Seminar talk, 29 March 2017: Difference between revisions
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| abstract = We consider four three-dimensional equations: (1) the rdDym equation <math>u_{ty} = u_x u_{xy} - u_y u_{xx}</math>, (2) the 3D Pavlov equation <math>u_{yy} = u_{tx} + u_y u_{xx} - u_x u_{xy}</math>; (3) the universal hierarchy equation <math>u_{yy} = u_t u_{xy} - u_y u_{tx}</math>, and (4) the modified Veronese web equation <math>u_{ty} = u_t u_{xy} - u_y u_{tx}</math>. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries. | | abstract = We consider four three-dimensional equations: (1) the rdDym equation <math>u_{ty} = u_x u_{xy} - u_y u_{xx}</math>, (2) the 3D Pavlov equation <math>u_{yy} = u_{tx} + u_y u_{xx} - u_x u_{xy}</math>; (3) the universal hierarchy equation <math>u_{yy} = u_t u_{xy} - u_y u_{tx}</math>, and (4) the modified Veronese web equation <math>u_{ty} = u_t u_{xy} - u_y u_{tx}</math>. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries. | ||
A joint work with [[Hynek Baran]], [[Oleg Morozov]], and [[Petr Vojčák]] | A joint work with [[Hynek Baran]], [[Oleg Morozov]], and [[Petr Vojčák]]. | ||
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Latest revision as of 17:38, 19 March 2017
Speaker: Joseph Krasil'shchik
Title: Nonlocal symmetries of Lax integrable equations: a comparative study
Abstract:
We consider four three-dimensional equations: (1) the rdDym equation , (2) the 3D Pavlov equation ; (3) the universal hierarchy equation , and (4) the modified Veronese web equation . For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.
A joint work with Hynek Baran, Oleg Morozov, and Petr Vojčák.
References:
arXiv:1611.04938