Abstract
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.
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