Секцiя 1. Актуальнi проблеми сучасної фiзики

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Two- and three-soliton solutions to one equation of the Korteweg-de Vries type
(КПІ ім. Ігоря Сікорського, 2024) Tsukanova, A. O.
In 1895 two Netherlandish applied mathematicians Korteweg and de Vries made a breakthrough in physics: they introduced a nonlinear partial differential equation describing motion of truly magic water wave, which is now called the Korteweg-de Vries equation. In the middle of 1960’s this equation was found out to be closely related to the Schröodinger equation. Due to this fact, the Korteweg-de Vries equation is interpreted as a condition for compatibility of an abstract system of two auxiliary linear equations, and the solution of this equation can be found using the Darboux transformation apparatus. This method allows us to construct new solutions to a nonlinear equation using a known one. General formulas for the Darboux transformation were obtained earlier. In the given paper we consider mathematical application of the Darboux transformation in order to obtain solutions of the Korteweg-de Vries equation in explicit form. The resulting formulas describe some interaction of multisoliton solutions, namely, two- and three- soliton solutions.