Reduction of weakly nonlinear parabolic partial differential equations

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2 Citations (Scopus)

Abstract

It is known that the Swift-Hohenberg equation ∂u/∂t=-(∂x2+1)2+1)u+e{open}(u-u3) can be reduced to the Ginzburg-Landau equation (amplitude equation) ∂A/∂t=4∂x2A+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 languageEnglish
Article number101501
JournalJournal of Mathematical Physics
Volume54
Issue number10
DOIs
Publication statusPublished - 2013 Oct 1
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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