Dissipative solitons

Abstract

Abstract The present review summarizes experimental and theoretical work dealing with self-organized solitary localized structures (LSs) that are observed in spatially extended nonlinear dissipative systems otherwise exhibiting translational and rotational symmetry. Thereby we focus on those LSs that essentially behave like particles and that we call dissipative solitons (DSs). Such objects are also solutions of corresponding nonlinear evolution equations and it turns out that they are rather robust with respect to interaction with each other, with impurities, and with the boundary; alternatively they are generated or annihilated as a whole. By reviewing the experimental results it turns out that the richest variety of DS phenomena has been observed in electrical transport systems and optical devices. Nevertheless, DSs show up also in many other systems, among which nerve pulses in living beings are of uppermost importance in practice. In most of these systems DSs behave very similarly. The experimental results strongly suggest that phenomenon of DSs is universal. On the background of the experimental findings models for a theoretical understanding are discussed. It turns out that in a limited number of cases a straightforward quantitative description of DS patterns can be carried out. However, for the overwhelming number of systems only a qualitative approach has been successful so far. In the present review particular emphasis is laid on reaction-diffusion systems for which a kind of 'normal form' can be written down that defines a relatively large universality class comprising e.g. important electrical transport, chemical, and biological systems. For the other large class of DS carrying systems, namely optical devices, the variety of model equations is much larger and one is far away, even from a universal qualitative description. Because of this, and due to the existence of several extensive reviews on optical systems, their theoretical treatment has been mentioned only shortly. Finally, it is demonstrated that in terms of a singular perturbation approach the interaction of DSs and important aspects of their bifurcation behaviour, under certain conditions, can be described by rather simple equations. This is also true when deriving from the underlying field equations a set of ordinary differential equations containing the position coordinates of the individual DSs. Such equations represent a theoretical foundation of the experimentally observed particle-like behaviour of DSs. Though at present there is little real practical application of DSs and related patterns in an outlook we point out in which respects this might change in future. A systematic summary of a large amount of experimental and theoretical results on reaction-diffusion systems, being rather close to the subject of the present review, can also be found on the website http://www.uni-muenster.de/Physik.AP/Purwins/Research-Summary. Acknowledgements The authors and in particular H.-G. Purwins are deeply indebted to the wonderful cooperation with many of the former collaborators of the Münster group on 'Nonlinear Systems and Pattern Formation', in particular to E. Ammelt, Yu.A. Astrov, J. Berkemeier, M. Bode, I. Brauer (Müller), R. Dohmen, E.L. Gurevich, S. Gurevich, G. Klempt, A.W. Liehr, V.M. Marchenko, A.S. Moskalenko, F.-J. Niedernostheide, M. Or-Guil, L.M. Portsel, C. Radehaus, M.C. Röttger, C.P. Schenk, R. Schmeling, P. Schütz, L. Stollenwerk, J.C. Strümpel, H. Willebrand, A.L. Zanin, and S. Zuccaro. All authors enjoyed the fruitful cooperation with R. Friedrich and W. Lange of the University of Münster and P. Boeuf from the CNRS at Toulouse. Fruitful discussions with N.N. Akhmediev, M.S. Benilov, U. Ebert, A.M. Ignatov, Yu.P. Raizer, N.N. Rosanov, L.D. Tsendin, A.G. Vladimirov are gratefully acknowledged. Many thanks are addressed to S. Kottemer for the careful preparation of drawings. Finally general support by C. Fallnich is highly appreciated. Notes Notes 1. Throughout the present review we will only give parameters in the cases when they are not easily accessible from the literature. 2. Here and for all electrical transport systems we use the term 'dark filament' in another sense as it is done in optical systems that are discussed in Section 2.5. 3. So that no confusion can occur, in the Sections 2.6.1, 2.6.2, here we use f to denote the frequency.