The dynamics  of small amplitude solutions of the Swift-Hohenberg equation on a large interval

Ling-Jun Wang

doi:10.3934/cpaa.2012.11.1129

Article Contents

2012, Volume 11, Issue 3: 1129-1156. Doi: 10.3934/cpaa.2012.11.1129

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The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval

Ling-Jun Wang^1,

1.
College of Science, Wuhan University of Science and Technology, Wuhan 430065

Received: November 2010

Revised: March 2011

Published: May 2012

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Abstract

We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval $(-l,l)$ with Dirichlet-Neumann boundary conditions, where $l>0$ is large and lies outside of some small neighborhoods of the points $n\pi$ and $(n+1/2)\pi,n \in N$. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter $O(l^{-2})$-close to its critical values when $l$ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on $R$ and admitting $2\pi$-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the $2\pi$ spatially periodic case.

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Mathematics Subject Classification: Primary: 37L10; Secondary: 35B32.

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