Generalized negative flows in hierarchies of integrable evolution equations

Generalized integrable hierarchies and Combescure symmetry transformations

Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure tr...

Finite Euler Hierarchies and Integrable Universal Equations

Recent work on Euler hierarchies of field theory Lagrangians iteratively constructed from their successive equations of motion is briefly reviewed. On the one hand, a certain triality structure is described, relating arbitrary field theories, classical topological field theories – whose classical solutions span topological classes of manifolds – and reparametri-sation invariant theories – gener...

Integrable hierarchies, dispersionless limit and string equations

The notion of string equations was discovered in the end of the eighties, and has been studied in the language of integrable hierarchies. String equations in the KP hierarchy are nowadays relatively well understood. Meanwhile, systematic studies of string equations in the Toda hierarchy started rather recently. This article presents the state of art of these issues from the author’s point of vi...

Integrable Evolution Equations in Time-Dependent Domains

Quasi-classical ∂̄-method: Generating equations for dispersionless integrable hierarchies

The quasi-classical ∂̄-dressing method is used to derive compact generating equations for dispersionless hierarchies. Dispersionless Kadomtsev-Petviashvili (KP) and two-dimensional Toda lattice (2DTL) hierarchies are considered as illustrative examples.