Nonlocal symmetries of integrable linearly degenerate equations: A~comparative study
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 169-192
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We continue the study of Lax integrable equations. We consider four three-dimensional equations{: (1)} the rdDym equation $u_{ty}=u_xu_{xy}- u_yu_{xx}$, $(2)$ the Pavlov equation $u_{yy}=u_{tx}+u_yu_{xx}-u_xu_{xy}$, $(3)$ the universal hierarchy equation $u_{yy}=u_tu_{xy}-u_yu_{tx}$, and $(4)$ the modified Veronese web equation $u_{ty}=u_tu_{xy}-u_yu_{tx}$. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.
Keywords:
partial differential equation, integrable linearly degenerate equation, nonlocal symmetry, recursion operator.
@article{TMF_2018_196_2_a0,
author = {H. Baran and I. S. Krasil'shchik and O. I. Morozov and P. Voj\v{c}\'ak},
title = {Nonlocal symmetries of integrable linearly degenerate equations: {A~comparative} study},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {169--192},
publisher = {mathdoc},
volume = {196},
number = {2},
year = {2018},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a0/}
}
TY - JOUR AU - H. Baran AU - I. S. Krasil'shchik AU - O. I. Morozov AU - P. Vojčák TI - Nonlocal symmetries of integrable linearly degenerate equations: A~comparative study JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2018 SP - 169 EP - 192 VL - 196 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a0/ LA - ru ID - TMF_2018_196_2_a0 ER -
%0 Journal Article %A H. Baran %A I. S. Krasil'shchik %A O. I. Morozov %A P. Vojčák %T Nonlocal symmetries of integrable linearly degenerate equations: A~comparative study %J Teoretičeskaâ i matematičeskaâ fizika %D 2018 %P 169-192 %V 196 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a0/ %G ru %F TMF_2018_196_2_a0
H. Baran; I. S. Krasil'shchik; O. I. Morozov; P. Vojčák. Nonlocal symmetries of integrable linearly degenerate equations: A~comparative study. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 169-192. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a0/