Journal ArticleDOI
Reduction of Time-Dependent Systems Admitting a Superposition Principle
TLDR
In this article, the problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective, and it is shown how it is possible to reduce the general solution of such a differential equation system defined by a Lie group to a pair of simpler problems, one in a subgroup H and another on a homogeneous space.Abstract:
The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.read more
Citations
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Journal ArticleDOI
Superposition rules, lie theorem, and partial differential equations
TL;DR: In this paper, a rigorous geometric proof of the Lie theorem on nonlinear superposition rules for solutions of nonautonomous ordinary differential equations is given filling in all the gaps present in the existing literature.
Journal ArticleDOI
Group Theoretical Approach to the Intertwined Hamiltonians
TL;DR: In this paper, the Backlund formula for the Schrodinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper, and a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator is given.
Journal ArticleDOI
Lie systems: theory, generalisations, and applications
José F. Cariñena,J. de Lucas +1 more
TL;DR: 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 mapping as mentioned in this paper.
Journal ArticleDOI
Riccati equation, Factorization Method and Shape Invariance
José F. Cariñena,Arturo Ramos +1 more
TL;DR: In this paper, the basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed, and the relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail.
Journal ArticleDOI
Shape Invariant potentials depending on n parameters transformed by translation
José F. Cariñena,Arturo Ramos +1 more
TL;DR: In this article, the problem of finding a class of shape-invariant potentials which depend on n parameters transformed by translation is solved using the properties of the Riccati equation.
References
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Book
Introduction to Lie Algebras and Representation Theory
TL;DR: In this paper, Semisimple Lie Algebras and root systems are used for representation theory, isomorphism and conjugacy theorem, and existence theorem for representation.
Book
Symplectic geometry and analytical mechanics
TL;DR: In this article, the authors introduce the notion of contact manifolds as a way to represent the local structure of a Poisson manifold and a Lie group on its cotangent bundle.
Journal ArticleDOI
Lie Algebraic Solution of Linear Differential Equations
James Wei,Edward Norman +1 more
TL;DR: In this paper, a finite product of exponential operators for the linear differential equation dU/dt = h(t)U can be represented as a Lie algebra and the number of terms in this expansion is equal to the dimension of the Lie algebra generated by H(t).
Journal ArticleDOI
On global representations of the solutions of linear differential equations as a product of exponentials
James Wei,E. Norman +1 more
TL;DR: In this article, it was shown that if U is a solution of (1), it can be represented in the form U(t) exp(Ei= gi(t), where the ai is scalar function of t, and the operators Xi are independent of t.
MonographDOI
Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen / Sophus Lie ; bearbeitet und herausgegeben von Georg Scheffers.
Sophus Lie,Georg Scheffers +1 more
TL;DR: In this article, the authors propose a model for projective transformation of a projective Gruppe in der Ebene, which is based on the lineare homogene Gruppen.
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Superposition rules, lie theorem, and partial differential equations
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