Geodesic
Fronts, traveling fronts, and their stability in the generalized Swift–Hohenberg equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 4, pp. 693-712

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The generalized Swift–Hohenberg equation with an additional quadratic term is studied. Time-stable localized stationary solutions of the pulse and front types are found. It is shown that stationary fronts give rise to traveling fronts, whose branches are also obtained. This study combines theoretical methods for dynamical systems (in particular, the theory of homo-and heteroclinic orbits) and numerical simulation. 
@article{ZVMMF_2008_48_4_a12,
     author = {N. E. Kulagin and L. M. Lerman and T. G. Shmakova},
     title = {Fronts, traveling fronts, and their stability in the generalized {Swift{\textendash}Hohenberg} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {48},
     number = {4},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_4_a12/}
}
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N. E. Kulagin; L. M. Lerman; T. G. Shmakova. Fronts, traveling fronts, and their stability in the generalized Swift–Hohenberg equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 4, pp. 693-712. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_4_a12/