N-Wave Equations with Orthogonal Algebras: Z2 and Z2 × Z2 Reductions and Soliton Solutions

Gerdjikov, Vladimir S.; Kostov, N. A.; Valchev, Tihomir I.

doi:10.3842/sigma.2007.039

Cited by 10 publications

(13 citation statements)

References 22 publications

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“…The consequences of these reductions and the constraints they impose on the FAS and the Gauss factors of the scattering matrix are well known, see [38,44,17,18,19,23]. It is important to note, that for the N-wave equations there are no reductions compatible with either P-or T-symmetry separately.…”

Section: Local and Non-local Reductionsmentioning

confidence: 99%

The N-wave equations with PT symmetry

Gerdjikov

Grahovski

Ivanov³

2016

Theor Math Phys

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We study extensions of N -wave systems with PT-symmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P-(spatial reflection) and T-(time reversal) symmetries is described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of 3-wave (related to the algebra sl(3, C)) and 4-wave (related to the algebra so(5, C)) systems, the properties of different types of 1-and 2-soliton solutions are discussed. It is shown that the PT symmetric 3-wave equations may have regular multi-soliton solutions for some specific choices of their parameters.

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Section: Local and Non-local Reductionsmentioning

confidence: 99%

The N-wave equations with PT symmetry

Gerdjikov

Grahovski

Ivanov³

2016

Theor Math Phys

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“…This purely algebraic reduction technique, first formulated by Mikhailov (see [Mik81]) and later developed in [MSY87], [LM05], has been successfully applied both in classical (e.g. [GKV07a], [GKV07b], [GGK01], [GGIK01], [HSAS84], [LM04]) and quantum integrable systems theory (e.g. [Bel80], [Bel81]) and it essentially consists in finding invariants elements of the Lie algebra over the ring of rational functions used in the Lax pair representation of an integrable equation.…”

Section: Algebraic Reductions and Automorphic Lie Algebrasmentioning

confidence: 99%

On the Classification of Automorphic Lie Algebras

Lombardo

Sanders

2010

Commun. Math. Phys.

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Abstract:The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that sl 2 (C)-based Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group D n are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl 2 (C)-based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.

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“…tion. Of course, one should consider also the numerous MNLS that can be obtained from (1) by applying Mikhailov reductions [29], see [15,13,14,16,17,19]. Some of these MNLS have applications to physics.…”

Section: Introductionmentioning

confidence: 99%

On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms

Gerdjikov

2017

Advanced Computing in Industrial Mathematics

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A special class of multicomponent NLS equations, generalizing the vector NLS and related to the BD.I-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a RiemannHilbert problem. We introduce the minimal sets of scattering data T which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of T can be viewed as the expansion coefficients of Q over the 'squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the mapping T → Q is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor (F = 1 and F = 2, respectively) BoseEinstein condensates.

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N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions

Cited by 10 publications

References 22 publications

The N-wave equations with PT symmetry

The N-wave equations with PT symmetry

On the Classification of Automorphic Lie Algebras

On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms

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