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Amplitude equations for systems with long-range interactions.

Klaus Kassner1, Chaouqi Misbah

  • 1Institut für Theoretische Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2002
PubMed
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This study develops amplitude equations for interface dynamics in pattern-forming systems with long-range interactions, revealing nonlocal and nonlinear terms that persist in the long-wave limit.

Area of Science:

  • Physics
  • Applied Mathematics

Background:

  • Pattern-forming systems often exhibit complex interface dynamics.
  • Understanding these dynamics is crucial for various scientific and engineering applications.
  • Long-range interactions introduce unique challenges in modeling these systems.

Purpose of the Study:

  • To derive amplitude equations for interface dynamics in systems with long-range interactions.
  • To investigate the persistence of long-range interactions in the long-wave limit.
  • To analyze the implications of nonlocal and nonlinear terms in interface equations.

Main Methods:

  • Derivation of amplitude equations using long-wave asymptotics.
  • Application of integral transforms to solve linear bulk equations.
  • Analysis of the Grinfeld and Saffman-Taylor instabilities as examples.

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Main Results:

  • Developed a method applicable to linear, integral-transform-solvable bulk equations.
  • Demonstrated that long-range interactions survive the long-wave limit.
  • Identified nonlocal and nonlinear terms in the resulting interface equation, a novel feature.

Conclusions:

  • The derived interface equation offers a new model for systems with persistent nonlocal effects.
  • This work provides insights into the universal dynamics of pattern-forming systems with long-range interactions.
  • The findings have implications for understanding pattern formation and interface stability in diverse physical systems.