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Localized Solutions of a Piecewise Linear Model of the Stationary Swift–Hohenberg Equation on the Line and on the Plane

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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.

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Authors and Affiliations

  1. State University of Management, Moscow, Russia

    N. E. Kulagin

  2. Research Institute for Applied Mathematics and Cybernetics, N. I. Lobachevski State University of Nizhnii Novgorod, Nizhnii Novgorod, Russia

    L. M. Lerman

  3. N. I. Lobachevski State University of Nizhni Novgorod, Faculty of Mechanics and Mathematics, Nizhnii Novgorod, Russia

    L. M. Lerman

Authors
  1. N. E. Kulagin
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  2. L. M. Lerman
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Correspondence to N. E. Kulagin.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 47, Proceedings of the Sixth International Conference on Differential and Functional Differential Equations and International Workshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 14–21 August, 2011). Part 3, 2013.

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Kulagin, N.E., Lerman, L.M. Localized Solutions of a Piecewise Linear Model of the Stationary Swift–Hohenberg Equation on the Line and on the Plane. J Math Sci 202, 684–702 (2014). https://doi.org/10.1007/s10958-014-2072-z

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