The semi-classical Korteweg–de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations.

中文翻译：

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Korteweg-de Vries 方程的惠瑟姆方程和相移

阶跃数据的半经典 Korteweg-de Vries 方程被认为在最高导数前有一个小参数。使用微扰分析，Whitham 理论被构建到更高阶。这允许获得阶一阶段和完整的超前阶解；结果得到了广泛的数值计算的证实。