Cauchy Problem for the Korteweg–De Vries Equation in the Class of Periodic Infinite-Gap Functions

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 283; no. 4; pp. 674 - 689
• Main Authors: Khasanov, A. B., Khasanov, T. G.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Cauchy problems; Differential equations; Functionals; Korteweg-Devries equation; Mathematics; Mathematics and Statistics
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07300-z

In this paper, the inverse spectral problem method is used to find a solution of the Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple derivation of the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of four times continuously differentiable periodic infinite-gap functions is proven. The sum of the uniformly convergent functional series constructed by solving the infinite system of Dubrovin equations and the first trace formula are proven to satisfy the nonlinear Korteweg–de Vries equation. In addition, it is proven that if the number π n is the period of the initial function, then the number π n is the period for the solution of the Cauchy problem with respect to the variable x. Here n ≥ 2 is a natural number.