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 TMF, 2000, Volume 122, Number 2, Pages 239–250 (Mi tmf566)

Binary Darboux transformations and $N$-wave systems in rings

S. B. Lebleab

b Technical University of Gdańsk

Abstract: The covariance theorems for elementary and binary Darboux transformations in rings are formulated and proved for generalized Zakharov–Shabat problems. The definition of the elementary Darboux transformation is extended to an arbitrary number of orthogonal idempotents. The binary transformation is defined as a sequence of elementary transformations for direct and conjugate problems. The heredity property for the reduction constraints is established for some $UV$ pairs in rings; hence, the transformation generates solutions and infinitesimal symmetries of the corresponding zero-curvature equations. The explicit expressions for the transformations, solitons, and infinitesimals are given in the general case and in physically significant cases of extended non-Abelian $N$-wave equations (with linear terms added).

DOI: https://doi.org/10.4213/tmf566

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English version:
Theoretical and Mathematical Physics, 2000, 122:2, 200–210

Bibliographic databases:

Citation: S. B. Leble, “Binary Darboux transformations and $N$-wave systems in rings”, TMF, 122:2 (2000), 239–250; Theoret. and Math. Phys., 122:2 (2000), 200–210

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf/v122/i2/p239

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. B. Leble, “Covariance of Lax Pairs and Integrability of the Compatibility Condition”, Theoret. and Math. Phys., 128:1 (2001), 890–905
2. Leble, SB, “Elementary, binary and Schlesinger transformations in differential ring geometry”, European Physical Journal B, 29:2 (2002), 189
3. A. A. Perelomova, S. B. Leble, “Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases”, Theoret. and Math. Phys., 144:1 (2005), 1030–1039
4. Haider B., Hassan M., Saleem U., “Binary Darboux Transformation and Quasideterminant Solutions of the Chiral Field”, J Nonlinear Math Phys, 18:2 (2011), 299–321
5. Bushra Haider, Mahmood-ul Hassan, “Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System”, SIGMA, 8 (2012), 084, 15 pp.
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