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Symmetry, Volume 5, Issue 3 (September 2013), Pages 233-270

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Research

Open AccessArticle Symmetries Shared by the Poincaré Group and the Poincaré Sphere
Symmetry 2013, 5(3), 233-252; doi:10.3390/sym5030233
Received: 29 May 2013 / Revised: 9 June 2013 / Accepted: 9 June 2013 / Published: 27 June 2013
Cited by 5 | PDF Full-text (345 KB) | HTML Full-text | XML Full-text
Abstract
Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s [...] Read more.
Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics. Full article
Open AccessArticle Supersymmetric Version of the Euler System and Its Invariant Solutions
Symmetry 2013, 5(3), 253-270; doi:10.3390/sym5030253
Received: 8 April 2013 / Revised: 14 June 2013 / Accepted: 3 July 2013 / Published: 12 July 2013
Cited by 1 | PDF Full-text (282 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of [...] Read more.
In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions. Full article

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