相关论文: Completely integrable systems: a generalization

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields…

After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a…

The complete integrability of a class of dynamical systems with the potential v(q)=q^{-2}+c q^2 is proved.

We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be…

All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…

We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular…

Standard (Arnold-Liouville) integrable systems are intimately related to complex rotations. One can define a generalization of these, sharing many of their properties, where complex rotations are replaced by quaternionic ones. Actually this…

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides…

We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a…

A study of harmonic maps into Lie groups as a generalisation of the study of other well-known integrable systems, particularly the Toda and self-dual Chern Simons theories.

The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…

We study the integrability of the general two-dimensional Zakharov-Shabat systems, which appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called…

In this paper we present the theorem on Lie integrability by quadratures for time-independent Hamiltonian systems on symplectic and contact manifolds, and for time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds. We…

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.

The geometric theory of Lie systems will be used to establish integrability conditions for several systems of differential equations, in particular Riccati equations and Ermakov systems. Many different integrability criteria in the…

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on…

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and…

We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R4 and R6. Furthermore, we construct some integrable and…

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…