Soliton surfaces in the generalized symmetry approach

We investigate some features of generalized symmetries of integrable systems aiming to obtain the Fokas–Gel’fand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. We show that if there exists a common symmetry of the zero-curvature representation of an integrable partial...

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Bibliographic Details
Published in:Theoretical and mathematical physics 2016-09, Vol.188 (3), p.1322-1333
Main Author: Grundland, A. M.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Elliptic differential equations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:We investigate some features of generalized symmetries of integrable systems aiming to obtain the Fokas–Gel’fand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. We show that if there exists a common symmetry of the zero-curvature representation of an integrable partial differential equation and its linear spectral problem, then the Fokas–Gel’fand immersion formula is applicable in its original form. In the general case, we show that when the symmetry of the zero-curvature representation is not a symmetry of its linear spectral problem, then the immersion function of the two-dimensional surface is determined by an extended formula involving additional terms in the expression for the tangent vectors. We illustrate these results with examples including the elliptic ordinary differential equation and the C P N −1 sigma-model equation.
ISSN:0040-5779
1573-9333
DOI:10.1134/S004057791609004X