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Уфимский математический журнал, 2021, том 13, выпуск 2, страницы 112–120
(Mi ufa556)
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Differential substitutions for non-Abelian equations of KdV type
V. E. Adler L.D. Landau Institute for Theoretical Physics,
Akademika Semenova av., 1A,
142432, Chernogolovka, Moscow Region, Russia
Аннотация:
The work is devoted to constructing differential substitutions connecting the non-Abelian KdV equation with other third-order evolution equations. One of the main results is the construction of a non-Abelian analog of the exponential Calogero–Degasperis equation in a rational form. Some generalizations of the Schwarzian KdV equation are also obtained. Equations and differential substitutions under study contain arbitrary non-Abelian parameters. The construction method is based on the auxiliary linear problem for KdV, in which the usual spectral parameter is replaced by a non-Abelian one. The wave function, corresponding to a fixed value of this parameter,
also satisfies a certain evolution equation. Passing to the left and right logarithmic derivatives of the wave function leads one to two versions of the modified KdV equation. In addition, a gauge transformation of the original linear problem leads to a linear problem for one of these versions, mKdV-2. After that, the described procedure is repeated, and the resulting evolution equation for the wave function contains already two arbitrary non-Abelian parameters. For the logarithmic derivative, we obtain an analog of the Calogero–Degasperis equation, which is thus a second modification of the KdV equation. Combining the found Miura-type transformations with discrete symmetries makes it possible to obtain chains of Bäcklund transformations for the modified equations.
Ключевые слова:
non-Abelian equation, Lax pair, Miura transformation.
Поступила в редакцию: 10.03.2021
Образец цитирования:
V. E. Adler, “Differential substitutions for non-Abelian equations of KdV type”, Уфимск. матем. журн., 13:2 (2021), 112–120; Ufa Math. J., 13:2 (2021), 107–114
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/ufa556 https://www.mathnet.ru/rus/ufa/v13/i2/p112
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Страница аннотации: | 120 | PDF русской версии: | 62 | PDF английской версии: | 15 | Список литературы: | 11 |
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