Invariant representation and symmetry reduction for differential equations with a small param\-eter.
The Lie group analysis of differential equations is one of the effective methods for constructing exact solutions of differential equations under consideration. A wide class of the group methods of integration is based on the theorem of invariant representation of equations. Due to this theorem, if an equation is invariant with respect to the Lie group \(G\), then the equation has an equivalent form defined by invariant functions of the group \(G\). In this paper, a similar theorem for equations with a small parameter is proved. Instead of the Lie transformation groups, the authors consider approximate groups. The features of approximate symmetry reduction are demonstrated with constructing travelling-wave solution for the shallow-water equations \[ \begin{gathered} h_t+ uh_x+ hu_x= 0,\\ u_t+ uu_x+ h_x+ \varepsilon b'(x)+ \varepsilon C_f(h){4^2\over h}= 0.\end{gathered} \]
scientific article; zbMATH DE number 2021944