
Zapiski Nauchnykh Seminarov POMI, 2021, Volume 506, Pages 258–278
(Mi znsl7154)




This article is cited in 7 scientific papers (total in 7 papers)
The Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinitegap functions
A. B. Hasanov^{a}, T. G. Hasanov^{b} ^{a} Samarkand State University
^{b} Urgench State University named after AlKhorezmi
Abstract:
In this paper, the method of the inverse spectral problem is applied to finding a solution to the Cauchy problem for the Kortewegde Vries equation in the class of periodic infinitegap functions. A simple derivation of the system of differential Dubrovin equations is proposed. The solvability of the Cauchy problem is proved for the infinite system of Dubrovin differential equations in the class of fourtimes continuously differentiable periodic infinitegap functions. It is shown that the sum of a uniformly convergent functional series constructed using the solution of the infinite system of Dubrovin equations and the formula for the first trace, does satisfy the nonlinear Korteweg–de Vries equation. In addition, it is proved that if the number $\frac{\pi }{n}$ is the period of the initial function, then the number $\frac{\pi }{n}$ is the period for the solution of the Cauchy problem with respect to the variable $x$. Here $n\ge 2$ is a positive integer.
Key words and phrases:
Kortewegde Vries equation, trace formulas , inverse spectral problem, Hill operator, system of Dubrovin equations.
Received: 09.10.2021
Citation:
A. B. Hasanov, T. G. Hasanov, “The Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinitegap functions”, Mathematical problems in the theory of wave propagation. Part 51, Zap. Nauchn. Sem. POMI, 506, POMI, St. Petersburg, 2021, 258–278
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