Related papers: Generalized Lie-B\"acklund theorem for Lie class \…

In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via…

Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems. In cases when the B\"acklund transformation admits a symmetry…

An exterior differential calculus in the general framework of generalized Lie algebroids is presented. A theorem of Maurer-Cartan type is obtained. All results with details proofs are presented and a new point of view over exterior…

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…

The notion of an exterior differential system (on a manifold) has recently been extended to the setting of a Lie algebroid. Here, we further develop the theory and we present two versions of the Cartan-K\"ahler theorem in the case where the…

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…

We will prove that the generalized Lie algebroid is a distinguished example by Lie algebroid. The generality of it with respect to the Lie algebroid is similar with the generality of the pull-back vector bundle with respect to the vector…

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2…

The geometry of an admissible B\"acklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal…

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal…

We proved a KAM theorem on existence of invariant tori in generalized Hamiltonian systems without action-angle variables. It is a generalization of the result of de la Llave et al. [Llave, 2005] that deals with canonical Hamiltonian system.

General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of B\"acklund transformations for Toda-type systems. Commutativity of B\"acklund transformations is shown to be equivalent to…

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation were found. In particular, the Lie algebra of the multiplication…

We generalize the submodel of nonlinear CP^1 models. The generalized models include higher order derivatives. For the systems of higher order equations, we construct a B\"acklund-like transformation of solutions and an infinite number of…

We show that the well-known formula by Ackermann and Utkin can be generalized to the case of higher-order sliding modes. By interpreting the eigenvalue assignment of the sliding dynamics as a zero-placement problem, the generalization…

We establish an explicit form of the Backlund transformation for the most known integrable systems.

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove…

In 1984 Milnor had shown how to deduce the Lie-Palais theorem on integration of infinitesimal actions of finite-dimensional Lie algebras on compact manifolds from general theory of regular Lie groups modelled on locally convex spaces. We…

We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the…