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

摘要: 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 stationary solutions of this considered, dispersion term $-\sigma u_{\xi\xi\xi}$ destroys spatial reversibility which plays an important role for studying localized patterns. In paper, we focus on its traveling wave directly apply dynamical approach to provide first rigorous proof existence periodic homoclinic bifurcating from origin without condition as parameters varied.