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PACS.47.54.+r – Pattern selection

By Hidetsugu Sakaguchi and Helmut R. Br

Abstract

Abstract. — We show that stable standing wave localized solutions of square symmetry are possible for a quintic Swift-Hohenberg type equation with complex coefficients. We point out that these localized solutions, which are surrounded by a pattern-free state exist for a range of values for the external stress parameter subcritically. We discuss similarities to recent experimental observations of standing wave localized squares in granular materials. The study of stable localized states in pattern forming nonequilibrium systems has attracted considerable attention over the last few years, both experimentally [1–3] and theoretically (Refs. [4–10] and references therein). From a modeling point of view, two main stabilizing mechanisms for these stable localized states have emerged so far. One can be traced back to a feedback loop [4]: the frequency of the traveling waves depends on the shape of the pulse and conversely, the frequency affects the pulse shape, while the other one is based on a trapping mechanism due to the interaction between large and small length scales [10] in equations of the Swift-Hohenberg type. The Swift-Hohenberg type equations studied in reference [10] have a Lyapunov functional, and the localized states can be interpreted as a local minimum of the Lyapunov functional. For the first mechanism breathing localized solutions can appea

Year: 1997
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