Localized radial solutions of the Swift-Hohenberg equation

Nonlinearity

Volumn 22, Issue 2, 2009, Pages 485-524

  Lloyd, David ^a   Sandstede, Björn ^a,b  

^a UNIVERSITY OF SURREY   (United Kingdom)

^b BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 65249131771     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/22/2/013     Document Type: Article

References (45)

1. Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)
2. Arbell H and Fineberg J 2000 Temporally harmonic oscillons in Newtonian fluids Phys. Rev. Lett. 85 756-9
3. Astrov Y A and Purwins H G 2001 Plasma spots in a gas discharge system: birth, scattering and formation of molecules Phys. Lett. A 283 349-54
4. Astrov Y A and Purwins H G 2006 Spontaneous division of dissipative solitons in a planar gas-discharge system with high ohmic electrode Phys. Lett. A 358 404-8
5. Budd C J, Hunt G W and Kuske R 2001 Asymptotics of cellular buckling close to the Maxwell load Proc. R. Soc. Lond. A 457 2935-64
6. Budd C J and Kuske R 2005 Localized periodic patterns for the non-symmetric generalized Swift-Hohenberg equation Physica D 208 73-95
7. Burke J and Knobloch E 2006 Localized states in the generalized Swift-Hohenberg equation Phys. Rev. E 73 056211
8. Burke J and Knobloch E 2007 Normal form for spatial dynamics in the Swift-Hohenberg equation Discrete Contin. Dyn. Syst. Suppl. (September issue) 170-80
9. Cox S M and Matthews P C 2002 Exponential time differencing for stiff systems J. Comput. Phys. 176 430-55
10. Cross M and Hohenberg P 1993 Pattern formation outside of equilibrium Rev. Mod. Phys. 65 851-1112 (Pubitemid 223000784)
11. Davies P, Blanchedeau P, Dulos E and Kepper P D 1998 Dividing blobs, chemical flowers and patterned islands in a reaction-diffusion system J. Phys. Chem. A 102 8236-44 (Pubitemid 128578174)
12. Doedel E J, Paffenroth R C, Champneys A R, Fairgrieve T F, Kuznetsov Y A, Oldeman B E, Sandstede B and Wang X J 2000 AUTO2000: continuation and bifurcation software for ordinary differential equations Technical Report Concordia University Available via http://www.dynamicalsystems.org/
13. Elphick C, Tirapegui E, Brachet M E, Coullet P and Iooss G 1987 A simple global characterization for normal forms of singular vector fields Physica D 29 95-127
14. Fife P C 2002 Pattern formation in gradient systems Handbook of Dynamical Systems 2 ed B Fiedler (Amsterdam: North-Holland) pp 679-719
15. Firth W J, Columbo L and Maggipinto T 2007 On homoclinic snaking in optical systems Chaos 17 037115
16. Groves M D 1998 Solitary-wave solutions to a class of fifth-order model equations Nonlinearity 11 341-53 (Pubitemid 128368462)
17. Hoyle R B 2006 Pattern Formation (Cambridge: Cambridge University Press)
18. Hunt G W, Peletier M A, Champneys A R, Woods P D, Wadee M A, Budd C J and Lord G J 2000 Cellular buckling in long structures Nonlinear Dyn. 21 3-29 (Pubitemid 30556689)
19. Kassam A-K and Trefethen L N 2005 Fourth-order time-stepping for stiff PDEs SIAM J. Sci. Comput. 26 1214-33
20. Kuzin I and Pohozaev S 1997 Entire Solutions of Semilinear Elliptic Equations (Basel: Birkhäuser)
21. Lee K J, McCormick W D, Pearson J E and Swinney H L 1994 Experimental observation of self-replicating spots in a reaction diffusion system Nature 369 215-8 (Pubitemid 24156121)
22. Lioubashevski O, Hamiel Y, Agnon A, Reches Z and Fineberg J 1999 Oscillons and propagating solitary waves in vertically vibrated colloidal suspension Phys. Rev. Lett. 83 3190-3 (Pubitemid 129582861)
23. Lloyd D J B and Champneys A R 2005 Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons SIAM J. Sci. Comput. 27 759-73
24. Lloyd D J B, Sandstede B, Avitabile D and Champneys A R 2008 Localized hexagon patterns of the planar Swift-Hohenberg equation SIAM J. Appl. Dynam. Syst. 7 1049-100
25. Menesguen Y, Barbay S, Hachair X, Leroy L, Sagnes I and Kuszelewicz R 2006 Optical self-organization and cavity solitons in optically pumped semiconductor microresonators Phys. Rev. A 74 023818
26. Meron E, Yizhaq H and Gilad E 2007 Localized structures in dryland vegetation: forms and functions Chaos 17 037109
27. Nasuno S 2003 Dancing 'atoms' and 'molecules' of luminous gas-discharge spots Chaos 13 1010-3
28. Peletier L A and Troy W C 2001 Spatial Patterns (Boston, MA: Birkhäuser)
29. Pismen L M 2006 Patterns and Interfaces in Dissipative Dynamics (Berlin: Springer)
30. Rabinovich M I, Ezersky A B and Weidman P D 2000 The Dynamics of Patterns (River Edge, NJ: World Scientific Publishing)
31. Richter R 2008 private communication
32. Richter R and Barashenkov I V 2005 Two-dimensional solitons on the surface of magnetic fluids Phys. Rev. Lett. 94 184503
33. Sandstede B 1997 Convergence estimates for the numerical approximation of homoclinic solutions IMA J. Numer. Anal. 17 437-62 (Pubitemid 127351706)
34. Sandstede B and Scheel A 2005 Absolute instabilities of standing pulses Nonlinearity 18 331-78 (Pubitemid 40097532)
35. Scheel A 1997 Subcritical bifurcation to infinitely many rotating waves J. Math. Anal. Appl. 215 252-61 (Pubitemid 127170552)
36. Scheel A 2003 Radially symmetric patterns of reaction-diffusion systems Mem. Am. Math. Soc. 165 (Pubitemid 37089275)
37. Sheffer E, Yizhaq H, Gilad E, Shachak M and Meron E 2007 Why do plants in resource-derived environments form rings? Ecol. Complexity 4 192-200 (Pubitemid 47419440)
38. Stenflo L and Yu M 2003 Oscillons at a plasma surface Phys. Plasmas 10 912-3
39. Swift J and Hohenberg P C 1977 Hydrodynamic fluctuations at the convective instability Phys. Rev. A 15 319-28
40. Umbanhowar P B, Melo F and Swinney H L 1996 Localized excitations in a vertically vibrated granular layer Nature 382 793-6 (Pubitemid 26299558)
41. Vanag V K and Epstein I R 2007 Localised patterns in reaction-diffusion systems Chaos 17 037110
42. Vladimirov A G, McSloy J M, Skryabin D V and Firth W J 2002 Two-dimensional clusters of solitary structures in driven optical cavities Phys. Rev. E 65 046606
43. Watson G N 1944 A Treatise on the Theory of Bessel Functions (Cambridge: Cambridge University Press)
44. Yochelis A, Burke J and Knobloch E 2006 Reciprocal oscillons and nonmonotonic fronts in forced nonequilibrium systems Phys. Rev. Lett. 97 254501
45. Zelik S and Mielke A Weak interaction in multi-pulse structures and space-time chaos in reaction-diffusion equations Mem. Am. Math. Soc. submitted