American Institute of Mathematical Sciences

December  2016, 9(6): 1647-1662. doi: 10.3934/dcdss.2016068

Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion

 1 Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048

Received  July 2015 Revised  August 2016 Published  November 2016

We investigate the 1D Swift-Hohenberg equation with dispersion $$u_t+2u_{\xi\xi}-\sigma u_{\xi\xi\xi}+u_{\xi\xi\xi\xi}=\alpha u+\beta u^2-\gamma u^3,$$ where $\sigma, \alpha, \beta$ and $\gamma$ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term $-\sigma u_{\xi\xi\xi}$ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the first rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.
Citation: Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
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