Lie Algebra Of Infinitesimal Symmetries Research Articles

Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

In this paper, the (2+1)-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of (1+1)-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

Null-forms of conic systems in R3 are determined by their symmetries

We address the problem of characterisation of null-forms of conic 3-dimensional systems, that is, control-affine systems whose field of admissible velocities forms a conic (without parameters) in the tangent space. Those systems have been previously identified as the simplest control systems under a conic nonholonomic constraint or as systems of zero curvature. In this work, we propose a direct characterisation of null-forms of conic systems among all control-affine systems by studying the Lie algebra of infinitesimal symmetries. Namely, we show that the Lie algebra of infinitesimal symmetries characterises uniquely null-forms of conic systems.

Regular F-manifolds: initial conditions and Frobenius metrics

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M. We obtain that any regular F-manifold admits a preferred system of local coordinates and we find conditions, in these coordinates, for a metric to be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular F-manifolds. We show that any regular F-manifold is locally isomorphic to the parameter space of a Malgrange universal connection. We prove an initial condition theorem for Frobenius metrics on regular F-manifolds.

Application of Lie groups to compressible model of two-phase flows

This paper presents some exact solutions for the drift-flux model of two-phase flows using Lie group analysis. The analysis involves an isentropic no-slip conservation of mass for each phase and the conservation of momentum for the mixture. The present analysis employs a complete Lie algebra of infinitesimal symmetries. Subsequent to these theoretical analysis a symmetry group is established. The symmetry generators are used for constructing similarity variables which reduce the model equations to a system of ordinary differential equations (ODEs). In particular, a general framework is discussed for solving the model equations analytically. As a consequence of this, new classes of exact group-invariant solutions are developed. This provides new insights into the fundamental properties of weak discontinuities and helps one to understand better on existence of solutions.

Differential Galois Theory and Lie Symmetries

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.

A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

In this paper,we applied the Painlevé property test on Krook‐Wu model of the nonlinear Boltzmann equation (p = 1). As a result, by using Bäcklund transformation, we obtained three solutions two of them were known earlier, while the third one is new and more general, we have also two reductions one of them is Abel's equation. Also, Lie‐group method is applied to the (p + 1)th Boltzmann equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub‐algebraic of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations for (p + 1)th Boltzmann equation; we obtained two general solutions for (p + 1)th Boltzmann equation and new solutions for Krook‐Wu model of Boltzmann equation (p = 1). Copyright © 2014 John Wiley & Sons, Ltd.

Chern–Moser operators and polynomial models in CR geometry

We consider the fundamental invariant of a real hypersurface in CN – its holomorphic symmetry group – and analyze its structure at a point of degenerate Levi form. Generalizing the Chern–Moser operator to hypersurfaces of finite multitype, we compute the Lie algebra of infinitesimal symmetries of the model and provide explicit description for each graded component. Compared with a hyperquadric, it may contain additional components consisting of nonlinear vector fields defined in terms of complex tangential variables.As a consequence, we obtain exact results on jet determination for hypersurfaces with such models. The results generalize directly the fundamental result of Chern and Moser from quadratic models to polynomials of higher degree.

Similarity Reductions and New Exact Solutions for B-Family Equations

In this paper, Lie-Group method is applied to the b-family equations which includes two important nonlinear partial differential equations Camassa--Holm (CH) equation and the Degasperis--Procesi (DP) equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub-algebras of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations (ODEs) for the b-family equations. The generalized He's Exp-Function method is used to drive exact solutions for the reduced nonlinear ODEs, some figures are given to show the properties of the solutions.

An Approximate Analytical Solution of Imploding Strong Shocks in a Non-Ideal Gas through Lie Group Analysis

The method of Lie group transformation is used to obtain an approximate analytical solution to the system of first-order quasilinear partial differential equations that govern a one-dimensional unsteady planer, cylindrically symmetric and spherically symmetric motion in a non-ideal gas, involving strong shock waves. Invariance groups admitted by the governing system of partial differential equations, which are indeed continuous group of transformations under which the system of partial differential equations remains invariant, are determined, and the complete Lie algebra of infinitesimal symmetries is established. The infinitesimal generators are used to construct the similarity variables. These similarity variables are used to reduce the governing system of partial differential equations into a system of ordinary differential equations.

