The Korteweg-de Vries Equation and Multisoliton Solutions
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Дата
2025
Автори
Науковий керівник
Назва журналу
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Назва тому
Видавець
КПІ ім. Ігоря Сікорського
Анотація
The Korteweg-de Vries equation is partial differential equation that describes certain types of wave phenomena. The solution to this equation describes one mysterious wave, discovered by John Scott Russell in 1834, and is named soliton. Soliton waves were later found in models of plasma, solid-state physics, biological systems, optic systems. In the middle of 1960's the Korteweg-de Vries equation was found out to be closely related to the Schrödinger equation. Due to this fact, this equation is interpreted as sort of condition for compatibility of an abstract system of two auxiliary linear equations, and the solution of this equation can be found with the help of the Darboux transformation apparatus. This method allows us to construct new solutions to a nonlinear equation with the help of 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.
Опис
Ключові слова
the Korteweg-de Vries equation, soliton, solitary solution, solitary wave, the Darboux transformation, the Darboux theorem, Schrödinger operator, Sturm-Liouville problem, one-soliton solution, multisoliton solution, two-soliton solution, three-soliton solution
Бібліографічний опис
Tsukanova, A. O. The Korteweg-de Vries Equation and Multisoliton Solutions / A. O. Tsukanova // Теоретичні і прикладні проблеми фізики, математики та інформатики : матерiали XXIII Всеукраїнської науково-практичної конференцiї студентiв, аспiрантiв та молодих вчених, [Київ], 14−17 травня 2025 р. / КПІ ім. Ігоря Сікорського. – Київ, 2025. – С. 87-93.