Geodesic
Differential substitutions for non-Abelian equations of KdV type
Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 107-114

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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.
Keywords: non-Abelian equation
Mots-clés : Lax pair, Miura transformation. 
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     author = {V. E. Adler},
     title = {Differential substitutions  for {non-Abelian} equations of {KdV} type},
     journal = {Ufa mathematical journal},
     pages = {107--114},
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     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a9/}
}
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V. E. Adler. Differential substitutions  for non-Abelian equations of KdV type. Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 107-114. http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a9/