### Abstract

The complex Ginzburg-Landau equation with weak noise, the normal form of the amplitude equation for the order parameter in a spatially distributed system undergoing a continuous Hopf bifurcation, is solved in certain limits for its time-independent probability distribution, which governs the steady state in one spatial dimension. The method used consists of solving the Hamilton-Jacobi equation of the nonequilibrium potential associated with the steady-state distribution. The solution is obtained in the limit of weak spatial diffusion of the order parameter. The nonequilibrium potential serves as a Lyapunov functional for the order-parameter field. We use our result to discuss the Newell-Kuramoto instability and the Eckhaus-Benjamin-Feir instability in one spatial dimension, and to calculate potential barriers of the saddles separating plane-wave attractors. The latter ones provide us with a global measure of stability for these attractors.

Original language | English |
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Pages (from-to) | 4661-4677 |

Number of pages | 17 |

Journal | Physical Review A |

Volume | 42 |

Issue number | 8 |

DOIs | |

Publication status | Published - Jan 1 1990 |

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

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## Cite this

*Physical Review A*,

*42*(8), 4661-4677. https://doi.org/10.1103/PhysRevA.42.4661