A Hamiltonian perturbation theory for coherent structures illustrated to wave problems

In wave problems with a Hamiltonian structure and some symmetry property, the relative equilibria are the physical coherent structures. They appear as families of states, parameterized by physical observables connected to the symmetry. A new abstract (and general) result about the relation between the kernel of the linearized Hamiltonian flow and its adjoint is the basis of a Hamiltonian perturbation theory. In this theory, the effect of a perturbation of the system is decomposed in an effect within the class of coherent structures and a transversal deviation. It is shown that, under mild conditions, the transversal deviation remains small (of the order of the pertnrbation) while the main effect is a (quasi-static) succession of various coherent structures. This means that, in a good approximation, the coherent structures can be used as base functions, and that the evolution of the parameters, as a set of collective coordinates, specifies the dynamics. Two examples illustrate the general result: dissipative perturbations of the Korteweg-de Vries equation, and a spatially inhomogeneous Sine-Gordon equation.