Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

Extended WKB analysis for the linear vectorial wave equation in the high-frequency regime

Pages: 687 – 706



Chunxiong Zheng (College of Mathematics and Systems Science, Xinjiang University, Urumqi, China; and Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Jiashun Hu (Department of Mathematical Sciences, Tsinghua University, Beijing, China)


We introduce an asymptotic solution form, termed as extended Wentzel–Kramers–Brillouin (E‑WKB), to solve the high-frequency vectorial wave equation when the initial Cauchy data are prescribed in the form of Wentzel–Kramers–Brillouin (WKB) function. The E‑WKB form, formulated as an integral of a family of Gaussian coherent states, can be regarded as an extension of the WKB form. The domain of the integral is the Lagrangian submanifold induced by the underlying Hamiltonian flow. Although the procedure of solving wave equations by using the E‑WKB form is parallel to that of the classical WKB analysis, the former can overcome the difficulty due to the presence of caustic points. We present numerical tests on vectorial Schrödinger equation and Helmholtz equation to validate the proposed asymptotic theory.


vectorial wave equation, high-frequency regime, caustics, WKB analysis, extended WKB analysis

2010 Mathematics Subject Classification

34E20, 35C20, 35S10

The first-named author’s work was supported by the Natural Science Foundation of Xinjiang Autonomous Region under No. 2019D01C026.

Received 18 November 2018

Accepted 18 November 2019

Published 30 June 2020