This article is cited in 1 paper

Localized solutions of a piecewise linear model of the stationary Swift–Hohenberg equation on the line and on the plane

N. E. Kulagin^a, L. M. Lerman<sup>bc

a</sup> State University of Management, Moscow, Russia
^b Research Institute for Applied Mathematics and Cybernetics, Lobachevski Nizhnii Novgorod State University, Nizhnii Novgorod, Russia
^c Lobachevski Nizhni Novgorod State University, Faculty of Mechanics and Mathematics, Nizhni Novgorod, Russia

Abstract: 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.

UDC: 517.925.41+517.956.25

 English version: Journal of Mathematical Sciences, 2014, 202:5, 684–702

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