Snakes, Ladders, and Isolas of Localized Patterns

Margaret Beck, Juergen Knobloch, David J. B. Lloyd, Bjoern Sandstede, Thomas Wagenknecht

Research output: Contribution to journalArticle

Abstract

Stable localized roll structures have been observed in many physical problems and model equations, notably in the one-dimensional (1D) Swift-Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.

Original languageEnglish
Pages (from-to)936-972
Number of pages37
JournalSIAM Journal on Mathematical Analysis
Volume41
Issue number3
DOIs
Publication statusPublished - 2009

Keywords

  • snaking
  • rolls
  • localized patterns
  • Swift-Hohenberg equation
  • SWIFT-HOHENBERG EQUATION
  • SOLITARY-WAVE SOLUTIONS
  • DIFFERENTIAL-EQUATIONS
  • HOMOCLINIC ORBITS
  • REVERSIBLE-SYSTEMS
  • SNAKING
  • BIFURCATION
  • STABILITY
  • ABSOLUTE
  • PULSES

Cite this

Beck, M., Knobloch, J., Lloyd, D. J. B., Sandstede, B., & Wagenknecht, T. (2009). Snakes, Ladders, and Isolas of Localized Patterns. SIAM Journal on Mathematical Analysis, 41(3), 936-972. https://doi.org/10.1137/080713306
Beck, Margaret ; Knobloch, Juergen ; Lloyd, David J. B. ; Sandstede, Bjoern ; Wagenknecht, Thomas. / Snakes, Ladders, and Isolas of Localized Patterns. In: SIAM Journal on Mathematical Analysis. 2009 ; Vol. 41, No. 3. pp. 936-972.
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abstract = "Stable localized roll structures have been observed in many physical problems and model equations, notably in the one-dimensional (1D) Swift-Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two {"}snaking{"} solution branches so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many {"}ladder{"} branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.",
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Beck, M, Knobloch, J, Lloyd, DJB, Sandstede, B & Wagenknecht, T 2009, 'Snakes, Ladders, and Isolas of Localized Patterns', SIAM Journal on Mathematical Analysis, vol. 41, no. 3, pp. 936-972. https://doi.org/10.1137/080713306

Snakes, Ladders, and Isolas of Localized Patterns. / Beck, Margaret; Knobloch, Juergen; Lloyd, David J. B.; Sandstede, Bjoern; Wagenknecht, Thomas.

In: SIAM Journal on Mathematical Analysis, Vol. 41, No. 3, 2009, p. 936-972.

Research output: Contribution to journalArticle

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AU - Beck, Margaret

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AU - Lloyd, David J. B.

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AU - Wagenknecht, Thomas

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Beck M, Knobloch J, Lloyd DJB, Sandstede B, Wagenknecht T. Snakes, Ladders, and Isolas of Localized Patterns. SIAM Journal on Mathematical Analysis. 2009;41(3):936-972. https://doi.org/10.1137/080713306