Abstract <p>Burgers hierarchy equations are considered. It is shown that the well-known Cole–Hopf transformation for linearizing the classical Burgers equation can be generalized to Burgers hierarchy equations of arbitrary order. This result is used to find solitary and periodic waves governed by Burgers hierarchy equations, which resemble the <i>N</i>-wave for the classical Burgers equation. The construction of solitary waves for the third-order Sharma–Tasso–Olver equation and for the fourth-order hierarchy equation is described in detail. For the third-order dispersion equation, we establish that mode-<i>N</i> solitary waves exhibit oscillations at the solution front. In the case of second- and fourth-order dissipative equations, there are no oscillations of this type.</p>