Dynamics of Codimension-One Defects

  • P. Coullet
  • C. Elphick
  • D. Repaux
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 41)


Pattern formation in nonequilibrium systems is associated with instabilities which break spatio-temporal symmetries [1], [2], The dynamics of a physical system is generally described by a set of partial differential equations of the type
where U(r, t) is a state vector, V stands for the gradient operator, r represents a position vector and {λ} is a set of parameters. We assume that this system possesses a stationary solution denoted U*(r). A perturbation u(r, t) to this solution obeys the following equation:
$$\frac{\partial U}{\partial t} = {L}_{\{\lambda\}}( \nabla ) u + N(u),$$
where L{λ}(∇) is a linear operator defined as \(L=\frac{\delta F}{\delta U}\vert_{U^*}\) stands for the non-linear terms. The stability of the stationary solution is determined by the spectrum of the linear operator L {λ} . We also assume that, for some {λ} = {λc} in the parameter space, the linear operator possesses an eigenvalue whose real part vanishes, while the others have a negative real part. Near the instability threshold, the perturbation of the stationary solution can be expressed as:
$$u(r, t) = A(r, t)\phi(r)+cc+V$$
where A(r,t) is the amplitude of the unstable mode, Φ(r) is the corresponding eigenvector which keeps track of the broken symmetries, and V contains the higher order contributions coming from the non-linear part of the unstable mode.


Chaotic Behavior Unstable Mode Instability Threshold Ferromagnetic Transition High Order Contribution 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • P. Coullet
    • 1
  • C. Elphick
    • 1
  • D. Repaux
    • 1
  1. 1.Laboratoire de Physique ThéoriqueUniversité de NiceNice CedexFrance

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