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On the dynamics of the piecewise-linear model of the Swift–Hohenberg equation

L. M. Lerman^ab, N. A. Slinyakova<sup>b

a</sup> Research Institute for Applied Mathematics and Cybernetics, N. I. Lobachevski State University of Nizhnii Novgorod
^b N. I. Lobachevski State University of Nizhni Novgorod, Faculty of Mechanics and Mathematics

Abstract: In this paper we study the piecewise-linear model of the stationary Swift–Hohenberg equation well known in mathematical physics, which provides explicit front type solutions. Due to the reversibility relative to two involutions of the corresponding Hamiltonian system, this involves the existence of a heteroclinic contour connecting two saddle-foci. Using methods of symbolic dynamics, we give a description of all solutions lying in the neighborhood of the contour at the level of the Hamiltonian containing the contour.

Keywords: Swift–Hohenberg equation, fronts, heteroclinic contour, Hamiltonian system, saddle focus, symbolic dynamics.

UDC: 517.938

MSC: 34C37, 34C28, 70K44, 70K55

Received: 06.10.2009