Solitary waves of the hierarchy equations Burgers

Burgers hierarchy equations are considered. It is shown that the well-known Cole–Hopf transformation for linearization of the classical Burgers equation generalizes to the case of equations of arbitrary order of the Burgers hierarchy. This fact allows us to find solitary and periodic waves described by the Burgers hierarchy equations, resembling the $N$-wave for the classical Burgers equation. A detailed consideration of the construction of solitary waves is presented for the third-order Sharma–Tasso–Olver equation and for the fourth-order hierarchy equation. It is found that for the third-order dissipative equation, $N$-wave type solitary waves have oscillations at the solution front. In the case of second and fourth order dissipative equations such oscillations are absent.