EXISTENCE OF GENERALIZED HOMOCLINIC SOLUTIONS FOR A MODIFIED SWIFT-HOHENBERG EQUATION

摘要

In this paper, we investigate the modified steady Swift-Hohenberg equation ku_(xxxx) + 2ku_(xx) + αu~2_x-εu + u~3 = 0, (1) where k ＞ 0, α and ε are constants. We obtain a homoclinic solution about the dominant system which will be proved to deform a reversible homoclinic solution approaching to a periodic solution of the whole equation with the aid of the Fourier series expansion method, the fixed point theorem, the reversibility and adjusting the phase shift. And the homoclinic solution approaching to a periodic solution of the equation are called generalized homoclinic solution.