Theory More theory Applications Acknowledgment ReferencesIOP PUBLISHING NONLINEARITYNonlinearity 21 (2008) T45–T60 doi:10.1088/0951-7715/21/4/T02OPEN PROBLEMSpatially localized structures in dissipative systems:open problemsE KnoblochDepartment of Physics, University of California, Berkeley CA 94720, USAReceived 19 December 2007, in final form 1 February 2008Published 28 February 2008Online atstacks.iop.org/Non/21/T45AbstractStationary spatially localized structures, sometimes called dissipative solitons,arise in many interesting and important applications, including buckling ofslender structures under compression, nonlinear optics, fluid flow, surfacecatalysis, neurobiology and many more. The recent resurgence in interest inthese structures has led to significant advances in our understanding of theorigin and properties of these states, and these in turn suggest new questions,both general and system-specific. This paper surveys these results focusingon open problems, both mathematical and computational, as well as on newapplications.Mathematics Subject Classification: 35B32, 35B60, 35G30(Some figures in this article are in colour only in the electronic version)TheoryThe study of stationary spatially localized states in driven dissipative systems has a long history.This paper is not intended as a review of this vast subject [13]. Instead it briefly summarizes,in nontechnical language, some of the key recent developments in our understanding of thesefascinating and often surprising structures, followed by a discussion of issues that the authorbelieves will be resolved in the near future, as well as some more challenging questions thatmay be around for a longer time. Propagating solitary waves or pulses are not considered.The paper focuses on continuum systems while acknowledging the existence of parallelphenomena in discrete dynamical systems, typically lattice systems. Throughout the term‘spatially localized structure’ will be used to refer to the presence of one state ‘embedded’ in abackground consisting of a different state. Examples include a finite amplitude homogeneousstate (hereafter, a nontrivial state) embedded in a background of the zero or trivial state, alocalized spatial or spatio-temporal oscillation in a background trivial state, etc. Figure 1shows a recent example from an experiment on a ferrofluid in an applied vertical dc magneticfield [57]. The figure shows a number of stable steady spatially localized ’solitons’ created bysuitable perturbation in the bistable region where both the flat surface and a periodic hexagonalarray of peaks coexist. These individual localized states may form different types of stablebound states resembling ‘molecules’.0951-7715/08/040045+16$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK T45T46 Open Problem(a) (b)Figure 1. (a) The surface energy Esof a ferrofluid layer for increasing (open squares, flat surface)and decreasing (open circles, periodic hexagonal array of peaks) magnetic induction B. The fullcircles indicate the energy of different numbers of spatially localized ‘solitons’, shown in (b), all atnominally identical parameter values (B = 8.91 mT) within the region of bistability between theflat surface and the periodic hexagonal pattern. The peaks along the boundary in (b) are an edgeeffect and are excluded from Es. Reprinted figure with permission from [57]. Copyright (2005)by the American Physical Society.The theory is simplest for reversible systems on unbounded domains in one spatialdimension, that is, translation-invariant systems that are also equivariant under spatialreflectionsR : x →−x. In this case the traditional picture associates time-independentspatially localized structures with homoclinic orbits on the real line. This formulation isusually referred to as spatial dynamics. Pomeau [53] pointed out that looking at localizedstates as approximate heteroclinic cycles, i.e. as connections from the background to theincluded state and back again, is more fruitful. This is particularly so for variational systems,i.e. systems that evolve at all times towards states of lower energy. In this picture one focusesattention on regions in parameter space with bistability between the two competing states, andin particular, on parameter values near the so-called Maxwell point where the two competingstates have the same energy. The heteroclinic cycle then becomes a bound state of two ‘fronts’,the structures that separate the background from the inclusion. Pomeau observed a crucialdifference between the case in which the two competing states are both spatially homogeneousand the case in which the localized state is structured. The former case is analogous to astandard first order phase transition, with the location of the phase transition determined bythe Maxwell construction. At this parameter value a front connecting the two states will bestationary: the two competing states coexist. Moreover, an infinite multiplicity of spatiallylocalized states of different lengths can be constructed by combining back-to-back fronts.However, this is not possible at other parameter values where one or other state is energeticallypreferred, and the fronts move to minimize energy by ‘expelling’ the higher energy state. Inthe case of bistability with a structured state the situation is quite different, however. Herethe fronts at either end can ‘lock’ or ‘pin’ to the structure within the localized state, and aninfinite number of localized states exists not only at the Maxwell point but in an entire intervalof parameter values around the Maxwell point. The resulting interval is called the pinningregion, or, for reasons explained below, the snaking region [53,67].The structure of solutions within the pinning region has been elucidated only recently,and the results are summarized in a recent paper [11] in a special issue of Chaos devotedto spatially localized structures. In variational systems on the real line, such as the bistableOpen Problem T47r||u||L2u(x) u(x)(a)(b)(c)–0.4 001r1r2uperu=0rM(b)(c)–30 0 3001.8–30 0 3001.8Figure 2. (a) Bifurcation diagram showing the snakes-and-ladders structure associated withhomoclinic snaking in the Swift–Hohenberg equation (1). The sidepanels show sample solutionprofiles at the locations (b,c) indicated in the left panel, one from each snaking branch. Heavylines indicate stable solutions. The pinning (snaking) region