Deformation of coherent structures

In this review we investigate the mathematical description of the distortion of clearly recognisable structures in phenomenological physics. The coherent structures we will explicitly deal with are surface waves on a layer of fluid, kink transitions in magnetic material, plane vortices, swirling flows in cylindrical pipes and periodic patterns in pattern formation equations. The deformation of such structures will be studied for perturbations of different kinds. Problems with dissipation as a perturbation include the decay of surface waves under the influence of uniform damping and viscosity, and the viscous decay of vortices along branches that connect to a Leith vortex. Inhomogeneity as a perturbative effect will be studied for waves above slowly varying topography, for the particle description of kinks in inhomogeneous magnetic materials and for swirling flows in slowly expanding pipes. Finally, slow variations in pattern formation equations will result in phase-diffusion or amplitude equations. The aim of this review is to show that the description of many perturbed evolutions can be understood from a unified mathematical point of view. Even in all cases where the perturbation, although assumed to be ‘small’, leads to distortions that are large on the temporal and spatial scales that we are interested in, a valid approximation will be found as a quasi-static or quasi-homogeneous succession of unperturbed states (an adiabatic evolution). In this review the main emphasis will be on conservative systems with a Hamiltonian (or, more general, Poisson) structure. We show that in many cases families of coherent structures can be characterised explicitly as constrained critical points of the energy. The parameters are the values of additional first integrals that are used as constraints, and the action of the symmetry flows corresponding to those integrals that are not Casimir functionals. Using the coherent structures as kinds of base function, and the parameters identifying them as evolutionary quantities, successive evolutions are a nonlinear variant of Fourier’s method. However, the smooth dependence on the parameters that characterise exact solutions leads to degeneracy of the linearised equation that governs the error of any approximation consisting of a succession of coherent structures. The asymptotically correct dynamics for the parameters is then found from solvability conditions for this equation. To be applicable in practice, the kernel of the adjoint linear equation should then be found † This research has been supported by the Netherlands Organization for Scientific Research, NWO, by contract 620–61–249. ‡ Part of the research is sponsored by the Commission of the European Communities, Directorate General XII-B, Joint Research Project CI1*-CT93-0018 between the Department of Mathematics, Institut Teknologi Bandung, Indonesia, and the Faculty of Applied Mathematics, University of Twente, The Netherlands. 0034-4885/96/040511+90$59.50 c © 1996 IOP Publishing Ltd 511 512 E R Fledderus and E van Groesen explicitly. Although a constructive algorithm does not seem to exist in general, it is shown that for the large class of Poisson systems there is a simple relation with the kernel of the linearised operator that enables the construction. The presentation of the mathematically intricate method is kept as transparent as possible in all places, but in particular in a separate introductory section where the main ideas and difficulties are explained. This review was received in October 1995 Deformation of coherent structures 513