Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Journal of Differential Equations

2010

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We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.

Section: 1)mentioning

confidence: 96%

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

Bozhkov

Journal of Differential Equations

2010

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“…(2) on flat manifolds, with some functions f (u) are performed in [11]. In [4,3,5,10] the Lie point symmetries, the Noether symmetries and the conservation laws of the Kohn-Laplace equations were studied. In [6] the symmetry analysis of Eq.…”

Section: Introductionmentioning

confidence: 99%

On the paper “Symmetry analysis of wave equation on sphere” by H. Azad and M.T. Mustafa

Journal of Mathematical Analysis and Applications

2010

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Keywords: Lie point symmetry Noether symmetry Conservation laws Wave equations on the sphere Scalar curvature Using the scalar curvature of the product manifold S 2 × R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M.T. Mustafa [H. Azad, M.T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl. 333 (2007) 1180-1188] to nonlinear Klein-Gordon equations on the two-dimensional sphere.

“…For instance, in [Bozhkov and Freire (2008a)], a complete group classification of the semilinear Kohn-Laplace equations was carried out. For that considered family of equations, when the nonlinear term is a power nonlinearity with the critical exponent, all Lie point symmetries are Noether symmetries and consequently, from Noether theorem, it is possible to find conservation laws for them, see [Bozhkov and Freire (2011)] and [Bozhkov and Freire (2008b)].…”

Section: Main Results and Preliminary Discussionmentioning

confidence: 99%

Symmetry analysis of a class of autonomous even-order ordinary differential equations

Silva

IMA Journal of Applied Mathematics

2015

A class of autonomous, even-order ordinary differential equations is discussed from the point of view of Lie symmetries. It is shown that for a certain power nonlinearity, the Noether symmetry group coincides with the Lie point symmetry group. First integrals are established and exact solutions are found. Furthermore, this paper complements, for the one-dimensional case, some results in the literature of Lie group analysis of poliharmonic equations and Noether symmetries obtained in the last twenty years. In particular, it is shown that the exceptional negative power discovered in [Bokhari, A. H., Mahomed, F. M. and Zaman, F. D. (2010) Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation. J. Math. Phys., 51, 053517] is a member of a one-parameter family of exceptional powers in which the Lie symmetry group coincides with the Noether symmetry group. Lie point symmetries, Noether symmetries, first integrals, exact solutions, ordinary differential equations, power nonlinearities.