This paper seems to be a review article but ignores almost all important references, misleading the audience to believe that only the author and his group have contributed much to the field. Of the 41 references, 26 belong to the author, 6 to his students, and the others are H. Poincaré (1), M. J. Lightthill (1), Y. H. Kuo (2), H. S. Tsien (1), and 4 references on computer algebra applications.
Dai and his group published their works only in Chinese Journals where Dai serves as a member of the editorial board except for Scientia Sinica Ser. A, this reflects that Dai’s theory has been completely refuted by the international community.
The so-called Poincaré-Lighthill-Kuo (PLK) method originates from the well-known Lindstedt-Poincaré method, which was first proposed by A. Lindstedt [Astron. Nachr. 103, 211–220 (1882; JFM 14.0926.02)] in 1882 to eliminate the secular terms by expanding the frequency in powers of epsilon, which is equivalent to introducing a linear transformation of time coordinate which was done by M. L. Lighthill. Y. H. Kuo applied the method to viscous flow. The Poincaré-Lighthill-Kuo method was named by H. S. Tsien [ Advances in applied mechanics, vol. IV, pp. 281–349. Academic Press Inc., New York, N.Y. (1956)], which , I thought, is inappropriate. Dai did not give a brief history of the Lindstedt-Poincaré method, deluding the audience that the PLK method is a method different from the Lindstedt-Poincaré method. Dai pointed out many researchers have further developed the PLK method, it is the fact. Many modified Lindstedt-Poincaré methods can be found in open literature. Dai wrote down some names but without any references, so the readers can never refer to the literature. The solution obtained by the Lindstedt-Poincaré method (or PLK method) is of asymptotic character, i.e., the obtained series is not always a convergent one, so in engineering applications, we always stop at few orders of approximation, higher order approximates, generally speaking, should be avoided. Symbolic computation is a powerful mathematical tool to complex calculation, so it is very easy to obtain a perturbation solution with very high orders, but this does not provide us with any advantages in obtaining perturbation series which is divergent, the higher the order, the worse the result, because the obtained asymptotic series might break down.
For the most recent development of the modified Lindstedt-Poincaré method, please refer to the 3-part article of the reviewer: 1. Int. J. Non-Linear Mech. 37, No. 2, 309–314 (2002; Zbl 1116.34320) and 315–320 (2002; Zbl 1116.34321); Int. J. Nonlinear Sci. Numer. Simul. 2, No. 4, 317–320 (2001; Zbl 1072.34507).