10,544 research outputs found
Linearization from complex Lie point transformations
Complex Lie point transformations are used to linearize a class of systems of
second order ordinary differential equations (ODEs) which have Lie algebras of
maximum dimension , with . We identify such a class by employing
complex structure on the manifold that defines the geometry of differential
equations. Furthermore we provide a geometrical construction of the procedure
adopted that provides an analogue in of the linearizability criteria
in .Comment: 17 Pages, to appear in Journal of Applied Mathematics. arXiv admin
note: substantial text overlap with arXiv:1104.383
Lie and Riccati Linearization of a Class of Liénard Type Equations
We construct a linearizing Riccati transformation by using an ansatz and a linearizing point transformation utilizing the Lie point symmetry generators for a three-parameter class of Liénard type nonlinear second-order ordinary differential equations. Since the class of equations also admits an eight-parameter Lie group of point transformations, we utilize the Lie-Tresse linearization theorem to obtain linearizing point transformations as well. The linearizing transformations are used to transform the underlying class of equations to linear third- and second-order ordinary differential equations, respectively. The general solution of this class of equations can then easily be obtained by integrating the linearized equations resulting from both of the linearization approaches. A comparison of the results deduced in this paper is made with the ones obtained by utilizing an approach of mapping the class of equations by a complex point transformation into the free particle equation. Moreover, we utilize the linearizing Riccati transformation to extend the underlying class of equations,
and the Lie-Tresse linearization theorem is also used to verify the conditions of linearizability of this new class of equations
Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations
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 implies the linearizability of systems of partial differential
equations corresponding to those complex ordinary differential equations. The
invertible complex transformations can be used to obtain invertible real
transformations that map a system of nonlinear partial differential equations
into a system of linear partial differential equation. Explicit invariant
criteria are given that provide procedures for writing down the solutions of
the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single
research paper "Linearizability criteria for systems of two second-order
differential equations by complex methods" which has been published in
Nonlinear Dynamics. Due to citations of both parts I and II these are not
replaced with the above published articl
Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis
Five equivalence classes had been found for systems of two second-order
ordinary differential equations, transformable to linear equations
(linearizable systems) by a change of variables. An "optimal (or simplest)
canonical form" of linear systems had been established to obtain the symmetry
structure, namely with 5, 6, 7, 8 and 15 dimensional Lie algebras. For those
systems that arise from a scalar complex second-order ordinary differential
equation, treated as a pair of real ordinary differential equations, a "reduced
optimal canonical form" is obtained. This form yields three of the five
equivalence classes of linearizable systems of two dimensions. We show that
there exist 6, 7 and 15-dimensional algebras for these systems and illustrate
our results with examples
Proper actions of Lie groups of dimension on -dimensional complex manifolds
In this paper we continue to study actions of high-dimensional Lie groups on
complex manifolds. We give a complete explicit description of all pairs
, where is a connected complex manifold of dimension ,
and is a connected Lie group of dimension acting effectively and
properly on by holomorphic transformations. This result complements a
classification obtained earlier by the first author for and a classical result due to W. Kaup for the maximal group dimension
.Comment: 60 page
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