Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations
The work is devoted to further applications of the algebraic geometry techniques to finding exact solutions for the Kadomtsev-Petviashvili (KP) equation. It is shown that a special Darboux transformation, called transference, acting on a pair of commuting differential operators of orders 4 and 6, implements the Abelian sum on their joint elliptic spectrum. Those transformations map KP solutions into solutions to the so-called Krichever-Novikov equations, which can be further mapped, by another transformation belonging to the same class, into solutions to the Korteweg-de Vries (KdV) equation. Reversing the sequence of the two mappings, one can construct exact solutions to the KP solutions, starting from those for the KdV equation. As an example, it is shown how a rational nondecaying solution to the KdV equation generates rational KP solutions.
scientific article; zbMATH DE number 790810
scientific article