For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel–Kramers–Brillouin (E-WKB) ansatz is an asymptotically valid solution. This ansatz is globally defined and formulated as an integral of coherent states over the displaced Lagrangian submanifold. This paper proves the optimal first order error estimate of the proposed E-WKB ansatz in \(L^2\) norm for the wave-type equation in the semi-classical regime. The key ingredients in the proof are the moving frame technique developed in Zheng (Commun Math Sci 11:105–140, 2013) and the deep relations between the E-WKB analysis and the classical WKB analysis. Numerical results on the linear KdV equation verify the theoretical analysis.

对于具有Wentzel–Kramers–Brillouin（WKB）类型初始数据的高频波动型方程的Cauchy问题，扩展的Wentzel–Kramers–Brillouin（E-WKB）ansatz是渐近有效的解决方案。该ansatz在全球范围内定义并表述为在拉格朗日子流形上相干状态的整体。本文证明了半经典状态下波动型方程在\（L ^ 2 \）范数下提出的E-WKB ansatz的最优一阶误差估计。证明中的关键要素是Zheng（Commun Math Sci 11：105–140，2013）开发的移动框架技术以及E-WKB分析和经典WKB分析之间的深层关系。线性KdV方程的数值结果验证了理论分析。