62 research outputs found
Conservations Laws for Critical Kohn-Laplace Equations on the Heisenberg Group
Using the complete group classification of semilinear differential equations
on the three-dimensional Heisenberg group carried out in a preceding work, we
establish the conservation laws for the critical Kohn-Laplace equations via the
Noether's Theorem.Comment: 9 pages, 1 table, submitted for publicatio
Special Conformal Groups of a Riemannian Manifold and Lie Point Symmetries of the Nonlinear Poisson Equation
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 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.Comment: Paper submitted for publicatio
The intrinsic geometry determined by the Cauchy problems of the Camassa-Holm equation
Pseudospherical surfaces determined by Cauchy problems involving the
Camassa-Holm equation are considered herein. We study how global solutions
influence the corresponding surface, as well as we investigate two sorts of
singularities of the metric: the first one is just when the co-frame of dual
form is not linearly independent. The second sort of singularity is that
arising from solutions blowing up. In particular, it is shown that the metric
blows up if and only if the solution breaks in finite time
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