Abstract
In this paper, five types of similarity reductions of Whitham-Broer-kaup equations in shallow water are given by both the classical Lie approach and the direct method. The Painlevé Ⅱ type reduction equation obtained by the classical Lie approach is only the special case of that obtained by the direct method. In the similarity reduction results of the direct method, three types of singularity points, i.e., poles, algebraic branch points and logarithmic branch points. are included.
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