Cauchy Problem for the Loaded Korteweg–de Vries Equation in the Class of Periodic Functions

Pub Date : 2024-02-26 DOI:10.1134/s001226612312008x

A. B. Khasanov, T. G. Khasanov

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Abstract

The inverse spectral problem method is applied to finding a solution of the Cauchy problem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded terms and a derivation of an analog of Dubrovin’s system of differential equations are proposed. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and the first trace formula actually satisfies the loaded nonlinear Korteweg–de Vries equation. In addition, we prove that if the initial function is a real \(\pi \)-periodic analytic function, then the solution of the Cauchy problem is a real analytic function in the variable \(x \) as well, and also that if the number \( {\pi }/{n} \), \(n\in \mathbb {N}\), \(n\ge 2 \), is the period of the initial function, then the number \({\pi }/{n} \) is the period for solving the Cauchy problem with respect to the variable \(x\).

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周期函数类中负载科特韦格-德-弗里斯方程的考奇问题

摘要应用逆谱问题方法寻找周期性无穷间隙函数类中加载 Korteweg-de Vries 方程的 Cauchyproblem 解，提出了构造带加载项的高阶 Korteweg-de Vries 方程的简单算法和 Dubrovin 微分方程系的推导。结果表明，通过求解杜布罗文方程组和第一迹公式构建的均匀收敛函数序列之和实际上满足加载非线性科特韦格-德弗里斯方程。此外，我们还证明了如果初始函数是实(\pi \)周期解析函数，那么考奇问题的解也是变量\(x \)中的实解析函数、而且，如果数\({\pi }/{n}\), \(n\in \mathbb {N}\), \(n\ge 2 \)，是初始函数的周期，那么数\({\pi }/{n}\) 就是相对于变量\(x\)求解考奇问题的周期。

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