Geodesic
Localized solutions of a~piecewise linear model of the stationary Swift--Hohenberg equation on the line and on the plane
Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 3, Tome 47 (2013), pp. 60-77

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In this paper we study a simplified model of the stationary Swift–Hohenberg equation, where the cubic nonlinearity is replaced by a piecewise linear function with similar properties. The main goal is to prove the existence of so-called localized solutions of this equation, i.e., solutions decaying to a homogeneous zero state with unbounded increase of the space variable. The following two cases of the space variable are considered: one-dimensional (on the whole line) and two-dimensional; in the latter case, radially symmetric solutions are studied. The existence of such solutions and increase of their number with change in the equation parameters are shown. 
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     author = {N. E. Kulagin and L. M. Lerman},
     title = {Localized solutions of a~piecewise linear model of the stationary {Swift--Hohenberg} equation on the line and on the plane},
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N. E. Kulagin; L. M. Lerman. Localized solutions of a~piecewise linear model of the stationary Swift--Hohenberg equation on the line and on the plane. Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 3, Tome 47 (2013), pp. 60-77. http://geodesic.mathdoc.fr/item/CMFD_2013_47_a4/