### Abstract

It is known that the Swift-Hohenberg equation ∂u/∂t=-(∂_{x}^{2}+1)^{2}+1)u+e{open}(u-u^{3}) can be reduced to the Ginzburg-Landau equation (amplitude equation) ∂A/∂t=4∂_{x}^{2}A+e{open}(A-3A|A|^{2}) by means of the singular perturbation method. This means that if e{open} > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations ∂u/∂t=Pu+e{open}f(u) is proposed. n amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.

Original language | English |
---|---|

Article number | 101501 |

Journal | Journal of Mathematical Physics |

Volume | 54 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2013 Oct 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics