We define Bäcklund–Darboux transformations in Sato’s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, tau-functions and spectral algebras.
The paper is an exposition of basic known local and global results on Lagrangian foliations such as the Theorem of Darboux-Lie, Weinstein, Arnold-Liouville, a global characterization of cotangent bundles, higher order Maslov classes, etc.
We consider the billiard map in the hypercube of . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that is the order of magnitude of the complexity.
This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. In an abstract setting, we establish the existence of traveling waves for discrete and continuous-time monotone semiflows in homogeneous and periodic habitats. The results are then extended to monotone semiflows with weak compactness. We also apply the theory to four classes of evolution systems.
It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms ϕ and ψ is equal to the number of coincidence points of ϕ̂ and ψ̂ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally, we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.
We give an existence theorem for bounded solutions to a system of second order ODEs. Dynamical applications are considered.