Nonlocal symmetries and interaction solutions for the KdV-type K(3,2) equation

The nonlocal symmetries for the special $K\left(m\text{,}n\right)$ equation, which is called KdV-type $K\left(3\text{,}2\right)$ equation, are obtained by means of the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables and the corresponding finite symmetry transformations are computed directly. The KdV-type $K\left(3\text{,}2\right)$ equation is also proved to be consistent tanh expansion solvable. New exact interaction excitations such as soliton–cnoidal wave solutions are given out analytically and graphically.