Similarity reductions of nearly concentric KdV equation

Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries (ncKdV) equation can be reduced to three types of (1+1)-dimensional variable coefficients partial differential equations (PDEs) and three types of variable coefficients ordinary differential equation. Furthermore, three types of (1+1)-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.