Exact solutions of Whitham-Broer-Kaup equations with variable coefficients

An equivalence transformation of Whitham-Broer-Kaup equations with variable coefficients (VCWBK) is obtainedby using modified Clarkson-Kruskal direct method. Further, the relationship between the solutions of VCWBK equationsand ones of the corresponding WBK equations with constant coefficients is obtained. In addition, by applying directsymmetry method, some symmetries and similarity reductions of the corresponding WBK equations with constantcoefficients are derived. Using an auxiliary function to solve some special cases, we obtain some new exact solutionsof VCWBK equations, including rational solutions, hyperbolic function solutions, trigonometric function solutions, andJacobi elliptic function solutions.