The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on and characterize Lie remarkable equations admitted by the …
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We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the…
Study 2D viscoelastic equations using Lie group theory.
problem Symmetry classification and reduction of 2D viscoelastic equations.
method Investigation through Lie group theory, including algebra of symmetries and optimal subalgebras.
result Classification of reductions of similarities related to Lie subalgebras.
Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.
problem Classifying Lie symmetry algebras for 2D quasilinear equations.
method Classification based on abelian Lie symmetry algebras of dimension and rank.
result Equations with specific symmetry algebras are linearizable.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.
problem Contact mechanical systems on Lie groups with symmetries.
method Reduction process using Lie group actions and symmetries.
result Euler-Poincaré-Herglotz equations on the reduced phase space.
This work analyses types of group actions on families of -dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to…
In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Study evolution equations on Lie groupoids using Fourier integral operators.
problem Solving evolution equations on Lie groupoids.
method Developed calculus of Fourier integral operators and used them to study the fundamental solution of the evolution equation.
result Developed a method to find the fundamental solution of the evolution equation on Lie groupoids.
Geometric framework for dissipative systems on Lie algebroids.
problem Formulating geometric framework for dissipative systems.
method Herglotz-type variational principle on Lie algebroids.
result Recover classical equations as special cases.
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
We discuss the correspondence between Nahm's equations, the Basu-Harvey-Terashima equations, and Lie superalgebras.
A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Li…
Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.
problem Investigate the Lax equation in infinite-dimensional Lie algebras and Lie groups.
method Derived integral expansions and generalized Baker-Campbell-Hausdorff formula for Lie groups.
result Explicit representation of product integral in terms of exponential map.
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
Quasi-Lie schemes for PDEsmath.AP
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining -dependent superposition rules and integrability conditions are analysed.…
The study analyzes stochastic Lie systems and their applications in various models.
problem Analyzing stochastic differential equations on manifolds.
method Coalgebra method for Hamiltonian stochastic Lie systems.
result New examples of stochastic Lie systems and Hamiltonian stochastic Lie systems are analyzed.
Constructs Lie-Rinehart algebra for Einstein's equations.
problem Initial value problem constraints for Einstein's equations.
method BV-BFV approach to boundary value problems, constructing -algebroid.
result Lie-Rinehart algebra comes from slight generalization of Lie algebroid.
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagr…
The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.
In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The red…
New Lie systems derived from Goursat distributions with applications to differential equations.
problem Analyzing Lie systems associated with Goursat distributions and their applications.
method Analyzing bracket-generating distributions and their relation to Lie systems, focusing on reductions and reconstructions.
result Lie systems associated with Goursat distributions can be reduced and solutions reconstructed from reduced systems.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the Poisson's equation, which has a subalgebra isomorphic to the dimensional special Euclidean group or group of rigid motions of . Looking the adjoint representation of ${\rm SE}(3)…
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
problem Quantum cohomology of minuscule flag manifolds.
method Combining Lie-theoretic treatments of tt*-Toda equations and quantum cohomology.
result Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
problem Understanding algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
method Algebraic characterization and differential Galois theory of rational connections.
result Equivalence of algebraic integrability to the triviality of the differential Galois group and demonstration of minimality under certain conditions.
We classify up to automorphisms all left-invariant non-Einstein solutions to the Einstein--Maxwell equations on 4-dimensional Lie algebras.
In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.
Investigates solving curvature equations on special Lie groups.
problem Solving curvature equations on non-compact simple Lie groups.
method Analyzes left-invariant naturally reductive metrics and conditions for solvability.
result Obtains conditions for the solvability of curvature equations.
Integrates Lax pair equations for a specific Lie algebra.
problem Low-dimensional Lie algebras of infinitesimal character .
method Shows complete integrability of Lax pair equations.
result Proves complete integrability for certain Lie algebras.
Lie Symmetry Classification and Numerical Analysis of KdV Equation with Power-law Nonlinearitymath.DG
In this paper, a complete Lie symmetry analysis of the damped wave equation with time-dependent coefficients is investigated. Then the invariant solutions and the exact solutions generated from the symmetries are presented. Moreover, a Lie algebraic classifications and the optimal system are discussed. Finally, using C…
Stochastic poisson equations associated to lie algebroids and some refinements of a principal bundlemath.DG
The aim of this paper is to present the stochastic Poisson equations associated to Lie algebroids. The stochastic Poisson equations associated to a refinement of a concrete principal bundle are determined.
Study exact Lie bialgebras from flat Lie groups, classifying them.
problem Classifying Lie bialgebras from flat Lie groups.
method Analyzing noncommutative deformations of Poisson-Lie groups with a metric, solving the classical Yang-Baxter equation.
result Complete classification of exact metaflat Lie bialgebras.
The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive…
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…
Paper corrects and expands stochastic Lie systems theory.
problem Stochastic Lie systems and their properties.
method Corrected stochastic Lie theorem, introduced new stochastic Lie systems.
result Stochastic Lie systems can differ significantly between Stratonovich and Itô approaches.
Study of symplectic groupoids from tt*-Toda equations.
problem Geometry of meromorphic connections with irregular singularities.
method Holomorphic symplectic groupoid structure over Steinberg cross section.
result Proves the space of tt*-Toda connections is a symplectic Lie groupoid.
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivative…
The -symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the -symplectic structures to investigate a type of systems of first-order ordinary differential equations, the -symplectic Lie systems. In part…
We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces …
The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy …