scholarly journals Lie systems and integrability conditions of differential equations and some of its applications

2008 ◽  
Author(s):  
J. F. Cariñena ◽  
J. de Lucas
Keyword(s):  
Differential Equations ◽  
Integrability Conditions ◽  
Lie Systems

scholarly journals QUANTUM LIE SYSTEMS AND INTEGRABILITY CONDITIONS

2009 ◽  
Vol 06 (08) ◽  
pp. 1235-1252 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
JAVIER DE LUCAS
Keyword(s):  
Quantum Mechanics ◽  
Differential Equations ◽  
Geometric Theory ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Integrable Quantum ◽  
Systems Of Differential Equations ◽  
Integrability Conditions ◽  
Mechanical Models ◽  
Lie Systems

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 analyzed from a geometric perspective. In this paper we use both developments to obtain a geometric theory of integrability in Quantum Mechanics and we use it to provide a series of non-trivial integrable quantum mechanical models and to recover some known results from our unifying point of view.

Quasi-Lie schemes for PDEs

2019 ◽  
Vol 16 (07) ◽  
pp. 1950096 ◽  
Author(s):  
J. F. Cariñena ◽  
J. Grabowski ◽  
J. de Lucas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformations ◽  
Partial Differential ◽  
First Order ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
Lie Systems ◽  
Abel Differential Equations

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.

Differential equations of smooth loops related to some space-time models: Integrability conditions and geometry

2018 ◽  
Vol 27 (07) ◽  
pp. 1841004
Author(s):  
L. Sbitneva
Keyword(s):  
Differential Equations ◽  
Homogeneous Spaces ◽  
Space Time ◽  
Lie Triple System ◽  
Lie Group Theory ◽  
Moufang Loops ◽  
Integrability Conditions ◽  
Curvature And Torsion ◽  
The Right ◽  
Bol Loops

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory. A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops. We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration. The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

THE CANONICAL PATH INTEGRAL QUANTIZATION OF CHIRAL SCHWINGER MODEL

2003 ◽  
Vol 18 (17) ◽  
pp. 1187-1196 ◽  
Author(s):  
S. I. MUSLIH
Keyword(s):  
Differential Equations ◽  
Path Integral ◽  
Integral Method ◽  
Equation Of Motion ◽  
Action Function ◽  
Schwinger Model ◽  
Integrability Conditions ◽  
Total Differential ◽  
Jacobi Formalism ◽  
Path Integral Method

We quantize the chiral Schwinger model by using the Hamilton–Jacobi formalism. We show that one can obtain the integrable set of equation of motion and the action function by using the integrability conditions of total differential equations and without any need to introduce unphysical auxiliary fields. The path integral for this model is obtained by using the canonical path integral method.

scholarly journals APPLICATIONS OF LIE SYSTEMS IN DISSIPATIVE MILNE–PINNEY EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 683-699 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
JAVIER DE LUCAS
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Geometric Approach ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Particular Solutions ◽  
Systems Of Differential Equations ◽  
Lie Systems ◽  
Linear Differential

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne–Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.

scholarly journals A new application of k-symplectic Lie systems

2015 ◽  
Vol 12 (07) ◽  
pp. 1550071 ◽  
Author(s):  
Javier de Lucas ◽  
Mariusz Tobolski ◽  
Silvia Vilariño
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Classical Field ◽  
Diffusion Equations ◽  
Field Theories ◽  
Symplectic Structures ◽  
First Order ◽  
Classical Field Theories ◽  
Lie Systems ◽  
Geometric Study

The k-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the k-symplectic structures to investigate a certain type of systems of first-order ordinary differential equations, the k-symplectic Lie systems. In particular, we analyze the properties, e.g., the superposition rules, of a new example of k-symplectic Lie system which occurs in the analysis of diffusion equations.

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