 Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems Physica D 134 1-47
46. Vega J M 1993 On the amplitude equations arising at the onset of the oscillatory instability in pattern formation SIAM J. Math. Anal. 24 603-17
47. Walgraef D 1997 Pattern selection and the effect of group velocity on interacting oscillatory and stationary instabilities Phys. Rev. E 55 6887-97