Seminar talk, 15 April 2026: Difference between revisions
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| abstract = For a system of differential equations and a symmetry, the framework of invariant reduction systematically computes how invariant geometric structures are inherited by the subsystem governing invariant solutions. In this setting, the reduction of structures invariant under a two-dimensional Lie algebra requires its commutativity. We extend this mechanism to the case where geometric structures are invariant under one symmetry <math>X</math> and are rescaled, by a factor of <math>-a</math>, by another symmetry <math>X_s</math> satisfying <math>[X_s, X] = aX</math>. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs. | | abstract = For a system of differential equations and a symmetry, the framework of invariant reduction systematically computes how invariant geometric structures are inherited by the subsystem governing invariant solutions. In this setting, the reduction of structures invariant under a two-dimensional Lie algebra requires its commutativity. We extend this mechanism to the case where geometric structures are invariant under one symmetry <math>X</math> and are rescaled, by a factor of <math>-a</math>, by another symmetry <math>X_s</math> satisfying <math>[X_s, X] = aX</math>. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs. | ||
| video = | | video = | ||
| slides = | | slides = [[Media:Inv_red_IV.pdf|Inv_red_IV.pdf]] | ||
| references = {{arXiv|2603.10131}} | | references = {{arXiv|2603.10131}} | ||
| 79YY-MM-DD = 7973-95-84 | | 79YY-MM-DD = 7973-95-84 | ||
}} | }} | ||
Revision as of 17:31, 15 April 2026
Speaker: Konstantin Druzhkov
Title: Invariant reduction for PDEs. IV: Symmetries that rescale geometric structures
Abstract:
For a system of differential equations and a symmetry, the framework of invariant reduction systematically computes how invariant geometric structures are inherited by the subsystem governing invariant solutions. In this setting, the reduction of structures invariant under a two-dimensional Lie algebra requires its commutativity. We extend this mechanism to the case where geometric structures are invariant under one symmetry and are rescaled, by a factor of , by another symmetry satisfying . As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs.
Slides: Inv_red_IV.pdf
References:
arXiv:2603.10131