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

Similarity solutions for three dimensional Euler equations using Lie group analysis

A similarity analysis of the nonlinear three dimensional unsteady Euler equations of gas dynamics is presented using Lie group of transformations with commuting infinitesimal operators. Symmetry groups admitted by the governing system of partial differential equations (PDEs) are obtained, and the complete Lie algebra of infinitesimal symmetries is established. The symmetry generators are used for constructing similarity variables which lead to a reduced system of equations with one independent variable less at each step and eventually to a system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly.

Variational character of the Cauchy-Riemann equations

Cauchy-Riemann equations for smooth maps ϕ: ℂ → ℂk are proved to be of variational type if and only if k is even. This fact is seen to be related to a complex differential form of degree (3, 0) on ℂ × ℂk, which exists only for an even k. The Lie algebra of infinitesimal symmetries of the Hamiltonian structure associated with Cauchy-Riemann equations is also determined.

Reduction of Hamiltonian systems with symmetry

This paper presents an alternative phase space reduction process for Hamiltonian systems with symmetry using a Lie algebra of infinitesimal symmetries instead of a Lie group of symmetries. This approach avoids the use of the various conditions on the action of the group or of the isotropy subgroup necessary for the usual construction of a reduced phase space (quotient) with a globally defined symplectic manifold structure. The existence of such a reduced phase space is proved without recourse to the existence of an equivariant moment map. Instead it is assumed that the orbits of the action associated to the Lie algebra can be described as the level sets of some analytic map, an assumption that holds in many of the known examples. The relation between the two reduction processes is discussed; in particular, some cases where the Marsden-Weinstein process leads to a quotient space which is not a smooth manifold are shown to be amenable to the method now proposed. The natural control theoretic interpretation of the techniques used is also discussed.

Symmetry groups and similarity solutions of MHD equations

A similarity analysis of the nonlinear nonstationary real MHD equations is presented. For several cases of real MHD the complete Lie algebra of infinitesimal symmetries is calculated. Using subalgebras of this Lie algebra, reduced systems for similarity solutions can be obtained. Some similarity solutions of the ideal MHD equations are discussed.

A note on the symmetry group and perturbation algebra of a parabolic partial differential equation

In this paper, the symmetry group of the parabolic partial differential equation ∂tf=Af is focused on (A is an analytic second-order elliptic linear differential operator on Rn). Following Olver [Applications of Lie Groups to Differential Equations (Springer, New York, 1986)], Ovsjannikov [Group Analysis of Differential Equations (Academic, New York, 1982)], and Rosencrans [J. Math. Anal. Appl. 56, 317 (1976); 61, 537 (1977)], the Lie algebra of infinitesimal symmetries is studied. It will be proven that it is finite dimensional (putting apart the trivial symmetries due to the linearity of the partial differential equation) by relating it to the Lie algebra of infinitesimal homothetic transformations of a Riemannian manifold (Rn,a), thus generalizing results of Ovsjannikov on elliptic equations and making use of the same Riemannian geometric techniques. Thanks to these techniques, the perturbation algebra of A, introduced by Rosencrans, is also shown to be related to geometric characteristics of (Rn,a).

Similarity solutions for the two-dimensional nonstationary ideal MHD equations

A similarity analysis of the non-linear two-dimensional non-stationary ideal MHD equations is presented. In the case of a magnetic field perpendicular to the isentropic motion of the plasma, the authors establish the complete Lie algebra of infinitesimal symmetries. The laws of conservation are mentioned. The similarity method for partial differential equations as a procedure for reducing the number of independent variables is applied repeatedly. Finally, they obtain systems of ordinary differential equations for similarity solutions of the MHD equations considered.

The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations

By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.

The Hamilton Cartan formalism for rth-order Lagrangians and the integrability of the KdV and modified KdV equations

The Hamilton Cartan formalism for rth order Lagrangians is presented in a form suited to dealing with higher-order conserved currents. Noether's Theorem and its converse are stated and Poisson brackets are defined for conserved charges. An isomorphism between the Lie algebra of conserved currents and a Lie algebra of infinitesimal symmetries of the Cartan form is established. This isomorphism, together with the commutativity of the Backlund transformations for the KdV and modified KdV equations, allows a simple geometric proof that the infinite collections of conserved charges for these equations are in involution with respect to the Poisson bracket determined by their Lagrangians. Thus, the formal complete integrability of these equations appears as a consequence of the properties of their Backlund transformations.