Symmetry and exact solutions of nonlinear spinor equations

Fushchich, W. I.; Zhdanov, Renat

doi:10.1016/0370-1573(89)90090-2

Cited by 62 publications

(69 citation statements)

References 38 publications

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“…Ovsiannikov [78] developed the method of partially invariant solutions. Bluman and Cole [9], in their study of symmetry reductions of the linear heat equation, proposed the so-called nonclassical method of group-invariant solutions (in the sequel referred to as the nonclassical method), which is also known as the "method of conditional symmetries" [33,34,36,37,59,76,81,82,103] and the "method of partial symmetries of the first type" [97]. In this method, the original pde (1.1) is augmented with the invariant surface condition ψ ≡ ξ(x, t, u)u x + τ (x, t, u)u t − φ(x, t, u) = 0, (1.4) which is associated with the vector field (1.3).…”

Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

254

290

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Classical and nonclassical symmetries of the nonlinear heat equationare considered. The method of differential Gröbner bases is used both to find the conditions on f (u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f (u) in terms of the roots of f (u) = 0. 0

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Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

254

290

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“…Nonlinear Dirac equations have a long history in the literature, particularly in the context of particle and nuclear theory [12,13,14,15], but also in applied mathematics and nonlinear dynamics [16,17,18,19,20]. As nonlinearity is a ubiquitous aspect of Nature, it is natural to ask how nonlinearity might appear in a relativistic setting.…”

Section: Introductionmentioning

confidence: 99%

The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

Haddad

Carr

2009

Physica D: Nonlinear Phenomena

109

120

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We show that Bose-Einstein condensates in a honeycomb optical lattice are described by a nonlinear Dirac equation in the long wavelength, mean field limit. Unlike nonlinear Dirac equations posited by particle theorists, which are designed to preserve the principle of relativity, i.e., Poincaré covariance, the nonlinear Dirac equation for Bose-Einstein condensates breaks this symmetry. We present a rigorous derivation of the nonlinear Dirac equation from first principles. We provide a thorough discussion of all symmetries broken and maintained.

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“…The algebraic-theoretical methods of constructing exact solutions for a wide class of nonlinear spinor equations, developed by W.I. Fushchych with coworkers [15], make it possible to find the so-called nongenerable solutions of the above-mentioned spinor equations. Just with this objective in view the subgroups of the group of generalized Lorentz transformations and their geometric invariants have been studied in the present work.…”

Section: Resultsmentioning

confidence: 99%

Subgroups of the Group of Generalized Lorentz Transformations and Their Geometric Invariants

Bogoslovsky

2005

SIGMA

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Abstract. It is shown that the group of generalized Lorentz transformations serves as relativistic symmetry group of a flat Finslerian event space. Being the generalization of Minkowski space, the Finslerian event space arises from the spontaneous breaking of initial gauge symmetry and from the formation of anisotropic fermion-antifermion condensate. The principle of generalized Lorentz invariance enables exact taking into account the influence of condensate on the dynamics of fundamental fields. In particular, the corresponding generalized Dirac equation turns out to be nonlinear. We have found two noncompact subgroups of the group of generalized Lorentz symmetry and their geometric invariants. These subgroups play a key role in constructing exact solutions of such equation.

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Symmetry and exact solutions of nonlinear spinor equations

Cited by 62 publications

References 38 publications

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Symmetry reductions and exact solutions of a class of nonlinear heat equations

The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

Subgroups of the Group of Generalized Lorentz Transformations and Their Geometric Invariants

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