This paper is the first part of our study the asymptotic behavior of the solutions of the cylindrical Korteweg–de Vries equation (cKdV). In this first part, we consider the solutions from Schwarz’s class and calculate the large time asymptotics using the method based on the asymptotic solution of the associated direct scattering problem. This approach was first suggested, in the cases of usual KdV and NLS equations, in the late 70s by Zakharov and Manakov. In a sequel to this paper, we plan to calculate the large time asymptotics of some other classes of solutions of the cKdV equation which exhibit the oscillatory type behavior, and we will also evaluate the short time asymptotics of the solutions of the cKdV equation. In the second part we will use the Defit–Zhou nonlinear steepest descent method for oscillatory Riemann–Hilbert problems.

中文翻译：

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圆柱Korteweg-de Vries方程的大时间渐近性。一世。

本文是我们研究的第一部分，它是圆柱Korteweg-de Vries方程（cKdV）解的渐近行为。在第一部分中，我们考虑Schwarz类的解，并使用基于相关直接散射问题的渐近解的方法来计算大时间渐近性。Zakharov和Manakov在70年代后期首次提出了通常的KdV和NLS方程的这种方法。在本文的续篇中，我们计划计算显示出振荡类型行为的cKdV方程的其他一些类别的解的大时间渐近性，并且还将评估cKdV方程的解的短时间渐近性。在第二部分中，我们将使用Defit-Zhou非线性最速下降方法求解振荡Riemann-Hilbert问题。